Error Estimate of Integral Deferred Correction Implicit Runge-Kutta ...

Error Estimate of Integral Deferred Correction Implicit Runge-Kutta method for Stiff Problems
Sebastiano Boscarino 1 , Jing-Mei Qiu 2
Abstract. In this paper, we present an error estimate of integral deferred correction (IDC) method constructed
with stiffly accurate implicit Runge-Kutta (R-K) method for singular perturbation problems containing a stiff
parameter ε. We focus our analysis on the IDC method using uniform distribution of quadrature nodes, but
excluding the left-most endpoint. The uniform distribution of nodes is important for high order accuracy
increase in correction loops [5], where as the use of quadrature nodes excluding the left-most endpoint lead to
an important stability condition for stiff problem, i.e. the method becomes L-stable if A-stable. In our error
estimate, we expand the global error in powers of ε and show convergence results for these error terms as well
as the remainder. Specifically, the order of convergence for the first term in global error (index 1) increase with
high order if a high order R-K method is applied in the IDC correction step; the order of convergence for the
second term (index 2) is determined by the stage order of the R-K method for the IDC prediction. Numerical
results for the stiff van der Pol equation are demonstrated to verify our error estimate.
Keywords: singular perturbation problem, Runge-Kutta method, integral deferred correction.
1 Introduction
Deferred correction (DC) methods for solving initial value problems
y ′ (t) = f(t, y(t)), y(y 0 ) = y 0 ∈ R N (1.1)
were investigated quite intensively, [2, 15, 1]. An advantage of DC methods is that one can use a simple numerical
method, for instance a first order method, to compute a solution with higher order accuracy accomplished by
using a numerical method to solve a series of correction equations during each time step. Each of the iteration
increases the order of accuracy of the solution. In [6], a new variation of the classical method of deferred correction
methods called spectral deferred method (SDC) was proposed. Essentially in SDC, the original differential
equation (1.1) is replaced with the corresponding Picard integral equation and a deferred correction procedure
is applied to an integral formulation of the error equation. Because the numerical integration have much better
stability and accuracy property than numerical differential, the SDC has been showed to outperformed DC in
many problems with promising numerical results [6]. In [6], the quadrature nodes in the proposed SDC methods
are chosen to be Gauss-Lobatto, Gauss-Radau or Gauss-Legendre points for high order of accuracy. When the
quadrature nodes are uniform, the SDC method is called integral deferred correction (IDC) method. There
are various SDC/IDC methods with different implementation strategies, e.g. in selecting time integrators in
prediction and correction steps [13, 12, 10, 9, 5, 4, 3] and in coupling with the Krylov subspace methods [9].
Among them, there is no an ‘alway optimal’ IDC method; each method has its strength and weakness; choices
of methods greatly depend on the characteristics of the system being solved. Under the IDC framework, it is
shown in [5, 4] that if an r th order integrator is used to solve the error equation, then the accuracy of the scheme
increases by r orders after each correction loop. This analysis has recently been extended in [3] for IDC methods
constructed with implicit and semi-implicit integrators. In [4], the IDC method constructed with high order
Runge-Kutta (R-K) method is showed to be a R-K method with corresponding Butcher tableau constructed.
The main goal of this paper is to study the convergence behavior for the IDC method constructed using
implicit R-K methods of different orders for the prediction and correction steps when applied to a special class
of problems called singular perturbation problems (SPPs) containing a parameter ε. An arbitrary SPP is given
by
y ′ = f(y, z),
εz ′ (1.2)
= g(y, z),
where y and z are vectors and ε > 0 is the stiff parameter. Classical books on this subject are [16, 14]. In
system (1.2) we suppose that 0 < ε ≪ 1 and f and g are sufficiently differentiable vector functions of the
1 Department of Mathematics and Computer Science, University of Catania, Catania, 95125, E-mail: boscarino@dmi.unict.it
2 Department of Mathematics, University of Houston, Houston, 77004. E-mail: jingqiu@math.uh.edu. Research supported by
Air Force Office of Scientific Computing YIP grant FA9550-12-0318, NSF grant DMS-0914852 and DMS-1217008 and University of
Houston.
