Error Estimate of Integral Deferred Correction Implicit Runge-Kutta ...
Error Estimate of Integral Deferred Correction Implicit Runge-Kutta ...
Error Estimate of Integral Deferred Correction Implicit Runge-Kutta ...
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<strong>Error</strong> <strong>Estimate</strong> <strong>of</strong> <strong>Integral</strong> <strong>Deferred</strong> <strong>Correction</strong> <strong>Implicit</strong> <strong>Runge</strong>-<strong>Kutta</strong> method for Stiff Problems<br />
Sebastiano Boscarino 1 , Jing-Mei Qiu 2<br />
Abstract. In this paper, we present an error estimate <strong>of</strong> integral deferred correction (IDC) method constructed<br />
with stiffly accurate implicit <strong>Runge</strong>-<strong>Kutta</strong> (R-K) method for singular perturbation problems containing a stiff<br />
parameter ε. We focus our analysis on the IDC method using uniform distribution <strong>of</strong> quadrature nodes, but<br />
excluding the left-most endpoint. The uniform distribution <strong>of</strong> nodes is important for high order accuracy<br />
increase in correction loops [5], where as the use <strong>of</strong> quadrature nodes excluding the left-most endpoint lead to<br />
an important stability condition for stiff problem, i.e. the method becomes L-stable if A-stable. In our error<br />
estimate, we expand the global error in powers <strong>of</strong> ε and show convergence results for these error terms as well<br />
as the remainder. Specifically, the order <strong>of</strong> convergence for the first term in global error (index 1) increase with<br />
high order if a high order R-K method is applied in the IDC correction step; the order <strong>of</strong> convergence for the<br />
second term (index 2) is determined by the stage order <strong>of</strong> the R-K method for the IDC prediction. Numerical<br />
results for the stiff van der Pol equation are demonstrated to verify our error estimate.<br />
Keywords: singular perturbation problem, <strong>Runge</strong>-<strong>Kutta</strong> method, integral deferred correction.<br />
1 Introduction<br />
<strong>Deferred</strong> correction (DC) methods for solving initial value problems<br />
y ′ (t) = f(t, y(t)), y(y 0 ) = y 0 ∈ R N (1.1)<br />
were investigated quite intensively, [2, 15, 1]. An advantage <strong>of</strong> DC methods is that one can use a simple numerical<br />
method, for instance a first order method, to compute a solution with higher order accuracy accomplished by<br />
using a numerical method to solve a series <strong>of</strong> correction equations during each time step. Each <strong>of</strong> the iteration<br />
increases the order <strong>of</strong> accuracy <strong>of</strong> the solution. In [6], a new variation <strong>of</strong> the classical method <strong>of</strong> deferred correction<br />
methods called spectral deferred method (SDC) was proposed. Essentially in SDC, the original differential<br />
equation (1.1) is replaced with the corresponding Picard integral equation and a deferred correction procedure<br />
is applied to an integral formulation <strong>of</strong> the error equation. Because the numerical integration have much better<br />
stability and accuracy property than numerical differential, the SDC has been showed to outperformed DC in<br />
many problems with promising numerical results [6]. In [6], the quadrature nodes in the proposed SDC methods<br />
are chosen to be Gauss-Lobatto, Gauss-Radau or Gauss-Legendre points for high order <strong>of</strong> accuracy. When the<br />
quadrature nodes are uniform, the SDC method is called integral deferred correction (IDC) method. There<br />
are various SDC/IDC methods with different implementation strategies, e.g. in selecting time integrators in<br />
prediction and correction steps [13, 12, 10, 9, 5, 4, 3] and in coupling with the Krylov subspace methods [9].<br />
Among them, there is no an ‘alway optimal’ IDC method; each method has its strength and weakness; choices<br />
<strong>of</strong> methods greatly depend on the characteristics <strong>of</strong> the system being solved. Under the IDC framework, it is<br />
shown in [5, 4] that if an r th order integrator is used to solve the error equation, then the accuracy <strong>of</strong> the scheme<br />
increases by r orders after each correction loop. This analysis has recently been extended in [3] for IDC methods<br />
constructed with implicit and semi-implicit integrators. In [4], the IDC method constructed with high order<br />
<strong>Runge</strong>-<strong>Kutta</strong> (R-K) method is showed to be a R-K method with corresponding Butcher tableau constructed.<br />
The main goal <strong>of</strong> this paper is to study the convergence behavior for the IDC method constructed using<br />
implicit R-K methods <strong>of</strong> different orders for the prediction and correction steps when applied to a special class<br />
<strong>of</strong> problems called singular perturbation problems (SPPs) containing a parameter ε. An arbitrary SPP is given<br />
by<br />
y ′ = f(y, z),<br />
εz ′ (1.2)<br />
= g(y, z),<br />
where y and z are vectors and ε > 0 is the stiff parameter. Classical books on this subject are [16, 14]. In<br />
system (1.2) we suppose that 0 < ε ≪ 1 and f and g are sufficiently differentiable vector functions <strong>of</strong> the<br />
1 Department <strong>of</strong> Mathematics and Computer Science, University <strong>of</strong> Catania, Catania, 95125, E-mail: boscarino@dmi.unict.it<br />
2 Department <strong>of</strong> Mathematics, University <strong>of</strong> Houston, Houston, 77004. E-mail: jingqiu@math.uh.edu. Research supported by<br />
Air Force Office <strong>of</strong> Scientific Computing YIP grant FA9550-12-0318, NSF grant DMS-0914852 and DMS-1217008 and University <strong>of</strong><br />
Houston.<br />
1
same dimensions as y and z respectively. The functions f, g and the initial values y(0), z(0) may depend<br />
smoothly on ε. For simplicity <strong>of</strong> notation we suppress this dependence. When the parameter ε in system (1.2)<br />
is small, the corresponding differential equation is stiff, and when ε tends to zero, the differential equation<br />
becomes differential algebraic. A sequence <strong>of</strong> differential-algebraic systems arises in the study <strong>of</strong> SPPs. This<br />
system allows us to understand many phenomena observed for very stiff problems. Indeed in Chap. VI. 3 <strong>of</strong><br />
[8] the authors show that most <strong>of</strong> the R-K methods presented in the literature suffer from the phenomenon <strong>of</strong><br />
order reduction in the stiff regime, i.e. ε small, when applied to (1.2). To this aim, we investigate the same<br />
phenomenon <strong>of</strong> the order reduction when it appears in the IDC framework. In the past, such order reduction<br />
phenomena have been numerically investigated without much theoretical justification for SPPs [13, 3]. Our<br />
analysis is based on the assumption <strong>of</strong> a smooth solution <strong>of</strong> system (1.2) and applies to the stiff case (H ≫ ε),<br />
where H is the time step size.<br />
We require that system (1.2) satisfies<br />
µ(g z (y, z)) ≤ −1, (1.3)<br />
in an ε-independent neighborhood <strong>of</strong> the solution, where µ denotes the logarithmic norm with respect to some<br />
inner product. Condition (1.3) guarantees the existence <strong>of</strong> an ε-expansion <strong>of</strong> the system (1.2) (see [8]). In other<br />
words we assume that the system (1.2) is dissipative. Furthermore, we suppose in our analysis that the initial<br />
values lie on a suitable manifold that allows smooth solutions even in the limit <strong>of</strong> infinite stiffness. In fact<br />
arbitrary initial values introduce in the solution a fast transient. One possible way to overcome this difficulty is<br />
simply to ensure that the numerical method resolves the transient phase by taking small step size <strong>of</strong> magnitude<br />
O(ε). Then the following results are obtained assuming that the transient phase is over. We note that the<br />
corresponding reduce system (ε = 0) is the differential algebraic equation (DAE)<br />
y ′ = f(y, z),<br />
0 = g(y, z),<br />
(1.4)<br />
whose initial values are consistent if 0 = g(y 0 , z 0 ). We assume that the Jacobian<br />
g z (y, z) is invertible (1.5)<br />
in a neighborhood <strong>of</strong> the solution <strong>of</strong> (1.4). This assumption guarantees the solvability <strong>of</strong> (1.4) and that the<br />
equation g(y, z) = 0 possesses a locally unique solution z = G(y) (implicit function theorem) which inserted into<br />
(1.4) gives<br />
y ′ = f(y, G(y)). (1.6)<br />
Furthermore, the same assumption guarantees that system (1.4) is a differential-algebraic one <strong>of</strong> index 1 [8].<br />
From now on we assume a Lipschitz conditions for G.<br />
In this paper, we consider an error estimate <strong>of</strong> IDC framework constructed with implicit R-K method for<br />
SPPs by presenting and proving two main Theorems in Section 3 and 4 respectively. We expand the error<br />
in powers <strong>of</strong> ɛ whose coefficients are error terms, and show convergence results for these error terms. Order<br />
reduction exists for both differential and algebraic components in the IDC framework. Especially, there is no<br />
order improvement for the ε ν (ν ≥ 1) error terms in IDC corrections, see Remark 4.7 and 4.15. We focus our<br />
analysis on the IDC method using all <strong>of</strong> the uniform quadrature nodes, but excluding the left-most endpoint.<br />
We also remark that an important property for implicit R-K method called stiffly accurate (we will define it<br />
in the next section), will be an important concept in our analysis both in the prediction and in the correction<br />
steps for IDC framework. If this property is not satisfied, the corresponding IDC method becomes unstable and<br />
the numerical solutions diverge.<br />
The paper is organized in the following way. In the rest <strong>of</strong> this introductory section, we briefly present<br />
existing classical local and global error estimate <strong>of</strong> general implicit R-K method for SPPs [8]. In Section 2,<br />
we introduce the IDC methods constructed with implicit backward Euler and high order implicit R-K method<br />
for SPPs (1.2). In Section 3, the main theoretical results are provided in the form <strong>of</strong> two main Theorems.<br />