1

same dimensions as y and z respectively. The functions f, g and the initial values y(0), z(0) may depend
smoothly on ε. For simplicity of notation we suppress this dependence. When the parameter ε in system (1.2)
is small, the corresponding differential equation is stiff, and when ε tends to zero, the differential equation
becomes differential algebraic. A sequence of differential-algebraic systems arises in the study of SPPs. This
system allows us to understand many phenomena observed for very stiff problems. Indeed in Chap. VI. 3 of
[8] the authors show that most of the R-K methods presented in the literature suffer from the phenomenon of
order reduction in the stiff regime, i.e. ε small, when applied to (1.2). To this aim, we investigate the same
phenomenon of the order reduction when it appears in the IDC framework. In the past, such order reduction
phenomena have been numerically investigated without much theoretical justification for SPPs [13, 3]. Our
analysis is based on the assumption of a smooth solution of system (1.2) and applies to the stiff case (H ≫ ε),
where H is the time step size.
We require that system (1.2) satisfies
µ(g z (y, z)) ≤ −1, (1.3)
in an ε-independent neighborhood of the solution, where µ denotes the logarithmic norm with respect to some
inner product. Condition (1.3) guarantees the existence of an ε-expansion of the system (1.2) (see [8]). In other
words we assume that the system (1.2) is dissipative. Furthermore, we suppose in our analysis that the initial
values lie on a suitable manifold that allows smooth solutions even in the limit of infinite stiffness. In fact
arbitrary initial values introduce in the solution a fast transient. One possible way to overcome this difficulty is
simply to ensure that the numerical method resolves the transient phase by taking small step size of magnitude
O(ε). Then the following results are obtained assuming that the transient phase is over. We note that the
corresponding reduce system (ε = 0) is the differential algebraic equation (DAE)
y ′ = f(y, z),
0 = g(y, z),
(1.4)
whose initial values are consistent if 0 = g(y 0 , z 0 ). We assume that the Jacobian
g z (y, z) is invertible (1.5)
in a neighborhood of the solution of (1.4). This assumption guarantees the solvability of (1.4) and that the
equation g(y, z) = 0 possesses a locally unique solution z = G(y) (implicit function theorem) which inserted into
(1.4) gives
y ′ = f(y, G(y)). (1.6)
Furthermore, the same assumption guarantees that system (1.4) is a differential-algebraic one of index 1 [8].
From now on we assume a Lipschitz conditions for G.
In this paper, we consider an error estimate of IDC framework constructed with implicit R-K method for
SPPs by presenting and proving two main Theorems in Section 3 and 4 respectively. We expand the error
in powers of ɛ whose coefficients are error terms, and show convergence results for these error terms. Order
reduction exists for both differential and algebraic components in the IDC framework. Especially, there is no
order improvement for the ε ν (ν ≥ 1) error terms in IDC corrections, see Remark 4.7 and 4.15. We focus our
analysis on the IDC method using all of the uniform quadrature nodes, but excluding the left-most endpoint.
We also remark that an important property for implicit R-K method called stiffly accurate (we will define it
in the next section), will be an important concept in our analysis both in the prediction and in the correction
steps for IDC framework. If this property is not satisfied, the corresponding IDC method becomes unstable and
the numerical solutions diverge.
The paper is organized in the following way. In the rest of this introductory section, we briefly present
existing classical local and global error estimate of general implicit R-K method for SPPs [8]. In Section 2,
we introduce the IDC methods constructed with implicit backward Euler and high order implicit R-K method
for SPPs (1.2). In Section 3, the main theoretical results are provided in the form of two main Theorems.
Numerical evidence supporting these theorems are summarized and presented. In Section 4, the main theorems
are proved based on the ε-expansion of numerical solutions of IDC methods constructed with backward Euler
and general implicit R-K methods. Finally, conclusions are given in Section 5.