Numerical evidence supporting these theorems are summarized and presented. In Section 4, the main theorems<br />
are proved based on the ε-expansion <strong>of</strong> numerical solutions <strong>of</strong> IDC methods constructed with backward Euler<br />
and general implicit R-K methods. Finally, conclusions are given in Section 5.<br />
1.1 <strong>Implicit</strong> R-K method applied to SPPs<br />
This section is a review <strong>of</strong> implicit R-K methods applied to SPPs (1.2). The IDC framework requires R-K<br />
methods in the prediction and correction steps. It is very important to know convergence results about R-<br />
2
K methods applied to (1.2) to investigate thoroughly the convergence estimates an IDC R-K method. Our<br />
discussion on these methods is based on notations introduced in ([8] Chap. VI.3).<br />
We consider an implicit R-K method applied to the SPP (1.2)<br />
where<br />
and the internal stages given by<br />
(<br />
yn+1<br />
)<br />
=<br />
z n+1<br />
(<br />
Yni<br />
(<br />
kni<br />
)<br />
=<br />
Z ni<br />
)<br />
=<br />
εl ni<br />
(<br />
yn<br />
z n<br />
)<br />
+ h<br />
(<br />
yn<br />
z n<br />
)<br />
+ h<br />
s∑<br />
i=1<br />
( f(Yni , Z ni )<br />
g(Y ni , Z ni )<br />
i∑<br />
j=1<br />
b i<br />
(<br />
kni<br />
l ni<br />
)<br />
)<br />
(1.7)<br />
(1.8)<br />
a ij<br />
(<br />
knj<br />
l nj<br />
)<br />
. (1.9)<br />
Such method is characterized by the coefficient matrix A = (a ij ) and vectors c = (c 1 , ..., c s ) T , b = (b 1 , ..., b s ) T .<br />
They can be represented by a tableau in the usual Butcher notation,<br />
c<br />
A<br />
b T . (1.10)<br />
The coefficients c are given by the usual relation c i = ∑ i<br />
j=1 a ij.<br />
In this paper we denote by p the classical order <strong>of</strong> the method when it is applied to a non-stiff equation.<br />
For stiff differential equations we consider an important concept the stage order q. It is defined by the relations<br />
(see and [8], Chap. IV.5), i.e.<br />
C(q) :<br />
s∑<br />
j=1<br />
a ij c k−1<br />
j<br />
= ck i<br />
, i = 1, · · · s, for k = 1, ..., q. (1.11)<br />
k<br />
This is equivalent to the fact that q = min(q 1 , ..., q s ) where, for a problem ẏ(t) = f(t, y(t)), with 0 ≤ t ≤ T<br />
and f a smooth function, the internal stages are O(h qi+1 )-approximations to the exact solution at c i h, i.e.<br />
y(t n + c i h) − Y i = O(h qi+1 ) where Y i = y(t n ) + h ∑ s<br />
j=1 a ijf(t n + c j h, Y j ), for 1 ≤ i ≤ s. For example for an<br />
s-stage diagonally implicit R-K (DIRK) method, the stage order is 1.<br />
Definition 1.1. (Stiffly accurate) An implicit R-K method is called stiffly accurate if b T = e T s A with e T s =<br />
(0, ..., 0, 1).<br />
Remark 1.2. Stiffly accurate methods are important for the solution <strong>of</strong> SPPs and differential algebraic equations.<br />
In particular this property is important for the L-stability <strong>of</strong> the method. To see this, let R(∞) =<br />
lim z→∞ R(z), with R(z) = 1 + zb T (I − zA) −1 1 being the stability function <strong>of</strong> an implicit scheme, where<br />
b T = (b 1 , ..., b s ) and 1= (1, ..., 1) T . If the A matrix is invertible, then R(∞) = 1− ∑ s<br />
i,j=1 b iω ij with ω ij elements<br />
<strong>of</strong> the inverse <strong>of</strong> (a ij ). Moreover, if the implicit method is stiffly accurate, then R(∞) = 0. This makes A-stable<br />
methods L-stable.<br />
Now supposing the matrix A invertible one obtains from (1.9)<br />
hl ni =<br />
s∑<br />
ω ij (Z nj − z n ), (1.12)<br />
j=1<br />
and inserting this into the numerical solution z n+1 it follows<br />
z n+1 = R(∞)z n +<br />
s∑<br />
b i ω ij Z nj (1.13)<br />
making the definition <strong>of</strong> z n+1 indepenent <strong>of</strong> ε. Now putting ε = 0 in (1.7) we get l ni = g(Y ni , Z ni ) = 0, then<br />
we obtain<br />
s∑<br />
Y ni = y n + a ij f(Y nj , Z nj ) (1.14)<br />
i=1<br />
3<br />
i=1
g(Y ni , Z ni ) = 0 (1.15)<br />
s∑<br />
y n+1 = y n + b i f(Y ni , Z ni ) (1.16)<br />
i=1<br />
z n+1 = R(∞)z n +<br />
s∑<br />
b i ω ij Z nj (1.17)<br />
We note that this represents the numerical method for solving the reduced system (1.4), i.e. ε = 0. From (1.15)<br />
we have Z ni = G(Y ni ) (<strong>Implicit</strong> Function Theorem). If the method is stiffly accurate we have that y n+1 = Y ns<br />
and z n+1 = Z ns = G(Y ns ) = G(y n+1 ). Then<br />
i=1<br />
g(y n+1 , z n+1 ) = 0. (1.18)<br />
In this case, the solution <strong>of</strong> (1.16)-(1.15)-(1.14) and (1.18) to system (1.4) is equivalent to the solution obtained<br />
by the same implicit R-K method applied to (1.6). Therefore, for the method (1.14)-(1.16)<br />
y n − y(t n ) = O(h p ). (1.19)<br />
Furthermore we have z n+1 = G(y n+1 ), then by assuming a Lipschitz condition for G<br />
We summarize these results by the following theorem.<br />
z n − z(t n ) = G(y n ) − G(y(t n )) = O(h p ). (1.20)<br />
Theorem 1.3. (Chap. VI. 1 Theorem 1.1 part (a) in [8]) Suppose that the system (1.4) satisfies (1.5) in a<br />
neighborhood <strong>of</strong> the exact solution and assume that the initial values are consistent. Consider a stiffly accurate<br />
R-K method <strong>of</strong> order p, and with invertible matrix A. Then the numerical solutions <strong>of</strong> (1.14)-(1.17) have global<br />
error<br />
y n − y(t n ) = O(h p ), z n − z(t n ) = O(h p ), (1.21)<br />
for t n − t 0 = nh ≤ Const.<br />
Now we review the main result obtained in Chap. VI.3 <strong>of</strong> [8] about the error analysis <strong>of</strong> implicit R-K methods<br />
for singular perturbation problem (1.2). We perform an asymptotic expansion <strong>of</strong> smooth solutions <strong>of</strong> system<br />
(1.2) and similarly for the numerical solutions <strong>of</strong> a R-K method applied to (1.2). The errors <strong>of</strong> the y and<br />
z-component are formally considered as<br />
y n − y(t n ) = ∑ ν≥0<br />
ɛ ν (y n,ν − y ν (t n )), z n − z(t n ) = ∑ ν≥0<br />
ɛ ν (z n,ν − z ν (t n )). (1.22)<br />
The first differences y n,0 − y 0 (t n ) and z n,0 − z 0 (t n ) in the expansion (1.22) are the global errors <strong>of</strong> the R-K<br />
method applied to the reduced system (1.4), i.e. system <strong>of</strong> index 1. The error estimates are summarized in<br />
Theorem 1.3. The second difference in (1.22) is related to the numerical solutions <strong>of</strong> the R-K method when<br />
applied to the differential algebraic system <strong>of</strong> index 2.<br />
Remark 1.4. A complete analysis for the convergence <strong>of</strong> implicit R-K methods for differential algebraic system<br />
<strong>of</strong> index 2 is given in [8] by Lemma 4.4, Theorem 4.5 and Theorem 4.6 in Chap. VII.4. Below we summarize<br />
optimal error estimates for R-K methods when applied to index 2 problem. From Lemma 4.4 in Chap. VII.4 <strong>of</strong><br />
[8], the local error estimate is<br />
δy h (t) . = y 1 − y(t n + h) = O(h q+1 ), δz h (t) . = z 1 − z(t n + h) = O(h q ). (1.23)<br />
If, in addition, the method is stiffly accurate<br />
δy h (t) = O(h min(p+1,q+2) ) with p ≥ q.<br />
From Theorem 4.5 and 4.6 in Chap. VII.4 <strong>of</strong> [8], the global convergence results follow. Some interesting results<br />
about local and global error for some important R-K methods are collected in Table 4.1 in Chap. VII.4 <strong>of</strong> [8].<br />
Here, as an example, we consider DIRK or SDIRK methods with p ≥ 2. Such methods have stage order q = 1<br />
implying that the local error<br />
δy h (t) = O(h 2 ), δz h (t) = O(h)<br />
δy h (t) = O(h 3 ), δz h (t) = O(h) if the method is stiffly accurate<br />
(1.24)<br />
4
and the global error<br />
y n − y(t n ) = O(h 2 ) z n − z(t n ) = O(h). (1.25)<br />
We note that concerning backward Euler method with p = q = 1, the local error is δy h (t) = O(h 2 ) and the<br />
global error y n − y(t n ) = O(h), i.e. the method, applied to a system <strong>of</strong> index 2, maintains the classical order.<br />
We note that backward Euler method is a first order Radau IIA method, then these estimates are a simple<br />
consequence <strong>of</strong> Theorem 4.9 in Chap. VII <strong>of</strong> [8] for the local error estimate and Theorem 4.5 in Chap. VII <strong>of</strong><br />
[8] for the global error estimate.<br />
Finally, the main result <strong>of</strong> global error estimate (1.22) <strong>of</strong> the R-K method when applied to SPP (1.2) is<br />
presented in Theorem 1.5 below. For details, see Chap. VI. 3, Theorem 3.3, 3.4, 3.8 and Corollary 3.10 <strong>of</strong> [8].<br />
Theorem 1.5. Consider the stiff problem (1.2), (1.3) with initial values y(0), z(0) admitting a smooth solution.<br />
Apply the R-K method (1.7)-(1.8)-(1.9) <strong>of</strong> classical order p and stage order q, (1 ≤ q < p). Assume that the<br />
method is A-stable, that the stability function satisfies |R(∞)| < 1 and that the eigenvalues <strong>of</strong> the coefficient<br />
matrix A have positve real parts. Then the global error <strong>of</strong> a R-K method satisfies<br />
y n − y(t n ) = O(h p ) + O(εh q+1 ), z n − z(t n ) = O(h q+1 ). (1.26)<br />
If in addition the method is stiffly accurate, we have<br />
The estimates hold uniformly for h ≤ h 0 and nh ≤ Const.<br />
2 IDC Formulations Applied to SPPs<br />
z n − z(t n ) = O(h p ) + O(εh q ) (1.27)<br />
In this section we consider IDC framework constructed with stiffly accurate implicit R-K method for SPPs. The<br />
use <strong>of</strong> uniform nodes is important for high order accuracy increase if high order R-K method is used in correction<br />
loops [5], where as the use <strong>of</strong> quadrature nodes excluding the left-most endpoint lead to an important stability<br />
condition for stiff problem R(∞) = 0 [11]. We remark the IDC method is not L-stable with R(∞) ≠ 0, if the<br />
quadrature nodes including the left-most endpoint is used. Moreover, when IDC R-K method using quadrature<br />
nodes excluding the left-most end point is represented in a Butcher Tableau, the corresponding A matrix would<br />
be invertible, see Section 4 and [4]. The invertibility <strong>of</strong> the A matrix is an important assumption in many <strong>of</strong><br />
the classical results in [8].<br />
2.1 IDC Framework<br />
We consider IDC procedure [6] applied to a singular pertubation problem<br />
y ′ (t) = f(y, z), y(t 0 ) = y 0 ,<br />
εz ′ (t) = g(y, z), z(t 0 ) = z 0 .<br />
(2.1)<br />
The time interval [0, T ] is discretized into intervals [t n , t n+1 ], n = 0, 1, ..., N − 1 such that<br />