1.1 Implicit R-K method applied to SPPs
This section is a review of implicit R-K methods applied to SPPs (1.2). The IDC framework requires R-K
methods in the prediction and correction steps. It is very important to know convergence results about R-
2

K methods applied to (1.2) to investigate thoroughly the convergence estimates an IDC R-K method. Our
discussion on these methods is based on notations introduced in ([8] Chap. VI.3).
We consider an implicit R-K method applied to the SPP (1.2)
where
and the internal stages given by
(
yn+1
)
=
z n+1
(
Yni
(
kni
)
=
Z ni
)
=
εl ni
(
yn
z n
)
+ h
(
yn
z n
)
+ h
s∑
i=1
( f(Yni , Z ni )
g(Y ni , Z ni )
i∑
j=1
b i
(
kni
l ni
)
)
(1.7)
(1.8)
a ij
(
knj
l nj
)
. (1.9)
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 .
They can be represented by a tableau in the usual Butcher notation,
c
A
b T . (1.10)
The coefficients c are given by the usual relation c i = ∑ i
j=1 a ij.
In this paper we denote by p the classical order of the method when it is applied to a non-stiff equation.
For stiff differential equations we consider an important concept the stage order q. It is defined by the relations
(see and [8], Chap. IV.5), i.e.
C(q) :
s∑
j=1
a ij c k−1
j
= ck i
, i = 1, · · · s, for k = 1, ..., q. (1.11)
k
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
and f a smooth function, the internal stages are O(h qi+1 )-approximations to the exact solution at c i h, i.e.
y(t n + c i h) − Y i = O(h qi+1 ) where Y i = y(t n ) + h ∑ s
j=1 a ijf(t n + c j h, Y j ), for 1 ≤ i ≤ s. For example for an
s-stage diagonally implicit R-K (DIRK) method, the stage order is 1.
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 =
(0, ..., 0, 1).
Remark 1.2. Stiffly accurate methods are important for the solution of SPPs and differential algebraic equations.
In particular this property is important for the L-stability of the method. To see this, let R(∞) =
lim z→∞ R(z), with R(z) = 1 + zb T (I − zA) −1 1 being the stability function of an implicit scheme, where
b T = (b 1 , ..., b s ) and 1= (1, ..., 1) T . If the A matrix is invertible, then R(∞) = 1− ∑ s
i,j=1 b iω ij with ω ij elements
of the inverse of (a ij ). Moreover, if the implicit method is stiffly accurate, then R(∞) = 0. This makes A-stable
methods L-stable.
Now supposing the matrix A invertible one obtains from (1.9)
hl ni =
s∑
ω ij (Z nj − z n ), (1.12)
j=1
and inserting this into the numerical solution z n+1 it follows
z n+1 = R(∞)z n +
s∑
b i ω ij Z nj (1.13)
making the definition of z n+1 indepenent of ε. Now putting ε = 0 in (1.7) we get l ni = g(Y ni , Z ni ) = 0, then
we obtain
s∑
Y ni = y n + a ij f(Y nj , Z nj ) (1.14)
i=1
3
i=1

g(Y ni , Z ni ) = 0 (1.15)
s∑
y n+1 = y n + b i f(Y ni , Z ni ) (1.16)
i=1
z n+1 = R(∞)z n +
s∑
b i ω ij Z nj (1.17)
We note that this represents the numerical method for solving the reduced system (1.4), i.e. ε = 0. From (1.15)
we have Z ni = G(Y ni ) (Implicit Function Theorem). If the method is stiffly accurate we have that y n+1 = Y ns
and z n+1 = Z ns = G(Y ns ) = G(y n+1 ). Then
i=1
g(y n+1 , z n+1 ) = 0. (1.18)
In this case, the solution of (1.16)-(1.15)-(1.14) and (1.18) to system (1.4) is equivalent to the solution obtained
by the same implicit R-K method applied to (1.6). Therefore, for the method (1.14)-(1.16)
y n − y(t n ) = O(h p ). (1.19)
Furthermore we have z n+1 = G(y n+1 ), then by assuming a Lipschitz condition for G
We summarize these results by the following theorem.
z n − z(t n ) = G(y n ) − G(y(t n )) = O(h p ). (1.20)
Theorem 1.3. (Chap. VI. 1 Theorem 1.1 part (a) in [8]) Suppose that the system (1.4) satisfies (1.5) in a
neighborhood of the exact solution and assume that the initial values are consistent. Consider a stiffly accurate
R-K method of order p, and with invertible matrix A. Then the numerical solutions of (1.14)-(1.17) have global
error
y n − y(t n ) = O(h p ), z n − z(t n ) = O(h p ), (1.21)
for t n − t 0 = nh ≤ Const.
Now we review the main result obtained in Chap. VI.3 of [8] about the error analysis of implicit R-K methods
for singular perturbation problem (1.2). We perform an asymptotic expansion of smooth solutions of system
(1.2) and similarly for the numerical solutions of a R-K method applied to (1.2). The errors of the y and
z-component are formally considered as
y n − y(t n ) = ∑ ν≥0
ɛ ν (y n,ν − y ν (t n )), z n − z(t n ) = ∑ ν≥0
ɛ ν (z n,ν − z ν (t n )). (1.22)
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 of the R-K
method applied to the reduced system (1.4), i.e. system of index 1. The error estimates are summarized in
Theorem 1.3. The second difference in (1.22) is related to the numerical solutions of the R-K method when
applied to the differential algebraic system of index 2.
Remark 1.4. A complete analysis for the convergence of implicit R-K methods for differential algebraic system
of index 2 is given in [8] by Lemma 4.4, Theorem 4.5 and Theorem 4.6 in Chap. VII.4. Below we summarize
optimal error estimates for R-K methods when applied to index 2 problem. From Lemma 4.4 in Chap. VII.4 of
[8], the local error estimate is
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)
If, in addition, the method is stiffly accurate
δy h (t) = O(h min(p+1,q+2) ) with p ≥ q.
From Theorem 4.5 and 4.6 in Chap. VII.4 of [8], the global convergence results follow. Some interesting results
about local and global error for some important R-K methods are collected in Table 4.1 in Chap. VII.4 of [8].
Here, as an example, we consider DIRK or SDIRK methods with p ≥ 2. Such methods have stage order q = 1
implying that the local error
δy h (t) = O(h 2 ), δz h (t) = O(h)
δy h (t) = O(h 3 ), δz h (t) = O(h) if the method is stiffly accurate
(1.24)
4

and the global error
y n − y(t n ) = O(h 2 ) z n − z(t n ) = O(h). (1.25)
We note that concerning backward Euler method with p = q = 1, the local error is δy h (t) = O(h 2 ) and the
global error y n − y(t n ) = O(h), i.e. the method, applied to a system of index 2, maintains the classical order.
We note that backward Euler method is a first order Radau IIA method, then these estimates are a simple
consequence of Theorem 4.9 in Chap. VII of [8] for the local error estimate and Theorem 4.5 in Chap. VII of
[8] for the global error estimate.
Finally, the main result of global error estimate (1.22) of the R-K method when applied to SPP (1.2) is
presented in Theorem 1.5 below. For details, see Chap. VI. 3, Theorem 3.3, 3.4, 3.8 and Corollary 3.10 of [8].
Theorem 1.5. Consider the stiff problem (1.2), (1.3) with initial values y(0), z(0) admitting a smooth solution.
Apply the R-K method (1.7)-(1.8)-(1.9) of classical order p and stage order q, (1 ≤ q < p). Assume that the
method is A-stable, that the stability function satisfies |R(∞)| < 1 and that the eigenvalues of the coefficient
matrix A have positve real parts. Then the global error of a R-K method satisfies
y n − y(t n ) = O(h p ) + O(εh q+1 ), z n − z(t n ) = O(h q+1 ). (1.26)
If in addition the method is stiffly accurate, we have
The estimates hold uniformly for h ≤ h 0 and nh ≤ Const.