0 = t 0 < t 1 < t 2 < ... < t n < ... < t N = T,<br />
with the step size H. Then, each interval [t n , t n+1 ] is discretized again into M uniform subintervals with<br />
quadrature nodes denoted by<br />
t n = t n,0 < t n,1 < · · · < t n,M = t n+1 , (2.2)<br />
with h = H M<br />
being the size <strong>of</strong> a substep. In this paper, the interval [t n, t n+1 ] will be referred to as a time step<br />
while a subinterval [t m , t m+1 ] (dropping the subscript n) will be referred to as a substep. We remark that the<br />
size <strong>of</strong> time interval [t n , t n+1 ] may vary as the IDC method is a one-step, multi-stage method. We assume the<br />
IDC quadrature nodes are uniform, which is a crucial assumption for high order improvement in accuracy, when<br />
consider applying general high order implicit R-K in prediction and correction steps for classical ODE system<br />
(1.1), see discussions in [5]. We also note that since h = H M , we will use O(hp ) and O(H p ) interchangeably<br />
throughout the paper.<br />
5
Integrating equations (2.1) with respect to t, we obtain the equivalent Picard equations<br />
{<br />
y(t) = y 0 + ∫ t<br />
t 0<br />
f(y(s), z(s))ds,<br />
εz(t) = εz 0 + ∫ t<br />
t 0<br />
g(y(s), z(s))ds,<br />
(2.3)<br />
Suppose that we have obtained approximate solutions ŷ m (0) and ẑ m<br />
(0) at t m by using a p th order numerical method<br />
for (2.1). We build a continuous polynomial interpolants ŷ (0) (t) and ẑ (0) (t) from these discrete values. Now we<br />
define the error functions<br />
e (0) (t) = y(t) − ŷ (0) (t), d (0) (t) = z(t) − ẑ (0) (t). (2.4)<br />
Note that e (0) (t) and d (0) (t) are not polynomials in general. We define the residual function<br />
{ (δ (0) ) ′ (t) = f(ŷ (0) (t), ẑ (0) (t)) − (ŷ (0) ) ′ (t)<br />
(ρ (0) ) ′ (t) = g(ŷ (0) (t), ẑ (0) (t)) − (εẑ (0) ) ′ (t).<br />
(2.5)<br />
Integrating (2.5) from t 0 to t gives,<br />
{<br />
δ (0) (t) = y 0 + ∫ t<br />
t 0<br />
f(ŷ (0) (s), ẑ (0) (s))ds − ŷ (0) (t),<br />
ρ (0) (t) = εz 0 + ∫ t<br />
t 0<br />
g(ŷ (0) (s), ẑ (0) (s))ds − εẑ (0) (t).<br />
Thus, the error equations about the error functions (2.4) by subtracting (2.6) from (2.3) become<br />
{<br />
e (0) (t) = ∫ t<br />
t 0<br />
f(e (0) (s) + ŷ 0 (s), d (0) (s) + ẑ (0) (s)) − f(ŷ (0) (s), ẑ (0) (s))ds + δ (0) (t),<br />
εd (0) (t) = ∫ t<br />
t 0<br />
g(e (0) (s) + ŷ 0 (s), d (0) (s) + ẑ (0) (s)) − g(ŷ (0) (s), ẑ (0) (s))ds + ρ (0) (t).<br />
(2.6)<br />
(2.7)<br />
Suppose that we have obtained approximate solutions ê (0)<br />
m and at t m by using a p th order numerical method<br />
for error equations (2.7). The numerical solution can then be improved as<br />
ˆd<br />
(0)<br />
m<br />
ŷ m (1) = ŷ m (0) + ê (0)<br />
m , ẑ m (1) = ẑ m (0) (0)<br />
+ ˆd m , ∀m = 0, · · · M.<br />
Such correction procedures can be repeated. In summary, the strategy <strong>of</strong> IDC methods is to use a simple<br />
numerical method to compute a previsional solution ŷ (0) (t) and ẑ (0) (t) on the interval [t n , t n+1 ] and then to<br />
solve a series <strong>of</strong> correction equations based on Equations (2.7), each <strong>of</strong> which improves the accuracy <strong>of</strong> the<br />
provisional solution.<br />
Remark 2.1. (About notations.) In our description <strong>of</strong> IDC, we let y (k)<br />
m<br />
and exact error functions (without hat); and let ŷ m (k) , ẑ m (k) , ê (k)<br />
m ,<br />
ˆd<br />
(k)<br />
m<br />
z m (k) , e (k)<br />
m , d (k)<br />
m denote the exact solutions<br />
denote the numerical approximations (with<br />
hat) to the exact solutions and error functions. We use subscript m to denote the location t = t m and use<br />
superscript (k) to denote the prediction (k = 0) and correction loops (k = 1, · · · ). We let ¯· denote the vector on<br />
IDC quadrature nodes. For example, ȳ = (y 1 , ·, y M ) excluding the left-most point.<br />
2.2 IDC Methods Based on Backward Euler Method<br />
In this subsection, we consider a simple backward Euler method for computing both the previsional solution<br />
and the corrections. As we mentioned in the introduction, we choose to work with the uniform nodes excluding<br />
the left-most endpoint for its stability property R(∞) = 0 [11].<br />
(Prediction step) Use a backward Euler discretization to compute an approximate solution ¯ŷ (0) = (ŷ (0)<br />
1<br />
to the exact solution ȳ = (y 1 , ..., y m , ..., y M ) for (2.1) at the nodes t 1 , ..., t M on the interval [t n , t n+1 ]. We make<br />
the same for the z-component. This gives<br />
{<br />
ŷ (0)<br />
m+1 = ŷ(0) m + hf(ŷ (0)<br />
m+1 , ẑ(0)<br />
m+1 ),<br />
εẑ (0)<br />
m+1 = εẑ(0) m + hg(ŷ (0)<br />
m+1 , ẑ(0) m+1 ) (2.8)<br />
for m = 0, 1, ...M − 1.<br />
(<strong>Correction</strong> loop). For k = 1, ..., K (K is the number <strong>of</strong> the correction step). Let ŷ (k−1) denote the k th<br />
sequence correction.<br />
, ..., ŷ(0) m , ..., ŷ (0)<br />
M )<br />
6
1. Denote the error function from the previous correction e (k−1) (t) = y(t) − ŷ (k−1) (t) where y(t) is the exact<br />
solution and ŷ (k−1) (t) is a (M − 1)-th polynomial interpolating ¯ŷ (k−1) . We compute the numerical error<br />
vector ¯ê (k−1) = (ê (k−1)<br />
1 , ..., ê (k−1)<br />
M<br />
) with ê(k−1) m approximates e (k−1) (t m ) by applying a backward Euler<br />
method to (2.7)<br />
⎧<br />
⎨ ê (k−1)<br />
m+1 = ê (k−1)<br />
m + h∆f (k−1)<br />
m+1 + ∫ t m+1<br />
t m<br />
δ (k−1) (s)ds,<br />
(k−1)<br />
(k−1)<br />
⎩ ε ˆd m+1 = ε ˆd m + h∆g (k−1)<br />
m+1 + ∫ t m+1<br />
(2.9)<br />
t m<br />
ρ (k−1) (s)ds<br />
where {<br />
and<br />
⎧<br />
⎨<br />
⎩<br />
∆f (k−1)<br />
m+1 = f(ŷ (k−1)<br />
m+1 + ê(k−1) m+1 , ẑ(k−1) (k−1)<br />
m+1 + ˆd m+1 ) − f(ŷ(k−1) m+1 , ẑ(k−1)<br />
∆g (k−1)<br />
m+1 = g(ŷ (k−1)<br />
m+1 + ê(k−1) m+1 , ẑ(k−1) m+1<br />
∫ tm+1<br />
m+1 )<br />
(k−1)<br />
+ ˆd m+1 ) − g(ŷ(k−1) m+1 , ẑ(k−1) m+1 ), (2.10)<br />
t m<br />
δ (k−1) (s)ds = ∫ t m+1<br />
t m<br />
f(ŷ (k−1) (s), ẑ (k−1) (s))ds − ŷ (k−1)<br />
m+1 + ŷ(k−1) m<br />
∫ tm+1<br />
t m<br />
ρ (k−1) (s)ds = ∫ t m+1<br />
t m<br />
g(ŷ (k−1) (s), ẑ (k−1) (s))ds − εẑ (k−1)<br />
m+1 + εẑ(k−1) m .<br />
(2.11)<br />
The integral term ∫ t m+1<br />
t m<br />
in equations (2.11) are approximated by a numerical quadrature. Especially, let<br />
S be the integration matrix with its (m, k) element<br />
where<br />
S m,k = 1 h<br />
∫ tm+1<br />
t m<br />
α k (s)ds, for m = 0, · · · , M − 1, k = 1, · · · M<br />
α k (s) =<br />
M∏<br />
i≠k,i=1<br />
is the Lagrangian basis function based on the node t k . Let<br />
with ∑ M<br />
j=1 Sm,j = 1. Then<br />
S m ( ¯f) =<br />
s − t k<br />
t i − t k<br />
(2.12)<br />
M∑<br />
S m,j f(y j , z j ), (2.13)<br />
j=1<br />
∫<br />
hS m ( ¯f)<br />
tm+1<br />
− f(y(s), z(s))ds = O(h M+1 ),<br />
t m<br />
for any smooth function f. In other words, the quadrature formula given by hS m ( ¯f) approximates the<br />
exact integration with (M + 1) order accuracy locally.<br />
Remark 2.2. Considering the following change <strong>of</strong> variable s = t 0 + σh in the interval [t n , t n+1 ] we get<br />
α k (t 0 + σh) = ∏ M<br />
, i.e. there is only a dependence on M and not on h. Then the integral<br />
i≠k σ−k<br />
i−k<br />
depends on M and not on h.<br />
S m,k = 1 h<br />
∫ tm+1<br />
t m<br />
α k (s)ds =<br />
∫ m+1<br />
m<br />
α k (t 0 + σh)dσ (2.14)<br />
2. Update the approximate solutions ¯ŷ (k) = ¯ŷ (k−1) + ¯ê (k−1) and ¯ẑ (k) = ¯ẑ (k−1) + ¯ˆd(k−1) .<br />
Remark 2.3. Using these notations, we get from eqs. (2.9)<br />
{<br />
ŷ (k)<br />
m+1 = ŷ(k) m + h∆f (k−1)<br />
εẑ (k)<br />
m+1 = εẑ(k) m<br />
m+1 + hSm ( ¯ˆf (k−1) ),<br />
+ h∆g (k−1)<br />
m+1 + hSm (¯ĝ (k−1) )<br />
(2.15)<br />
where we use the S m notation introduced in (2.13).<br />
Remark 2.4. Since we consider the nodes excluding the left most quadrature point t 0 , the order <strong>of</strong> approximation<br />
for integration/interpolation will be one order lower than the usual one considered in [6, 5].<br />
7
2.3 IDC Methods Based on General <strong>Implicit</strong> R-K<br />
Below, we describe how we apply an s-stage implicit R-K methods in correction loops for IDC framework. For<br />
the internal stages in the R-K method, we introduce the integration matrix and interpolation matrix as following<br />
hS cmi,k =<br />
∫ tm+c mih<br />
t m<br />
α k (s)ds, P cmi,k = α k (t m + c mi h), (2.16)<br />
∀m = 0, · · · , M − 1, ∀k = 1, · · · M, ∀mi = 1, · · · s, where α j (s) as introduced in equation (2.12) is the<br />
Lagrangian basis function based on the node t j . Let<br />
Then<br />
S cmi ( ¯f)<br />
M∑<br />
= S cmi,j f(y j , z j ), P cmi ( ¯f)<br />
M∑<br />
= P cmi,j f(y j , z j )<br />
j=1<br />
j=1<br />
∫ tm+c ih<br />
hS cmi ( ¯f) − f(y(s), z(s))ds = O(h M+1 )<br />
t m<br />
P cmi ( ¯f) − f(y(t m + c i h), z(t m + c i h)) = O(h M )<br />
for any smooth function f. In other words, the quadrature formula given by hS cmi ( ¯f) approximates the exact<br />
integration with (M +1) th order accuracy locally, while the interpolation formula given by P cmi ( ¯f) approximates<br />
the exact solution at R-K internal stages with M th order accuracy.<br />
To compute the numerical error approximating the error function e (k−1) (t m ), d (k−1) (t m ) with a general<br />
implicit R-K method to (2.7),<br />
with<br />
(<br />
(<br />
ê (k−1)<br />
m+1<br />
ˆd (k−1)<br />
m+1<br />
(k−1)<br />
∆ ˆK<br />
ε∆<br />
mi<br />
(k−1) ˆL mi<br />
)<br />
)<br />
=<br />
.<br />
=<br />
=<br />
(<br />
ê (k−1)<br />
m + h ∫ 1<br />
0 δ(t m + τh)dτ<br />
ˆd (k−1)<br />
m + h/ε ∫ 1<br />
0 ρ(t m + τh)dτ<br />
(<br />
(<br />
)<br />
f(Ŷ (k)<br />
mi , Ẑ(k) mi ) − P cmi ( ¯ˆf (k−1) )<br />
g(Ŷ (k)<br />
mi , Ẑ(k) mi ) − P cmi (¯ĝ (k−1) )<br />
+ h<br />
)<br />
(<br />
s∑<br />
b i<br />
i=1<br />
f(Ŷ (k)<br />
mi , Ẑ(k) mi ) − f(P cmi (¯ŷ (k−1) ), P cmi (¯ẑ (k−1) ))<br />
g(Ŷ (k)<br />
mi , Ẑ(k) mi ) − g(P cmi (¯ŷ (k−1) ), P cmi (¯ẑ (k−1) ))<br />
(k−1)<br />
∆ ˆK<br />
∆<br />
mi<br />
(k−1) ˆL mi<br />
)<br />
)<br />
(2.17)<br />
(2.18)<br />
+ O(h M ), (2.19)<br />
where the last equation above is due to the high order interpolation accuracy <strong>of</strong> P cmi . We note that equation<br />
(2.18) is for numerical implementation, whereas equation (2.19) is preparation for our analysis. Here we put<br />
with ( Ê(k−1) mi<br />
ˆD (k−1)<br />
mi<br />
Ŷ (k)<br />
mi<br />
)<br />
= P cmi (¯ŷ (k−1) ) + Ê(k−1) mi , Ẑ (k)<br />
mi = P cmi (¯ẑ (k−1) ) +<br />
=<br />
(<br />
ê (k−1)<br />
m<br />
ˆd (k−1)<br />
m<br />
We can rewrite the previous system (2.17) as<br />
⎛<br />
with<br />
⎛<br />
⎝ ŷ(k) m+1 − hSm,(k−1) ¯ˆf<br />
ẑ (k)<br />
m+1 − hSm,(k−1) ¯ĝ<br />
⎛<br />
⎝ Ŷ (k)<br />
mi<br />
− hS cmi,(k−1)<br />
¯ˆf<br />
Ẑ (k)<br />
mi − hScmi,(k−1)<br />
¯ĝ<br />
⎝ Sm,(k−1) ¯ˆf<br />
εS m,(k−1)<br />
¯ĝ<br />
+ h ∫ c mi<br />
δ(t<br />
0 m + τh)dτ<br />
+ h/ε ∫ c mi<br />
ρ(t<br />
0 m + τh)dτ<br />
⎞<br />
⎠ =<br />
⎞<br />
= S m ( ¯ˆf (k−1) )<br />
⎠ =<br />
= S m (¯ĝ (k−1) )<br />
⎞<br />
(<br />
(<br />
⎠ ,<br />
ŷ (k)<br />
m<br />
ẑ (k)<br />
m<br />
ŷ (k)<br />
m<br />
ẑ (k)<br />
m<br />
⎛<br />
)<br />
)<br />
+ h<br />
+ h<br />
)<br />
+ h<br />
(<br />
s∑<br />
b i<br />
i=1<br />
(<br />
i∑<br />
a ij<br />
j=1<br />
⎝ Scmi,(k−1)<br />
¯ˆf<br />
εS cmi,(k−1)<br />
¯ĝ<br />
(<br />
i∑<br />
a ij<br />
j=1<br />
(k−1)<br />
∆ ˆK<br />
∆<br />
mi<br />
(k−1) ˆL mi<br />
(k−1)<br />
∆ ˆK mj<br />
∆<br />
ˆD<br />
(k−1)<br />
mi (2.20)<br />
ˆL<br />
(k−1)<br />
mj<br />
= S cmi ( ¯ˆf (k−1) )<br />
= S cmi (¯ĝ (k−1) )<br />
(k−1)<br />
∆ ˆK mj<br />
∆<br />
ˆL<br />
(k−1)<br />
mj<br />
)<br />
)<br />
)<br />
. (2.21)<br />
(2.22)<br />
, (2.23)<br />
⎞<br />
⎠ . (2.24)<br />
8
Remark 2.5. Assuming that A is invertible, following a similar procedure as in equation (1.12) and (1.13), we<br />
get in the vectorial form from the second equation <strong>of</strong> (2.23)<br />
ˆL<br />
(k−1)<br />
m1<br />
with ∆ ¯ˆL(k−1) = (∆ , · · · , ∆<br />
Plug this into the second equation <strong>of</strong> (2.22), we get<br />
h∆ ¯ˆL(k−1) = A −1 ( ¯Ẑ (k) − ẑ (k)<br />
m 1 − hS¯c (¯ĝ (k−1) )),<br />
ˆL<br />
(k−1)<br />
ms ) T , 1 = (1, · · · , 1) T and ¯c = (c m1 , · · · , c ms ) are R-K internal stages.<br />
ẑ (k)<br />
m+1 = ẑ m<br />
(k) + hS m (¯ĝ (k−1) ) + b T A −1 ( ¯Ẑ (k) − ẑ m (k) 1 − hS¯c (¯ĝ (k−1) )).<br />
Especially for stiffly accurate R-K method with b T A −1 = e T s , we have<br />
ẑ (k)<br />
m+1 = R(∞)ẑ(k) m + b T A −1 ¯Ẑ(k) = Ẑ(k) ms, (2.25)<br />
by R(∞) = 0 and S m (¯ĝ (k−1) ) = e T s S¯c (¯ĝ (k−1) ) = b T A −1 S¯c (¯ĝ (k−1) ). We remark that equation (2.25) is in a<br />
similar spirit to (1.17) for implicit R-K method.<br />
2.4 ε-asymptotic expansion<br />
In this paper we want to use the approach proposed in [8] considering the ε-expansion <strong>of</strong> the exact and numerical<br />
solution to study the behavior <strong>of</strong> the local error for the IDC method. To prove the main results, ε-expansion<br />
<strong>of</strong> the exact solutions and numerical solutions are performed; this is a preparation for local error estimates in<br />
Section 4.<br />
• The ε-expansion <strong>of</strong> the exact solution <strong>of</strong> problem (2.1) is the following,<br />
( ) ( ∑ ∞ y(t)<br />
=<br />
ν=0 y )<br />
∑ ν(t)ε ν<br />
∞<br />
z(t)<br />
ν=0 z ν(t)ε ν<br />
(2.26)<br />
where y ν (t) and z ν (t) are ε-independent functions, which are solutions <strong>of</strong> a sequence <strong>of</strong> differential algebraic<br />
equations <strong>of</strong> arbitrary index [8]. Evaluating (2.26) at t m this gives<br />
( ) ( ∑ ∞ ym<br />
=<br />
ν=0 y )<br />
∑<br />
m,νε ν<br />
∞<br />
z m ν=0 z m,νε ν (2.27)<br />
Inserting (2.26) into (2.1) and collecting terms <strong>of</strong> equal powers <strong>of</strong> ε yields<br />
ε 0 :<br />
{ y<br />
′<br />
0 = f(y 0 , z 0 )<br />
0 = g(y 0 , z 0 )<br />
(2.28)<br />
ε 1 :<br />
{ y<br />
′<br />
1 = f y (y 0 , z 0 )y 1 + f z (y 0 , z 0 )z 1<br />
. = F1<br />
z ′ 0 = g y (y 0 , z 0 )y 1 + g z (y 0 , z 0 )z 1<br />
. = G1<br />
(2.29)<br />
· · · (2.30)<br />
ε ν :<br />
{ y<br />
′<br />
ν = f y (y 0 , z 0 )y ν + f z (y 0 , z 0 )z ν + φ ν (y 0 , z 0 , · · · , y ν−1 , z ν−1 ) . = F ν<br />
z ′ ν−1 = g y (y 0 , z 0 )y ν + g z (y 0 , z 0 )z ν + ψ ν (y 0 , z 0 , · · · , y ν−1 , z ν−1 ) . = G ν<br />
(2.31)<br />
with initial values y ν (0), z ν (0) known from (2.26). We observe that system (2.28) under the condition<br />
(1.3) is a differential algebraic system <strong>of</strong> index 1. According to [8], if we consider (2.28) and (2.29) together<br />
we have a differential algebraic system <strong>of</strong> index 2. In general (2.28), (2.29) and (2.31) is a system <strong>of</strong> index<br />
ν.<br />
• The ε-expansion <strong>of</strong> the numerical solution at k th iteration is the following. The case <strong>of</strong> k = 0 is for<br />
numerical solution at prediction.<br />
( ) (<br />
ŷ m<br />
(k) ∑∞<br />
)<br />
ẑ m<br />
(k) =<br />
ν=0 ŷ(k) m,νε ν<br />
∑ ∞<br />
. (2.32)<br />
ν=0 ẑ(k) m,νε ν<br />
Backward Euler in prediction and correction steps <strong>of</strong> IDC.<br />
By plugging the ε-expansion <strong>of</strong> numerical solution (2.32) into the numerical scheme (2.8)-(2.11), and<br />
matching and collecting terms that are <strong>of</strong> equal powers <strong>of</strong> ɛ, one obtains the following.<br />
9
– For the prediction step k = 0<br />
where<br />
⎧<br />
⎨<br />
⎩<br />
ˆF (0)<br />
m+1,1<br />
Ĝ (0)<br />
m+1,1<br />
ε 0 :<br />
ε 1 :<br />
{<br />
{<br />
(<br />
.<br />
= f y (ŷ (0)<br />
.<br />
=<br />
ŷ (0)<br />
m+1,0 = ŷ(0) m,0 + hf(ŷ(0) m+1,0 , ẑ(0) m+1,0 )<br />
0 = g(ŷ (0)<br />
m+1,0 , ẑ(0) m+1,0 ). (2.33)<br />
ŷ (0)<br />
m+1,1 = ŷ(0) (0)<br />
m,1 + hˆF m+1,1<br />
ẑ (0)<br />
m+1,0 = ẑ(0) m,0 + hĜ(0) m+1,1<br />
m+1,0 , ẑ(0) m+1,0 )ŷ(0) m+1,1 + f z(ŷ (0)<br />
m+1,0 , ẑ(0) m+1,0 )ẑ(0) m+1,1<br />
(<br />
g y (ŷ (0)<br />
m+1,0 , ẑ(0) m+1,0 )ŷ(0) m+1,1 + g z(ŷ (0)<br />
m+1,0 , ẑ(0) m+1,0 )ẑ(0) m+1,1<br />
Equation (2.33)-(2.34) are consistent discretizations <strong>of</strong> equation (2.28)-(2.29).<br />
– For the correction steps k ≥ 1,<br />
⎧<br />
⎪⎨<br />
ε 0 :<br />
⎪⎩<br />
⎧<br />
⎪⎨<br />
ε 1 :<br />
⎪⎩<br />
ŷ (k)<br />
m+1,0 = ŷ (k)<br />
(k−1)<br />
m,0 + h∆ ˆf m+1,0 + hSm ( ¯ˆf (k−1)<br />
0 )<br />
0 = h∆ĝ (k−1)<br />
m+1,0 + hSm (¯ĝ (k−1)<br />
0 )<br />
ŷ (k)<br />
m+1,1<br />
ẑ (k)<br />
m+1,0<br />
= ŷ (k)<br />
(k−1)<br />
m,1 + h∆ˆF m+1,1 + hSm (¯ˆF(k−1) 1 )<br />
= ẑ (k)<br />
m,0 + h∆Ĝ(k−1) m+1,1 + hSm (k−1)<br />
( ¯Ĝ 1 )<br />
)<br />
)<br />
.<br />
(2.34)<br />
(2.35)<br />
(2.36)<br />
(2.37)<br />
In equation (2.36)<br />
and in equation (2.37)<br />
{<br />
(k−1)<br />
∆ ˆf m+1,0 = f(ŷ (k)<br />
m+1,0 , ẑ(k) m+1,0 ) − f(ŷ(k−1) m+1,0 , ẑ(k−1) m+1,0 )<br />
∆ĝ (k−1)<br />
m+1,0 = g(ŷ (k)<br />
m+1,0 , ẑ(k) m+1,0 ) − g(ŷ(k−1) m+1,0 , ẑ(k−1) m+1,0 ) (2.38)<br />
where<br />
∆ˆF (k−1)<br />
m+1,1 =<br />
=<br />
(k) ˆF m+1,1<br />
(<br />
f y (ŷ (k)<br />
−<br />
− ˆF<br />
(k−1)<br />
m+1,1<br />
m+1,0 , ẑ(k) m+1,0 )ŷ(k) m+1,1 + f z(t m+1 , ŷ (k)<br />
(<br />
f y (ŷ (k−1)<br />
m+1,0 , ẑ(k−1) m+1,0 )ŷ(k−1) m+1,1 + f z(ŷ (k−1)<br />
m+1,0 , ẑ(k−1) m+1,0 )ẑ(k−1) m+1,1<br />
= f y (y m+1,0 , z m+1,0 )ê (k−1)<br />
m+1,1 + f z(y m+1,0 , z m+1,0 )<br />
.<br />
= f y ê (k−1)<br />
m+1,1 + f (k−1)<br />
z ˆd m+1,1 + O(hsk−1+1 ),<br />
m+1,0 , ẑ(k) m+1,0 )ẑ(k) m+1,1<br />
)<br />
)<br />
ˆd<br />
(k−1)<br />
m+1,1 + O(hs k−1+1 )<br />
(2.39)<br />
ˆF (k)<br />
m,1 = f y(ŷ (k)<br />
m,0 , ẑ(k) m,0 )ŷ(k) m,1 + f z(ŷ (k)<br />
m,0 , ẑ(k) m,0 )ẑ(k) m,1 . (2.40)<br />
Here we assume y m,0 − ŷ (k)<br />
m,0 = O(hs k+1 ) locally for all k. For simplicity <strong>of</strong> notations we let<br />
f y (y m+1,0 , z m+1,0 ) = f y , and similarly for f z . Similarly, we have<br />
∆Ĝ(k−1) m+1,1 = g y ê (k−1)<br />
m+1,1 + g (k−1)<br />
z ˆd m+1,1 + O(hsk−1+1 ). (2.41)<br />
Equation (2.36)-(2.37) are consistent discretizations <strong>of</strong> equation (2.28)-(2.29) respectively.<br />
<strong>Implicit</strong> R-K in prediction and correction steps <strong>of</strong> IDC.<br />
ˆK<br />
(k−1)<br />
mi<br />
ˆL<br />
(k−1)<br />
mi<br />
We formally expand the quantities ∆ , ∆<br />
(2.22) with k ≥ 1 into power <strong>of</strong> ε with ε-independent coefficients<br />
ŷ (k)<br />
m<br />
Ŷ (k)<br />
mi<br />
∆<br />
ẑ (k)<br />
m<br />
= ŷ (k)<br />
m,0 + εŷ(k) m,1 + ε2 ŷ (k)<br />
= Ŷ (k)<br />
mi,0<br />
= ∆<br />
ˆK<br />
(k−1)<br />
mi<br />
m,2 + ...<br />
+ εŶ<br />
(k)<br />
mi,1 + ε2 Ŷ (k)<br />
ˆK<br />
(k−1)<br />
mi,0<br />
mi,2 + ...<br />
(k−1)<br />
+ ε∆ ˆK + ε2 ∆<br />
mi,1<br />
= ẑ (k)<br />
m,0 + εẑ(k) m,1 + ε2 ẑ (k)<br />
m,2 + ...<br />
Ẑ (k)<br />
mi = Ẑ(k) mi,0 + εẐ(k) mi,1 + ε2 Ẑ (k)<br />
∆ = ε −1 ∆<br />
ˆL<br />
(k−1)<br />
mi<br />
ˆL<br />
(k−1)<br />
mi,−1<br />
from (2.18) and Ŷ (k)<br />
mi , Ẑ(k) mi , ŷ(k) m+1 , ẑ(k) m+1 from (2.20)<br />
ˆK<br />
(k−1)<br />
mi,2 + ...<br />
mi,2 + ...<br />
(k−1)<br />
+ ε∆ ˆL<br />
+ ∆ ˆL<br />
(k−1)<br />
mi,0<br />
mi,1<br />
+ ε2 (k−1)<br />
∆ ˆL mi,2 + · · · . (2.42)<br />
10
Inserting (2.42) into (2.18) we obtain the following<br />
ε 0 :<br />
ε 1 :<br />
ε ν :<br />
Similarly, we have<br />
(k−1)<br />
∆ ˆK mi,0 = f(Ŷ (k)<br />
(k−1)<br />
∆ ˆK mi,1 =<br />
(k−1)<br />
∆ ˆK mi,ν =<br />
(<br />
f y (Ŷ (k)<br />
−<br />
(k−1)<br />
∆ ˆL mi,−1 = g(Ŷ (k)<br />
(k−1)<br />
∆ ˆL mi,0 =<br />
mi,0 , Ẑ(k) mi,0 ) − f(P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 )) + O(h M ) (2.43)<br />
)<br />
mi,0 , Ẑ(k) (k)<br />
mi,0 )Ŷ mi,1 + f z(Ŷ (k)<br />
mi,0 , Ẑ(k) mi,0 )Ẑ(k) mi,1<br />
(<br />
f y (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ))P cmi (¯ŷ (k−1)<br />
1 )<br />
)<br />
+f z (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ))P cmi (¯ẑ (k−1)<br />
1 )<br />
(<br />
f y (Ŷ (k)<br />
mi,0 , Ẑ(k) (k)<br />
mi,0 )Ŷ mi,ν + f z(Ŷ (k)<br />
mi,0 , Ẑ(k) mi,0 )Ẑ(k) mi,ν<br />
(<br />
− f y (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ))P cmi (¯ŷ ν (k−1) )<br />
)<br />
+f z (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ))P cmi (¯ẑ (k−1)<br />
ν )<br />
+ψ ν (Ŷ (k)<br />
mi,0 , Ẑ(k) (k)<br />
mi,0 , ..., Ŷ mi,ν−1 , Ẑ(k) mi,ν−1 )<br />
+ O(h M ) (2.44)<br />
)<br />
+ψ ν (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ), ..., P cmi (¯ŷ (k−1)<br />
ν−1 ), P cmi (¯ẑ (k−1)<br />
ν−1 ))<br />
+O(h M ). (2.45)<br />
(<br />
g y (Ŷ (k)<br />
−<br />
mi,0 , Ẑ(k) mi,0 ) − g(P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 )) + O(h M ) (2.46)<br />
)<br />
mi,0 , Ẑ(k) (k)<br />
mi,0 )Ŷ mi,1 + g z(Ŷ (k)<br />
mi,0 , Ẑ(k) mi,0 )Ẑ(k) mi,1<br />
(<br />
g y (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ))P cmi (ȳ (k−1)<br />
1 )<br />
)<br />
+g z (P cmi (¯ŷ (k−1)<br />
0 ), P cmi (¯ẑ (k−1)<br />
0 ))P cmi (¯z (k−1)<br />
1 )<br />
+ O(h M ).<br />
Because <strong>of</strong> the linearity <strong>of</strong> relations (2.22) and (2.23), we have to order ε ν with ν = −1 in vectorial form<br />
and for ν ≥ 0,<br />
where<br />
with S m,(k−1)<br />
¯ˆF ν<br />
⎛<br />
(k−1)<br />
hA∆ ˆL ¯m,−1 + hS−→c (¯ĝ) = 0,<br />
⎝ ŷ(k) m+1,ν − hSm,(k−1) ¯ˆF ν<br />