2 IDC Formulations Applied to SPPs
z n − z(t n ) = O(h p ) + O(εh q ) (1.27)
In this section we consider IDC framework constructed with stiffly accurate implicit R-K method for SPPs. The
use of uniform nodes is important for high order accuracy increase if high order R-K method is used in correction
loops [5], where as the use of quadrature nodes excluding the left-most endpoint lead to an important stability
condition for stiff problem R(∞) = 0 [11]. We remark the IDC method is not L-stable with R(∞) ≠ 0, if the
quadrature nodes including the left-most endpoint is used. Moreover, when IDC R-K method using quadrature
nodes excluding the left-most end point is represented in a Butcher Tableau, the corresponding A matrix would
be invertible, see Section 4 and [4]. The invertibility of the A matrix is an important assumption in many of
the classical results in [8].
2.1 IDC Framework
We consider IDC procedure [6] applied to a singular pertubation problem
y ′ (t) = f(y, z), y(t 0 ) = y 0 ,
εz ′ (t) = g(y, z), z(t 0 ) = z 0 .
(2.1)
The time interval [0, T ] is discretized into intervals [t n , t n+1 ], n = 0, 1, ..., N − 1 such that
0 = t 0 < t 1 < t 2 < ... < t n < ... < t N = T,
with the step size H. Then, each interval [t n , t n+1 ] is discretized again into M uniform subintervals with
quadrature nodes denoted by
t n = t n,0 < t n,1 < · · · < t n,M = t n+1 , (2.2)
with h = H M
being the size of a substep. In this paper, the interval [t n, t n+1 ] will be referred to as a time step
while a subinterval [t m , t m+1 ] (dropping the subscript n) will be referred to as a substep. We remark that the
size of time interval [t n , t n+1 ] may vary as the IDC method is a one-step, multi-stage method. We assume the
IDC quadrature nodes are uniform, which is a crucial assumption for high order improvement in accuracy, when
consider applying general high order implicit R-K in prediction and correction steps for classical ODE system
(1.1), see discussions in [5]. We also note that since h = H M , we will use O(hp ) and O(H p ) interchangeably
throughout the paper.
5

Integrating equations (2.1) with respect to t, we obtain the equivalent Picard equations
{
y(t) = y 0 + ∫ t
t 0
f(y(s), z(s))ds,
εz(t) = εz 0 + ∫ t
t 0
g(y(s), z(s))ds,
(2.3)
Suppose that we have obtained approximate solutions ŷ m (0) and ẑ m
(0) at t m by using a p th order numerical method
for (2.1). We build a continuous polynomial interpolants ŷ (0) (t) and ẑ (0) (t) from these discrete values. Now we
define the error functions
e (0) (t) = y(t) − ŷ (0) (t), d (0) (t) = z(t) − ẑ (0) (t). (2.4)
Note that e (0) (t) and d (0) (t) are not polynomials in general. We define the residual function
{ (δ (0) ) ′ (t) = f(ŷ (0) (t), ẑ (0) (t)) − (ŷ (0) ) ′ (t)
(ρ (0) ) ′ (t) = g(ŷ (0) (t), ẑ (0) (t)) − (εẑ (0) ) ′ (t).
(2.5)
Integrating (2.5) from t 0 to t gives,
{
δ (0) (t) = y 0 + ∫ t
t 0
f(ŷ (0) (s), ẑ (0) (s))ds − ŷ (0) (t),
ρ (0) (t) = εz 0 + ∫ t
t 0
g(ŷ (0) (s), ẑ (0) (s))ds − εẑ (0) (t).
Thus, the error equations about the error functions (2.4) by subtracting (2.6) from (2.3) become
{
e (0) (t) = ∫ t
t 0
f(e (0) (s) + ŷ 0 (s), d (0) (s) + ẑ (0) (s)) − f(ŷ (0) (s), ẑ (0) (s))ds + δ (0) (t),
εd (0) (t) = ∫ t
t 0
g(e (0) (s) + ŷ 0 (s), d (0) (s) + ẑ (0) (s)) − g(ŷ (0) (s), ẑ (0) (s))ds + ρ (0) (t).