ẑ (k)<br />
m+1,ν − hSm,(k−1) ¯Ĝ ν<br />
⎛<br />
⎝ Ŷ (k)<br />
mi,ν − hScmi,(k−1)<br />
¯ˆF ν<br />
Ẑ (k)<br />
mi,ν − hScmi,(k−1)<br />
¯Ĝ ν<br />
⎛<br />
⎞<br />
⎠ =<br />
⎞<br />
⎠ =<br />
⎝ Sm,(k−1) ¯ˆF ν<br />
εS m,(k−1)<br />
¯Ĝ ν<br />
(<br />
⎞<br />
(<br />
⎠ =<br />
hb T ∆L (k−1)<br />
¯m,−1 + hSm (¯ĝ) = 0, (2.47)<br />
ŷ (k)<br />
m,ν<br />
ẑ (k)<br />
m,ν<br />
ŷ (k)<br />
m,ν<br />
ẑ (k)<br />
m,ν<br />
(<br />
)<br />
)<br />
+ h<br />
+ h<br />
(<br />
s∑<br />
b i<br />
i=1<br />
(<br />
i∑<br />
a ij<br />
j=1<br />
S m (¯ˆF(k−1) ν )<br />
S m (<br />
¯Ĝ<br />
(k−1)<br />
ν )<br />
)<br />
(k−1)<br />
∆ ˆK mi,ν<br />
∆<br />
ˆL<br />
(k−1)<br />
mi,ν<br />
(k−1)<br />
∆ ˆK mj,ν<br />
∆<br />
ˆL<br />
(k−1)<br />
mj,ν<br />
)<br />
)<br />
(2.48)<br />
. (2.49)<br />
= S m (¯ˆF(k−1) ν ) and S m,(k−1) = S ¯Ĝ m (k−1)<br />
( ¯Ĝ ν ). Similarly for S cmi,(k−1) and S cmi,(k−1) .<br />
ν<br />
¯ˆF ν<br />
¯Ĝ ν<br />
(2.50)<br />
• Let the ε-expansion <strong>of</strong> error function e (k) (t), d (k) (t) at the k th iteration be the following.<br />
( ) (<br />
e (k) ∑∞<br />
) ( ∑∞<br />
)<br />
m<br />
d (k) =<br />
ν=0 e(k) m,νε ν<br />
∑ ∞<br />
=<br />
ν=0 (y m,ν − ŷ m,ν)ε (k) ν<br />
m<br />
ν=0 d(k) m,νε ν ∑ ∞<br />
ν=0 (z m,ν − ẑ m,ν)ε (k) . (2.51)<br />
ν<br />
Let the ε-expansion <strong>of</strong> numerical approximations <strong>of</strong> error functions ê (k) (t), ˆd (k) (t) at the k th iteration be<br />
the following. ( ) (<br />
ê (k) ∑∞ ) ( ∑∞ )<br />
m<br />
ˆd (k)<br />
l=0<br />
=<br />
ê(k) m,l<br />
∑ εl<br />
l=0<br />
∞ (k) =<br />
(ŷ(k+1) m,l<br />
− ŷ (k)<br />
m,l<br />
∑ )εl<br />
∞<br />
m<br />
l=0<br />
ˆd<br />
m,l εl l=0 (ẑ(k+1) m,l<br />
− ẑ (k) . (2.52)<br />
m,l )εl<br />
Combining (2.51) and (2.52), we observe with k, ν ≥ 0, m = 0, · · · M<br />
e (k)<br />
m,ν = ê (k)<br />
m,ν + e (k+1)<br />
m,ν , d (k)<br />
m,ν =<br />
ˆd<br />
(k)<br />
m,ν + d (k+1)<br />
m,ν . (2.53)<br />
11
3 Main results and numerical evidence<br />
In this section, we present the main theoretical results in the form <strong>of</strong> theorems, and provide numerical evidence<br />
supporting the main theorems. We will provide a rigorous mathematical pro<strong>of</strong> in the next section.<br />
3.1 Main results<br />
The aim <strong>of</strong> this section is to present convergence results <strong>of</strong> IDC framework based on backward Euler method<br />
and general implicit R-K method when applied to (1.2).<br />
Theorem 3.1. Consider the stiff system (1.2) with the condition (1.3) holds with initial values y(0), z(0)<br />
admitting a smooth solution. Consider the IDC method constructed with M uniformly distributed quadrature<br />
nodes excluding the left-most point, and backward Euler method for the prediction and correction loops k =<br />
1, · · · K. Then for any fixed constant c > 0 the global error after K correction loops satisfies the following<br />
estimates<br />
e (K)<br />
n<br />
d (K)<br />
n<br />
= y n (K) − y(t n ) = O(H min{K+1,M} ) + O(εH) + O(ε 2 ) + O(ε 3 /H)<br />
= z n (K) − z(t n ) = O(H min{K+1,M} ) + O(εH) + O(ε 2 ) + O(ε 3 /H),<br />
for ε ≤ cH, where H = Mh is one IDC time step as in equation (2.2).<br />
H ≤ H 0 and nH ≤ Const.<br />
(3.1)<br />
The estimates hold uniformly for<br />
Theorem 3.2. Consider the stiff system (1.2) with the condition (1.3) holds with initial values y(0), z(0)<br />
admitting a smooth solution. Consider the IDC method constructed with M uniformly distributed quadrature<br />
nodes excluding the left-most point, an implicit stiffly accurate R-K method <strong>of</strong> order p (0) , stage order q (0) with<br />
(q (0) < p (0) ) for the prediction step. Apply implicit R-K methods <strong>of</strong> different classical orders (p (1) , p (2) , . . . , p (K) )<br />
in the correction loops k = 1, · · · K. Assume that each <strong>of</strong> these implicit R-K methods in the correction loops are<br />
stiffly accurate. Then for any fixed constant c > 0 the global error after K correction loops satisfies the following<br />
estimates<br />
e (K)<br />
n<br />
d (K)<br />
n<br />
= y n (K) − y(t n ) = O(H min{sK,M} ) + O(εH q(0) ) + · · · + O(ε ν H q(0) +1−ν ) + O(ε ν+1 /H)<br />
= z n (K) − z(t n ) = O(H min{SK,M} ) + O(εH q(0) ) + · · · + O(ε ν H q(0) +1−ν ) + O(ε ν+1 /H)<br />
for ε ≤ cH, 1 ≤ ν ≤ q (0) + 1, where s K = ∑ K<br />
k=0 p(k) , and H = Mh is one IDC time step as in equation (2.2).<br />
The estimates hold uniformly for H ≤ H 0 and nH ≤ Const.<br />
Remark 3.3. We note that the estimates (3.1) and (3.2) can be rewritten as<br />
(3.2)<br />
e (K)<br />
n = e (K)<br />
n,0 + εe(K) n,1 + · · · + εν e (K)<br />
n,ν + O(ε ν+1 /H)<br />
d (K)<br />
n = d (K)<br />
n,0 + εd(K) n,1 + · · · + εν d (K)<br />
n,ν + O(ε ν+1 /H)<br />
(3.3)<br />
It represents the global error functions e (K)<br />
n<br />
and d (K)<br />
n<br />
at the K-th correction as an ε-expansion <strong>of</strong> e (K)<br />
n,ν , ν =<br />
0, 1, · · · , which are the global errors <strong>of</strong> the IDC implicit R-K method applied to the differential algebraic systems<br />
systems (2.28), (2.29) and (2.31) <strong>of</strong> different indexs. Then the terms O(· · · ) in (3.1) and (3.2) are the estimates<br />
<strong>of</strong> such global errors. An estimate <strong>of</strong> the remainder is given by O(ε ν+1 /H). We will justify these estimates in<br />
the next section.<br />
3.2 Numerical evidence<br />
We present numerical evidence <strong>of</strong> Theorem 3.1 and Theorem 3.2. Below, we consider the IDC method embedded<br />
with the following implicit methods.<br />
• The first order backward Euler method (BE). The order p = 1 and the stage order q = 1.<br />
• The second order stiffly accurate DIRK method (DIRK2-SA) with the Butcher tableau<br />
γ γ 0<br />
1 1 − γ γ<br />
1 − γ γ<br />
(3.4)<br />
where γ = 1 − √ 2<br />
2<br />
. This method is stiffly accurate with the order p = 2 and the stage order q = 1.<br />
12
• The second order not stiffly accurate midpoint method (DIRK2-NSA)<br />
1/2 1/2<br />
1<br />
(3.5)<br />
This method is not stiffly accurate with the order p = 2 and the stage order q = 1.<br />
• The third order Radau IIA method (Radau) with the Butcher tableau<br />
1/3 5/12 −1/12<br />
1 3/4 1/4<br />
3/4 1/4<br />
(3.6)<br />
This method is stiffly accurate with the order p = 3 and the stage order q = 2.<br />
The indicated order <strong>of</strong> convergence by the Theorems for the y- z- component in the singular perturbation system<br />
(1.2) are summarized in Table 3.1. Below is a discussion <strong>of</strong> the Table.<br />
• When the first order backward Euler method is used in both prediction and k correction steps in an<br />
IDC framework with M quadrature points (IDC-BE-M-k), the order <strong>of</strong> convergence will increase by 1 for<br />
index 1 problem leading to a term <strong>of</strong> H min(M,k+1) ; the order <strong>of</strong> convergence for index 2 problem will be<br />
dominated by the stage order <strong>of</strong> the prediction, leading to a term <strong>of</strong> εH.<br />
• When the second order stiffly accurate DIRK method is used in both prediction and k correction steps<br />
in an IDC framework with M quadrature points (IDC-DIRK2-SA-M-k), the order <strong>of</strong> convergence will<br />
increase by 2 for index 1 problem leading to a term <strong>of</strong> H min(M,2(k+1)) ; the order <strong>of</strong> convergence for index<br />
2 problem will be dominated by the stage order <strong>of</strong> the prediction, leading to a term <strong>of</strong> εH.<br />
• An important ingredient, suggested by the analysis is the property <strong>of</strong> stiffly accuracy for implicit R-<br />
K method. Such a choice provides a significant benefit for the convergence <strong>of</strong> the numerical solution,<br />
without which would lead to the divergence <strong>of</strong> numerical solution. For example, if we consider using the<br />
second order non stiffly accurate DIRK method in both the prediction and k correction steps <strong>of</strong> an IDC<br />
framework with M quadrature points (IDC-DIRK2-NSA-M-k), divergence results are expected. Note that<br />
in Sect. 4.2, a satisfactory theoretical explanation <strong>of</strong> this fact is given.<br />
• When the third order stiffly accurate Radau IIA method (with stage order q = 2) is used in prediction<br />
step and the first order backward Euler method is used in k correction steps in an IDC framework with M<br />
quadrature points (IDC-Radau-BE-M-k), the order <strong>of</strong> convergence will increase by 1 for index 1 problem<br />
leading to a term <strong>of</strong> H min(M,3+k) ; the order <strong>of</strong> convergence for index 2 problem will be dominated by the<br />
stage order <strong>of</strong> the prediction leading to a term <strong>of</strong> εH 2 .<br />
Table 3.1: Global error predicted by Theorem 3.1 and Theorem 3.2 with H ≫ ε. Note that ‘SA’/‘NSA’ means<br />
stiffly accurate/not stiffly accurate.<br />
Method y−comp z−comp<br />
IDC-BE-M-k H min(M,k+1) + εH H min(M,k+1) + εH<br />
IDC-DIRK2-SA-M-k H min(M,2(k+1)) + εH H min(M,2(k+1)) + εH<br />
IDC-DIRK2-NSA-M-k diverge diverge<br />
IDC-Radau-BE-M-k H min(M,3+k) + εH 2 H min(M,3+k) + εH 2<br />
For numerical verification, we first consider a scalar example [8]<br />
εz ′ = −z + cos(t) (3.7)<br />
with the analytical solution<br />
z(t) =<br />
cos(t) + ε sin(t)<br />
1 + ε 2 + Cexp(−t/ε),<br />
13
where C = z(0) − 1 is determined by the initial condition. For a well-prepared initial condition, let C = 0. This<br />
is a good example to investigate the order <strong>of</strong> convergence for the ε 1 term in equation (1.22), as the error for<br />
ε 0 is 0. Indeed, for stiff parameter ε = 10 −6 only a region <strong>of</strong> first order convergence is observed for the stiffly<br />
accurate backward Euler method where the global and local error given for the z-component in Theorem 1.5 is<br />
O(εH). Figure 3.1 gives the one step error (local error) and global error <strong>of</strong> backward Euler method; expected<br />
O(εH) is observed. We also test the IDC embedded with second order but not stiffly accurate midpoint method<br />
with three quadrature nodes in Figure 3.2. Divergence is observed when time step is large compared to ε if an<br />
IDC-correction is performed.<br />
Figure 3.1: Scalar example. Local, i.e. one step error (left plot) and global error at T = 0.5 (right plot) <strong>of</strong><br />
backward Euler method. O(ɛH) is observed in both plots.<br />
Then we consider the van der Pol equation [8] with the well-prepared initial data up to O(ε 3 )<br />
{ {<br />
y ′ = z<br />
y(0) = 2<br />
εz ′ = (1 − y 2 )z − y , z(0) = − 2 3 + 10<br />
81 ε − 292<br />
(3.8)<br />
2187 ε2<br />
• The numerical results <strong>of</strong> the IDC methods embedded with the first order stiffly accurate backward Euler<br />
method in both prediction and two correction steps are presented in the upper row <strong>of</strong> Figure 3.3. The<br />
order <strong>of</strong> convergence for ε 0 term would increase with the correction loops. The ε 1 term <strong>of</strong> error behaves<br />
like O(εH) for both y- z-components.<br />
• The numerical results <strong>of</strong> IDC methods embedded with the second order stiffly accurate DIRK method<br />
in both prediction and one correction step are presented in the middle row <strong>of</strong> Figure 3.3. The order <strong>of</strong><br />
convergence for ε 0 term would increase with second order with the correction loop. The ε 1 term <strong>of</strong> error<br />
behaves like O(εH) for both y- z-components.<br />
• The numerical results <strong>of</strong> IDC methods embedded with the third order stiffly accurate Radau IIA method<br />
in the prediction step and the first order backward Euler method in two correction steps are presented in<br />
Figure 3.2: Scalar example. Global error (T = 0.1) <strong>of</strong> the IDC-second order DIRK method that is not stiffly<br />
accurate with three quadrature points and one correction step. ε = 10 −4 .<br />
14
the bottown row <strong>of</strong> Figure 3.3. The order <strong>of</strong> convergence for ε 0 term would increase with first order with<br />
the correction loop and is observed to be O(H 5 ). The ε 1 term <strong>of</strong> error behaves like O(εH 2 ) for both y-<br />
z-components.<br />
Numerical observations in Figure 3.3 are consistent with Theorem 3.1 and 3.2 and Table 3.1. Especially,<br />
it is observed that the IDC method embedded with implicit R-K methods exhibits order reduction both in the<br />
differential and algebraic component. They produce an estimate for the y and z component <strong>of</strong> the following<br />
form<br />
e (k)<br />
n = y n − y (k) (t n ) = O(H s k<br />
) + εO(H q(0) ) + ε 2 O(H q(0) −1 ) + · · · ,<br />
d (k)<br />
(3.9)<br />
n = z n − z (k) (t n ) = O(H s k<br />
) + εO(H q(0) ) + ε 2 O(H q(0) −1 ) + · · · ,<br />
after k correction steps. For example, in Figure 3.3, we observe a behavior like e (k)<br />
n = O(H 3 ) + O(εH) + O(ε 2 )<br />
where the term O(ε 2 s<br />
) can be neglected since ε ≪ H. Furthermore, if the step size H > ε k −q (0) , O(H s k<br />
)<br />
is dominant; otherwise the term O(εH q(0) ) is observed. A singularity may appear in the neighborhood <strong>of</strong><br />
1<br />
s<br />
H ≈ ε k −q (0)<br />
where we have a cancellation <strong>of</strong> error terms between O(H s k<br />
) and εO(H q(0) ) with error constants<br />
<strong>of</strong> an opposite sign, see for example Figure 3.3.<br />
4 Pro<strong>of</strong>s <strong>of</strong> main results<br />
In this section, we prove Theorems 3.1 and 3.2. Theorem 3.1 is a special case for Theorem 3.2, yet we present<br />
the pro<strong>of</strong> for Theorem 3.1 first to demonstrate the basic ingredients <strong>of</strong> the pro<strong>of</strong>. The pro<strong>of</strong> is then generalized<br />
for Theorem 3.2. Our error estimate is based on the ε-expansion outlined in Section 2.4.<br />
4.1 <strong>Error</strong> estimates for Theorem 3.1.<br />
We perform local error estimate for Theorem 3.1 by four Lemmas. We again note that since h = H M<br />
, we use<br />
O(h p ) and O(H p ) interchangeably below in our pro<strong>of</strong>. We then prove the global error estimate in Proposition 4.8<br />
based on the four Lemmas.<br />
Lemma 4.1. (Prediction step, ε 0 ) Suppose that the reduce system (1.4) satisfies (1.3) and that the initial<br />
values are consistent. Consider the backward Euler method for the prediction step (2.33). Then the numerical<br />
solutions have the following local error estimate at each interior node <strong>of</strong> IDC t m with m = 0, ..., M<br />
and<br />
ŷ (0)<br />
m,0 − y m,0 = O(h 2 ), ẑ (0)<br />
m,0 − z m,0 = O(h 2 ),<br />
g(ŷ (0)<br />
m,0 , ẑ(0) m,0 ) = 0.<br />
Pro<strong>of</strong>. For the exact solution, we have (2.28), indicating that y 0 (t) and z 0 (t) lies on the manifold g(y 0 (t), z 0 (t)) =<br />
0 and z 0 (t) = G(y 0 (t)). For the numerical solution, we have (2.33), indicating a similar behavior <strong>of</strong> the numerical<br />
solution<br />
0 = g(ŷ (0)<br />
m,0 , ẑ(0) m,0 ), (4.1)<br />
with m = 0, ..., M. Now by (4.1), we get ẑ (0)<br />
m,0 = G(ŷ(0) m,0 ). This implies that ŷ(0) m,0 represents the numerical<br />
solution <strong>of</strong> the ordinary differential equation y 0(t) ′ = ˆf(y 0 (t)) with ˆf = . f(y 0 (t), G(y 0 (t))). Then from the<br />
backward Euler method we have for the local truncation error |ŷ (0)<br />
m,0 − y m,0| ≤ C m h 2 with m = 0, ..., M and for<br />
some constant C m independent <strong>of</strong> H. Therefore, ŷ (0)<br />
m,0 − y m,0 = O(h 2 ). By ẑ (0)<br />
m,0 = G(ŷ(0) m,0 ) and the Lipschitz<br />
condition <strong>of</strong> G, it follows that ẑ (0)<br />
m,0 − z m,0 = O(h 2 ) with m = 0, ..., M.<br />
Lemma 4.2. (<strong>Correction</strong> steps: ε 0 ) Under the same assumptions <strong>of</strong> Lemma 4.1 we consider the backward Euler<br />
method for the correction steps (2.9)-(2.11). Assume that the numerical solutions after k − 1 correction loops<br />
have the local error estimate with m = 0, ..., M<br />
e (k−1)<br />
m,0 = y m,0 − ŷ (k−1)<br />
m,0 = O(h min(k+1,M+1) ), d (k−1)<br />
m,0 = z m,0 − ẑ (k−1)<br />
m,0 = O(h min(k+1,M+1) ), (4.2)<br />
1<br />
15
,<br />
,<br />
,<br />
Figure 3.3: Van der Pol equation. Global error (T = 0.5) <strong>of</strong> the IDC-BE method with M = 3 quadrature<br />
points and two correction steps (upper row); and <strong>of</strong> the IDC method with stiffly accurate DIRK with M = 4<br />
quadrature points and one correction step (middle row); and <strong>of</strong> the IDC method with the third order stiffly<br />
accurate Radau IIA method for prediction and first order backward Euler for two correction steps and with<br />
M = 6 quadrature points (bottom row). ε = 10 −6 .<br />
16
and<br />
g(ŷ (k−1)<br />
m,0 , ẑ (k−1)<br />
m,0 ) = 0.<br />
Then the numerical solutions after k correction loops have the local error estimate at the interior nodes <strong>of</strong> IDC<br />
with m = 0, ..., M<br />
and<br />
e (k)<br />
m,0 = y m,0 − ŷ (k)<br />
m,0 = O(hmin(k+2,M+1) ), d (k)<br />
m,0 = z m,0 − ẑ (k)<br />
m,0 = O(hmin(k+2,M+1) ), (4.3)<br />
g(ŷ (k)<br />
m,0 , ẑ(k) m,0 ) = 0. (4.4)<br />
Pro<strong>of</strong>. Without loss <strong>of</strong> generosity and for simplicity, we let k = 1 and assume a fixed M ≥ 1. Consider equation<br />
(2.36) with k = 1 be the numerical scheme for the first correction. From the prediction step (2.33) we have<br />
g(ŷ (0)<br />
m,0 , ẑ(0) m,0 ) = 0. From (2.36), we have g(ŷ(1) m,0 , ẑ(1) m,0 ) = 0, with m = 0, ..., M, i.e. equation (4.4) with k = 1.<br />
From the invertibility condition <strong>of</strong> the function g z in equation (1.3), we get<br />
ŷ (1)<br />
m+1,0 = ŷ(1)<br />
(1)<br />
(0)<br />
m,0 + h( ˆf(ŷ m+1,0 ) − ˆf(ŷ m+1,0 )) + hSm ( ¯ˆf (0)<br />
0 ), (4.5)<br />
(1)<br />
where ˆf(ŷ m+1,0 ) = f(ŷ(1) m+1,0 , G(ŷ(1) m+1,0 )), ẑ(1) m,0 = G(y(1) m,0 ), and Sm ( ¯ˆf (0)<br />
0 ) = Sm (0) (0)<br />
( ˆf(¯ŷ 0 , G(¯ŷ 0 ))). The scheme<br />
(4.5) <strong>of</strong> updating ŷ (1)<br />
m,0 can be interpreted as the the applying a correction step in the IDC framework to the<br />
non-stiff ordinary differential equation (1.6). Therefore applying classical results as [17] <strong>of</strong> IDC frameworks using<br />
backward Euler method applied to a classical ordinary differential equation we can have for the local truncation<br />
error after one correction |y m,0 − ŷ (1)<br />
m,0 | ≤ C mh 3 for some constant C m independent <strong>of</strong> H and with h ≤ h 0 .<br />
Therefore y m,0 − ŷ (1)<br />
m,0 = O(h3 ). By ẑ (1)<br />
m,0 = G(ŷ(1) m,0 ) and Lipschitz condition <strong>of</strong> G, we get z m,0 − ẑ (1)<br />
m,0 = O(h3 ),<br />
∀m = 1, · · · M. The estimate for general k > 1 can be proved by mathematical induction in a similar fashion.<br />
Lemma 4.3. (Prediction step, ε 1 ) Assume the condition (1.3) holds and initial values <strong>of</strong> the differential algebraic<br />
system (2.28) -(2.29) are consistent, then the local error estimate at the interior nodes <strong>of</strong> IDC <strong>of</strong> the backward<br />
Euler method (2.33)-(2.35) for the prediction step, holds<br />
ŷ (0)<br />
m,1 − y m,1 = O(h 2 ), ẑ (0)<br />
m,1 − z m,1 = O(h). (4.6)<br />
Pro<strong>of</strong>. The pro<strong>of</strong> is a special case <strong>of</strong> Theorem 3.4 in Chap. VI with ν = 1 and Lemma 4.4 in Chap. VII <strong>of</strong> [8].<br />
Lemma 4.4. (<strong>Correction</strong> steps, ε 1 ) Under the same assumptions <strong>of</strong> Lemma 4.3 we consider the backward Euler<br />
method for the correction steps (2.9)-(2.11). Assume that the numerical solutions at the interior nodes <strong>of</strong> IDC<br />
after k correction loops have the local error estimate (4.2) for ε 0 term. Assume after k − 1 correction loops have<br />
the local error estimate<br />
e (k−1)<br />
m,1 = y m,1 − ŷ (k−1)<br />
m,1 = O(h 2 ), d (k−1)<br />
m,1 = z m,1 − ẑ (k−1)<br />
m,1 = O(h), (4.7)<br />
holds for m = 1, · · · M. Then the numerical solutions after k correction loops have the local error at the interior<br />
nodes <strong>of</strong> IDC<br />
e (k)<br />
m,1 = y m,1 − ŷ (k)<br />
m,1 = O(h2 ), d (k)<br />
m,1 = z m,1 − ẑ (k)<br />
m,1 = O(h), (4.8)<br />
holds for m = 1, · · · M.<br />
Pro<strong>of</strong>. Without loss <strong>of</strong> generosity and for simplicity, we let k = 1 and assume M ≥ 1 fixed. We prove (4.8) by<br />
mathematical induction w.r.t. m. Especially, we know e (k)<br />
m,1 = d(k) m,1 = 0, with m = 0. We assume (4.8) is valid<br />
for 0, · · · , m. We will prove that (4.8) is valid for m + 1. We consider the ε-expansion <strong>of</strong> the exact solution.<br />
Integration (2.29) over [t m , t m+1 ] gives<br />
{<br />
ε 1 y m+1,1 = y m,1 + ∫ t m+1<br />
t<br />
:<br />
m<br />
F 1 (τ)dτ,<br />
z m+1,0 = z m,0 + ∫ t m+1<br />
(4.9)<br />
t m<br />
G 1 (τ)dτ,<br />
with F 1 and G 1 defined in (2.29). We consider now<br />
e (1)<br />
m+1,1 = y m+1,1 − ŷ (1)<br />
m+1,1 ,<br />
d(1) m+1,1 = z m+1,1 − ẑ (1)<br />
m+1,1 (4.10)<br />
17
the differences with the exact solutions and numerical ones.<br />
From (2.39) and (2.41) we have<br />
⎧<br />
⎨<br />
Subtract equation (2.37) from equation (4.9) gives<br />
ε 1 :<br />
⎧<br />
⎨<br />
⎩<br />
⎩<br />
∆ˆF (0)<br />
m+1,1 = ( f y ê (0)<br />
m+1,1 + f (0)<br />
z ˆd m+1,1 ) + O(h2 )<br />
∆Ĝ(0) m+1,1 = ( g y ê (0)<br />
m+1,1 + g (0)<br />
z ˆd m+1,1 ) + O(h2 ).<br />
e (1)<br />
m+1,1 = e(1)<br />
(0)<br />
m,1 − h∆ˆF m+1,1 − hSm (¯ˆF(0) 1 ) + ∫ t m+1<br />
t m<br />
F 1 (τ)dτ<br />
d (1)<br />
m+1,0 = d(1) m,0 − h∆Ĝ(0) m+1,1 − hSm (0)<br />
( ¯Ĝ 1 ) + ∫ t m+1<br />
t m<br />
G 1 (τ)dτ.<br />
(4.11)<br />
(4.12)<br />
On the right-hand side <strong>of</strong> the equations in (4.12) we add and subtract the following quantities: hS m (¯F 1 )<br />
and hS m (Ḡ1), these are the integrals <strong>of</strong> the M-th degree interpolating polynomials on (t m , F 1 (t m )) M m=1 and<br />
(t m , G 1 (t m )) M m=1 over the subinterval [t m , t m+1 ], hence they are accurate to the order O(h M+1 ), i.e. ∫ t m+1<br />
t m<br />
F 1 (τ)dτ−<br />
hS m (¯F 1 ) = O(h M+1 ). By the assumption <strong>of</strong> local error estimate <strong>of</strong> (4.2) and (4.7), S m (¯F 1 ) − S m (¯ˆF1 ) and<br />
S m (Ḡ1) − S m ( ¯Ĝ 1 ) are accurate to the order O(h). Thus,<br />
⎧<br />
(<br />
)<br />
⎪⎨ e (1)<br />
m+1,1 = e (1)<br />
m,1 − h f y ê (0)<br />
m+1,1 + f (0)<br />
z ˆd m+1,1 + O(h 2 ),<br />
(<br />
)<br />
(4.13)<br />
⎪⎩ d (1)<br />
m+1,0 = d (1)<br />
m,0 − h g y ê (0)<br />
m+1,1 + g (0)<br />
z ˆd m+1,1 + O(h 2 )<br />
From (2.53) and (4.2), we have<br />
⎧<br />
⎨<br />
⎩<br />
ê (0)<br />
m,1 = ŷ(1) m,1 − ŷ(0) m,1 = e(0) m,1 − e(1) m,1 = −e(1) m,1 + O(h2 ),<br />
ˆd (0)<br />
m,1 = ẑ(1) m,1 − ẑ(0) m,1 = d(0) m,1 − d(1) m,1 = −d(1) m,1 + O(h),<br />
and put it into equation (4.13) gives,<br />
⎧<br />
(<br />
)<br />
⎪⎨ e (1)<br />
m+1,1 = e(1) m,1 + h f y e (1)<br />
m+1,1 + f zd (1)<br />
m+1,1 + O(h 2 ),<br />
(<br />
)<br />
⎪⎩ d (1)<br />
m+1,0 = d(1) m,0 + h g y e (1)<br />
m+1,1 + g zd (1)<br />
m+1,1 + O(h 2 )<br />
(4.14)<br />
Now using the estimate (4.2) about d (1)<br />
m,0 , by the second equation in (4.14) we obtain<br />
d (1)<br />
m+1,1 = −g−1 z g y e (1)<br />
m+1,1 + O(h) (4.15)<br />
with the invertibility <strong>of</strong> g z . Inserting this into the first equation in (4.14) gives<br />
e (1)<br />
m+1,1 = (1 − h(f y − f z g −1<br />
z g y )) −1 e (1)<br />
m,1 + O(h2 ) (4.16)<br />
Finally e (1)<br />
m+1,1 = O(h2 ) follows from (4.16), and d (1)<br />
m+1,1 = O(h) follows from (4.15). We note that the pro<strong>of</strong> <strong>of</strong><br />
the general k is similar.<br />
Remark 4.5. In [4], the IDC with explicit R-K in the prediction and correction loops is incorporated as<br />
a high-order explicit R-K method. Similarly, the IDC-BE can be viewed as an implicit R-K method with<br />
a corresponding Butcher tableau. Below, we present the Butcher tableau for the IDC-BE with one loop <strong>of</strong><br />
correction. The Butcher tableau can be generated by a similar fashion if there are more than one correction<br />
loops. The Butcher tableau takes the form<br />
⃗c T Z<br />
⃗c P T<br />
⃗ b<br />
T<br />
1<br />
⃗ b<br />
T<br />
2<br />
(4.17)<br />
18
where ⃗c = 1 M [1, · · · , M]T , Z is a M × M matrix <strong>of</strong> zeros, T and P are M × M matrices, with<br />
⎡<br />
⎤<br />
1 0 0 . . . 0<br />
T = 1 1 1 0 . . . 0<br />
⎢<br />
M ⎣<br />
.<br />
. . ..<br />
⎥<br />
. . ⎦ .<br />
1 1 1 . . . 1<br />
⎡<br />
P = ⎢<br />
⎣<br />
( ˜S 11 − 1 M ) ⎤<br />
˜S12 . . . ˜S1,M−1 ˜S1,M<br />
( ˜S 21 − 1 M ) ( ˜S 22 − 1 M ) . . . ˜S2,M−1 ˜S2,M<br />
. ⎥<br />
.<br />
. .. .<br />
. ⎦ ,<br />
( ˜S M,1 − 1 M ) ( ˜S M,2 − 1 M ) . . . ( ˜S M,M−1 − 1 M ) ( ˜S M,M − 1 M )<br />
where the term ˜S ij = ∫ t i<br />
t 0<br />
α j (s)ds with α j (s) as defined in equation (2.12). The vectors<br />
⃗ b<br />
T<br />
1 =<br />
((<br />
˜S M,1 − 1 M<br />
) (<br />
, ˜S M,2 − 1 ) (<br />
, · · · , ˜S M,M − 1 ))<br />
M<br />
M ) , ⃗ b<br />
T<br />
2 = 1 (1, 1, · · · , 1).<br />
M<br />
Now by the Butcher table constructed, the following Proposition follows.<br />
Proposition 4.6. The IDC-BE is an implicit stiffly accurate RK method with the matrix<br />
( ) T Z<br />
A =<br />
P T<br />
in (1.10) invertible.<br />
Remark 4.7. In the estimates above, we show that there is no improvement in the order <strong>of</strong> convergence for<br />
approximating y 1 and z 1 in IDC corrections. This is consistent with our numerical evidence presented in the<br />
previous section. The reason is that both the local and global error for ẑ (k)<br />
1 approximating z 1 in the prediction<br />
and correction steps is <strong>of</strong> first order. This sets the bottleneck for order increase for the term O(h 2 ) in equation<br />
(4.13).<br />
The Proposition below brings the local estimate from four Lemmas above to a global error estimate for terms<br />
in equation (3.3).<br />
Proposition 4.8. Under the same assumption <strong>of</strong> Theorem 3.1, then for any fixed constant c > 0, the following<br />
global error estimate for equation (3.3) holds with<br />
e (K)<br />
n,0 = O(Hmin(k+1,M) ), d (K)<br />
n,0 = O(Hmin(k+1,M) )<br />
e (K)<br />
n,1<br />
= O(H), d(K) n,1 = O(H),<br />
for ε ≤ cH and ν ≤ 2. The estimates hold uniformly for H ≤ H 0 and nH ≤ Const.<br />
Pro<strong>of</strong>. From Lemma 4.2, we have g(ŷ (k)<br />
m,0 , ẑ(k) m,0 ) = 0, ∀m. Hence, the IDC-BE method is a state space form<br />
method (for Definition see Chap. VI in [8]). In fact, from the above Proposition, it is also a stiffly accurate<br />
method. By the mathematical induction with respect to k, Lemma 4.1 and 4.2 give the local estimate<br />
y 0 (t 1 ) − ŷ (k)<br />
M,0 = O(Hmin(k+2,M+1) ). (4.18)<br />
With the help <strong>of</strong> Theorem 3.4 in Chap. II [7] in estimating global error from local error, we have the global<br />
error estimate<br />
e (K)<br />
n,0 = y 0(nH) − ŷ (k)<br />
n,0 = O(Hmin(k+1,M) ).<br />
As IDC-BE method is a stiffly accurate method, z = G(y) for both exact solution and numerical solution. By<br />
the Lipschitz condition <strong>of</strong> G, we have the global error estimate<br />
d (K)<br />
n,0 = z 0(nH) − ẑ (k)<br />
n,0 = O(Hmin(k+1,M) )<br />
19
Lemma 4.4 gives the local estimate<br />
y 1 (t 1 ) − ŷ (k)<br />
M,1 = O(H2 ). (4.19)<br />
From the Remark 1.4 for stiffly accurate method, and with the help <strong>of</strong> Theorem 4.5, 4.6 in Chap. VII.4 <strong>of</strong> [8],<br />
we have the global error estimate for y and z<br />
e (K)<br />
n,1 = y 1(nH) − ŷ (k)<br />
n,1 = O(H), d(K) n,1 = z 1(nH) − ẑ (k)<br />
n,1 = O(H).<br />
4.2 <strong>Error</strong> estimates for Theorem 3.2<br />
In this subsection, we extend the above result to the general case <strong>of</strong> using implicit R-K methods in the IDC<br />
framework. We remark that the crucial assumption in Theorem 3.2 is that the implicit RK method is stiffly<br />
accurate. In the case that this property is not satisfied, the method becomes unstable and the numerical solutions<br />
diverge, see Figure 3.2. In order to justify this, from the invertibility <strong>of</strong> matrix A and by the first formula <strong>of</strong><br />
(2.47) we get<br />
(k)<br />
∆ ˆL m,−1 = −A−1 S −→c (¯ĝ (k) ), (4.20)<br />
substituting which into the second formula <strong>of</strong> (2.47) yields<br />
−b T A −1 S¯c (¯ĝ (k−1) ) + S m (¯ĝ (k−1) ) = 0. (4.21)<br />
Proposition 4.9. Equation (4.21) is automatically satisfied, if the implicit RK method is stiffly accurate in the<br />
IDC framework.<br />
Pro<strong>of</strong>. An implicit RK method is stiffly accurate if<br />
with e s = (0, · · · , 0, 1) T . From (4.21) we get<br />
b T A −1 = e T s , (4.22)<br />
−e T s S¯c (¯ĝ (k−1) ) + S m (¯ĝ (k−1) ) = 0. (4.23)<br />
Since the last row <strong>of</strong> the spectral integration matrix is s m,k = ∫ t m+c sh<br />
α k (τ)dτ by (4.22) we get c s = 1 and then<br />
∫ tm+c sh<br />
t m<br />
α k (τ)dτ = ∫ t m+1<br />
t m<br />
α k (τ)dτ. This yields that e T s S¯c (¯ĝ (k−1) ) = S m (¯ĝ (k−1) ) and then the equation (4.23) is<br />
satisfied.<br />
In fact, similar to Prop. 4.6, we have the following Proposition for the IDC embedded with stiffly accurate<br />
implicit R-K methods. The pro<strong>of</strong> is omitted for brevity.<br />
Proposition 4.10. The IDC method embedded with stiffly accurate implicit RK methods is an implicit stiffly<br />
accurate R-K method with a corresponding Butcher Tableau that has the matrix A in (1.10) invertible.<br />
We prove the error estimate in Theorem 3.2 by two Lemmas. Lemma 4.11 is about the local truncation<br />
error estimate in the case ε 0 , i.e., R-K methods applied to the reduced system (1.4) in the correction steps.<br />
Lemma 4.14 is about the local truncation error estimate in the case ε ν with ν ≥ 1.<br />
Lemma 4.11. (The case <strong>of</strong> ε 0 .) Consider the same assumptions as in Theorem 3.2. Consider the limiting case<br />
<strong>of</strong> ε = 0. The numerical solutions after k correction loops have the local error estimate at the interior nodes <strong>of</strong><br />
IDC with m = 0, · · · M<br />
t m<br />
e (k)<br />
m,0 = O(hmin(s k+1,M+1) ),<br />
d (k)<br />
m,0 = O(hmin(s k+1,M+1) ).<br />
(4.24)<br />
Pro<strong>of</strong>. As done in Lemma 4.4, without loss <strong>of</strong> generosity and for simplicity, we let k = 1 and assume M ≥ 1.<br />
The pro<strong>of</strong> for the general k can be proved by mathematical induction. We will omit the superscript (k) for<br />
different R-K methods when there is no confusion.<br />
Since the R-K method for the prediction is stiffly accurate, by Remark 1.2, we have R(∞) = 0 and b T A −1 =<br />
e T s . This implies by (1.17) and (1.16) ẑ (0)<br />
m+1,0 = Ẑ(0) ms,0<br />
for all i and, in particular, Ẑ(0) ms,0<br />
= G(Ŷ<br />
(0)<br />
ms,0<br />
and ŷ(0) m+1,0 = Ŷ (0)<br />
ms,0<br />
). Then this gives ẑ(0)<br />
20<br />
m+1,0 = G(ŷ(0) m+1,0 ).<br />
. By (1.15) we get Ẑ(0)<br />
mi,0<br />
= G(Ŷ<br />
(0)<br />
mi,0 )
Now for the correction step k = 1, by the stiffly accurate property <strong>of</strong> the R-K method applied in the first<br />
(0)<br />
correction step and ¯ĝ 0 = (g(ŷ (0)<br />
1,0 , ẑ(0) 1,0 ), · · · g(ŷ(0)<br />
M,0 , ẑ(0) M,0 )) = ⃗0 from the prediction step, it follows<br />
{<br />
ŷ (1)<br />
m+1,0 = ŷ(1) m,0 + hSm ( ¯ˆf (0)<br />
0 ) + h ∑ s<br />
i=1 b (0)<br />
i∆ ˆK mi,0<br />
g(ŷ (1)<br />
m+1,0 , ẑ(1) m+1,0 ) = 0 (4.25)<br />
The internal stages are given by<br />
{<br />
(1) Ŷ mi,0 = ŷ(1) m,0 + hScmi ( ¯ˆf (0)<br />
0 ) + h ∑ i<br />
j=1 a ij∆<br />
ˆK<br />
(0)<br />
mj,0<br />
g(Ŷ (1)<br />
mi,0 , Ẑ(1) mi,0 ) = 0 (4.26)<br />
Now, from the invertibility <strong>of</strong> function g z , by (4.26) we get Ẑ(1) mi,0<br />
R-K method reads<br />
Ŷ (1)<br />
mi,0 = ŷ(1) m,0 + h ∑ s<br />
j=1 a (0)<br />
ij∆ ˆK mj,0 + hScmi ( ¯ˆf (0)<br />
ŷ (1)<br />
m+1,0 = ŷ(1) m,0 + h ∑ s<br />
i=1 b i∆<br />
(1)<br />
= G(Ŷ mi,0 ) and ẑ(1) m+1,0 = G(ŷ(1) m+1,0 ). Thus the<br />
0 )<br />
ˆK<br />
(0)<br />
mi,0 + hSm ( ¯ˆf (0)<br />
0 ) (4.27)<br />
and ¯ˆf (0)<br />
0 = (f(ŷ (0)<br />
0,0 , G(ŷ(0) 0,0 )), · · · f(ŷ(0)<br />
M,0 , G(ŷ(0) M,0<br />
)). The scheme (4.27) <strong>of</strong> updating ŷ(1) m,0 can be interpreted as the<br />
applying a correction step in the IDC framework to the ordinary differential equation (1.6). Therefore applying<br />
classical results <strong>of</strong> local truncation error as in [4, 5] <strong>of</strong> IDC frameworks using R-K method applied to a classical<br />
ordinary differential equation, we obtain the local error estimate for m = 0, · · · M<br />
with s 2 = p (0) + p (1) , and<br />
e (1)<br />
m,0 = O(hmin(s2+1,M+1) ), (4.28)<br />
E (1)<br />
m,0 = y 0(t m + c i h) − Ŷ (1)<br />
mi,0 = O(hmin(s1+q(1) +1,M+1) ). (4.29)<br />
By ẑ (1)<br />
m,0 = G(ŷ(1)<br />
(1)<br />
m,0 ), Ẑ(1) mi,0 = G(Ŷ mi,0 ) and using the Lipschitz condition <strong>of</strong> G, we get<br />
and<br />
d (1)<br />
m,0 = z m,0 − ẑ (1)<br />
m,0 = O(hmin(s2+1,M+1) ).<br />
D (1)<br />
mi,0 = z 0(t m + c i h) − Ẑ(1) mi,0 = O(hmin(s1+q(1) +1,M+1) ),<br />
where q (1) is the stage order for the R-K method applied to the first correction loop. We note that the pro<strong>of</strong> <strong>of</strong><br />
the general k is similar.<br />
Remark 4.12. The estimate (4.28) is from [5] via estimating the smoothness <strong>of</strong> the rescaled error function.<br />
The estimate (4.29) follows a similar fashion.<br />
Remark 4.13. With the estimates in the above Lemma, and from equation (2.44)<br />
where ∆K (k−1)<br />
mi,1<br />
(k−1)<br />
∆ ˆK mi,1 = f y (y mi,0 , z mi,0 )Ê(k−1) mi,1<br />
+ f (k−1)<br />
z(y mi,0 , z mi,0 ) ˆD mi,1 + O(hs k−1+1 ).<br />
.<br />
= ∆K (k−1)<br />
mi,1 + O(hs k−1+1 ) (4.30)<br />
.<br />
= f y (y mi,0 , z mi,0 )Ê(k−1) mi,1 + f z (y mi,0 , z mi,0 )<br />
(k−1) ˆD mi,1<br />
(k)<br />
. Here, we replace Ŷ mi,0 and P cmi (¯ŷ (k−1)<br />
0 ) by<br />
y mi,0 with an error <strong>of</strong> O(h s k−1+q (k) +1 ) and O(h s k−1+1 ) at the position t = t m + c i h respectively.<br />
Similarly,<br />
where ∆L (k−1)<br />
mi,0<br />
(k−1)<br />
∆ ˆL mi,0 = g y (y mi,0 , z mi,0 )Ê(k−1) mi,1<br />
+ g (k−1)<br />
z(y mi,0 , z mi,0 ) ˆD mi,1 + O(hs k−1+1 ).<br />
.<br />
= ∆L (k−1)<br />
mi,0 + O(hs k−1+1 ), (4.31)<br />
.<br />
= g y (y mi,0 , z mi,0 )Ê(k−1) mi,1 + g (k−1)<br />
z(y mi,0 , z mi,0 ) ˆD mi,1 .<br />
21
Lemma 4.14. (The case <strong>of</strong> ε ν (ν ≥ 1).) Consider the same assumptions as in Theorem 3.2 with 0 < ε
In a similar fashion as in equations (4.39), written the internal stages in a vectorial form, we have<br />
where Ē(0) 1 = (E (0)<br />
m1,1 , · · · , E(0) ms,1<br />
Ē (0)<br />
1 = e (1)<br />
(0)<br />
m,11 − hA∆ ¯K 1 + O(h q(0) +1 )<br />
¯D (0)<br />
0 = d (1)<br />
(0)<br />
m,01 − hA∆ ¯L 0 + O(h q(0) +1 )<br />
(4.42)<br />
(0)<br />
), ¯D 0 = (D (0)<br />
m1,0 , · · · , D(0) ms,0 ), where s is the number <strong>of</strong> internal stages in a R-K<br />
method and 1 = (1, 1, · · · , 1) T is a vector <strong>of</strong> size s. Then from the second equation in (4.42) and using (4.24),<br />
we get<br />
A(g y (t m + c i h)E (1)<br />
mi,1 + g z(t m + c i h)D (1)<br />
mi,1 ) = O(hq(0) ) (4.43)<br />
where we replace Ê(0) mi,1 by E(1) mi,1 with +2 O(hq(0) ) error, and replace<br />
(4.41). Thus, from the invertibility <strong>of</strong> A<br />
(0) ˆD mi,1 by D(1) mi,1 with O(hq(0) ) error due to<br />
D (1)<br />
mi,1 = −(g−1 z g y )(t m + c i h)E (1)<br />
mi,1 + O(hq(0) ), (4.44)<br />
for all mi. Plug the above equation (4.44) into the first equation <strong>of</strong> (4.42)<br />
∆K (0)<br />
mi,1 = (f y − f z g −1<br />
z g y )(t m + c i h)E (1)<br />
mi,1 + O(hq(0) ). (4.45)<br />
Next, we prove the local error e (1)<br />
m,1 = +1 O(hq(0) ) by mathematical induction w.r.t. m. Especially, we would like<br />
to show that e (1)<br />
m+1,1 = +1 O(hq(0) ), if we assume the local error e (1)<br />
l,1 = +1 O(hq(0) ), ∀l ≤ m. To show this, we<br />
plug equation (4.45) into the first equation <strong>of</strong> the vectorial form (4.42) and obtain E (1)<br />
mi,1 = +1 ). Thus,<br />
O(hq(0)<br />
from (4.44),<br />
D (1)<br />
mi,1 = ). (4.46)<br />
O(hq(0)<br />
From (4.45), ∆K (0)<br />
mi,1 = O(hq(0) ). Plug this estimate into the first equation <strong>of</strong> (4.39), we obtain the desired<br />
estimate <strong>of</strong><br />
e (1)<br />
m+1,1 = O(hq(0) +1 ). (4.47)<br />
Now in order to prove the estimate d (1)<br />
m,1 = ), we start to considering equation (2.25). Since the R-K<br />
O(hq(0)<br />
method is stiffly accurate, from Remark 2.5, we have ẑ (1)<br />
m+1,1 = Ẑ(1) ms,1 . Hence<br />
z 1 (t m+1 ) − ẑ (1)<br />
m+1,1 = d(1) m+1 = D(1)<br />
(4.46)<br />
ms,1 = O(h q(0) ), m = 0, · · · M − 1 (4.48)<br />
The above pro<strong>of</strong> can be generalized for the IDC method with different RK methods applied to k correction<br />
loops. The local error estimate at the interior nodes <strong>of</strong> the IDC method with m = 0, · · · M is<br />
e (k)<br />
m,1 = O(hq(0) +1 ), d (k)<br />
m,1 = O(hq(0) ).<br />
We have thus proved the case ν = 1. The general estimates for ν > 1 (4.32) can be obtained in a similar<br />
fashion to the case <strong>of</strong> ν = 1, as in the Theorem 3.4 in Chap.VI <strong>of</strong> [8], then we have for the local errors<br />
e (k)<br />
m,ν = O(h q(0) +2−ν ), d (k)<br />
m,ν = O(h q(0) +1−ν ).<br />
Remark 4.15. We remark that we can not improve the estimate <strong>of</strong> the global error for the y-component as<br />
done in Theorem 3.4 in [8] for high-indices. Indeed the reason <strong>of</strong> this lost <strong>of</strong> accuracy for the y-component is<br />
represented <strong>of</strong> the evaluation <strong>of</strong> the integrals by (2.50). These integrals contain the estimate <strong>of</strong> the algebraic<br />
variable z obtained in the prediction step that reduces the order <strong>of</strong> the differential variable y, then a definition<br />
<strong>of</strong> a new variable as done in Theorem 3.4 in [8] can not produce any benefit to the y-component. This can be<br />
seen in the evaluation from (4.37) to (4.39) due to (4.38). We note that a similar conclusion for the remainder<br />
can be drawn.<br />
Similar to Proposition 4.8, we have the following Proposition for global error estimates <strong>of</strong> IDC implicit R-K<br />
method. The pro<strong>of</strong> follows from Lemma 4.11 and 4.14, in a similar spirit to that <strong>of</strong> Proposition 4.8. We omit<br />
the pro<strong>of</strong> for brevity.<br />
Proposition 4.16. Under the same assumption <strong>of</strong> Theorem 3.2, then for any fixed constant c, the following<br />
global error estimate for equation (3.3) holds with<br />
e (K)<br />
n,0 = O(Hmin(s k,M) ), d (K)<br />
n,0 = O(Hmin(s k,M) )<br />
e (K)<br />
n,ν = O(H q(0) +1−ν ), d (K)<br />
n,1 = O(Hq(0) +1−ν ),<br />
for ε ≤ cH and 1 ≤ ν ≤ q (0) + 1. The estimates hold uniformly for H ≤ H 0 and nH ≤ Const.<br />
23
4.3 Pro<strong>of</strong> <strong>of</strong> main Theorems and estimation <strong>of</strong> the remainder<br />
The pro<strong>of</strong> <strong>of</strong> Theorem 3.1 and 3.2 follow directly from Prop. 4.8, 4.16 and the estimate about the remainder in<br />
Prop. 4.17 below. We justify that the term O(ε ν+1 /H) representing the remainder in the expansion (3.3).<br />
Proposition 4.17. Under the same hypothesis as those <strong>of</strong> theorem 3.2 for any fixed constant c > 0, the global<br />
error satisfies for ε ≤ cH<br />
q (0) ∑+1<br />
e (k)<br />
n = e (k)<br />
n,ν + O(ε ν+1 /H),<br />
ν=0<br />
q (0) ∑+1<br />
d (k)<br />
n = d (k)<br />
n,ν + O(ε ν+1 /H). (4.49)<br />
ν=0<br />
Here e (k)<br />
n,ν = y n,ν − ŷ n,ν, (k) d (k)<br />
n,ν = z n,ν − ẑ n,ν (k) are the global errors <strong>of</strong> the IDC method applied to (2.28), (2.29) and<br />
(2.31). The estimates (4.49) hold uniformly for H ≤ H 0 and nH ≤ Const.<br />
Pro<strong>of</strong>. In order to prove the theorem we consider the truncated series for e (k)<br />
n<br />
∑<br />
q (0) +1<br />
e (k)<br />
n = e (k)<br />
n,ν,<br />
ν=0<br />
Using (2.51), then the statement (4.49) is equivalent to prove<br />
e (k)<br />
n<br />
∑<br />
q (0) +1<br />
d (k)<br />
n =<br />
ν=0<br />
and d (k)<br />
n<br />
as<br />
d (k)<br />
n,ν (4.50)<br />
− e (k)<br />
n = O(εν+1 /H), d (k)<br />
n − d (k)<br />
n = O(εν+1 /H). (4.51)<br />
In this situation, the same conclusions <strong>of</strong> Theorem 3.8 in Chap. VI <strong>of</strong> [8] hold. The use <strong>of</strong> stiffly accurate implicit<br />
<strong>Runge</strong>-<strong>Kutta</strong> methods with matrix A invertible from Prop. 4.10 guarantees the hypothesis <strong>of</strong> Theorem 3.6 in<br />
Chap. I <strong>of</strong> [8]. It is worth commenting that the estimate for the y component in (4.51) is not optimal as in<br />
Theorem 3.8 in [8]. We obtain this estimate by considering remark 4.15, and by the estimates in Prop. 4.16.<br />
5 Conclusion<br />
Global errors are studied for an IDC framework with uniform distribution <strong>of</strong> quadrature points excluding the<br />
leftmost point, embedded with high order implicit R-K method for a class <strong>of</strong> singular perturbation problems.<br />
Two Theorems on the estimate <strong>of</strong> global error in the form <strong>of</strong> an ɛ expansion are presented and proved. The<br />
asymptotic analysis enables us to understand the phenomenon <strong>of</strong> order reduction for IDC methods when applied<br />
to stiff problems. Numerical results on van der Pol equations are presented to reveal the convergence results.<br />
In the future, our goal is to extend a similar analysis in order to study the global error and stability property<br />
<strong>of</strong> IDC framework embedded with high order implicit-explicit (IMEX) R-K methods.<br />
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