(2.6)
(2.7)
Suppose that we have obtained approximate solutions ê (0)
m and at t m by using a p th order numerical method
for error equations (2.7). The numerical solution can then be improved as
ˆd
(0)
m
ŷ m (1) = ŷ m (0) + ê (0)
m , ẑ m (1) = ẑ m (0) (0)
+ ˆd m , ∀m = 0, · · · M.
Such correction procedures can be repeated. In summary, the strategy of IDC methods is to use a simple
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
solve a series of correction equations based on Equations (2.7), each of which improves the accuracy of the
provisional solution.
Remark 2.1. (About notations.) In our description of IDC, we let y (k)
m
and exact error functions (without hat); and let ŷ m (k) , ẑ m (k) , ê (k)
m ,
ˆd
(k)
m
z m (k) , e (k)
m , d (k)
m denote the exact solutions
denote the numerical approximations (with
hat) to the exact solutions and error functions. We use subscript m to denote the location t = t m and use
superscript (k) to denote the prediction (k = 0) and correction loops (k = 1, · · · ). We let ¯· denote the vector on
IDC quadrature nodes. For example, ȳ = (y 1 , ·, y M ) excluding the left-most point.
2.2 IDC Methods Based on Backward Euler Method
In this subsection, we consider a simple backward Euler method for computing both the previsional solution
and the corrections. As we mentioned in the introduction, we choose to work with the uniform nodes excluding
the left-most endpoint for its stability property R(∞) = 0 [11].
(Prediction step) Use a backward Euler discretization to compute an approximate solution ¯ŷ (0) = (ŷ (0)
1
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
the same for the z-component. This gives
{
ŷ (0)
m+1 = ŷ(0) m + hf(ŷ (0)
m+1 , ẑ(0)
m+1 ),
εẑ (0)
m+1 = εẑ(0) m + hg(ŷ (0)
m+1 , ẑ(0) m+1 ) (2.8)
for m = 0, 1, ...M − 1.
(Correction loop). For k = 1, ..., K (K is the number of the correction step). Let ŷ (k−1) denote the k th
sequence correction.
, ..., ŷ(0) m , ..., ŷ (0)
M )
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
solution and ŷ (k−1) (t) is a (M − 1)-th polynomial interpolating ¯ŷ (k−1) . We compute the numerical error
vector ¯ê (k−1) = (ê (k−1)
1 , ..., ê (k−1)
M
) with ê(k−1) m approximates e (k−1) (t m ) by applying a backward Euler
method to (2.7)
⎧
⎨ ê (k−1)
m+1 = ê (k−1)
m + h∆f (k−1)
m+1 + ∫ t m+1
t m
δ (k−1) (s)ds,
(k−1)
(k−1)
⎩ ε ˆd m+1 = ε ˆd m + h∆g (k−1)
m+1 + ∫ t m+1
(2.9)
t m
ρ (k−1) (s)ds
where {
and
⎧
⎨
⎩
∆f (k−1)
m+1 = f(ŷ (k−1)
m+1 + ê(k−1) m+1 , ẑ(k−1) (k−1)
m+1 + ˆd m+1 ) − f(ŷ(k−1) m+1 , ẑ(k−1)
∆g (k−1)
m+1 = g(ŷ (k−1)
m+1 + ê(k−1) m+1 , ẑ(k−1) m+1
∫ tm+1
m+1 )
(k−1)
+ ˆd m+1 ) − g(ŷ(k−1) m+1 , ẑ(k−1) m+1 ), (2.10)
t m
δ (k−1) (s)ds = ∫ t m+1
t m
f(ŷ (k−1) (s), ẑ (k−1) (s))ds − ŷ (k−1)
m+1 + ŷ(k−1) m
∫ tm+1
t m
ρ (k−1) (s)ds = ∫ t m+1
t m
g(ŷ (k−1) (s), ẑ (k−1) (s))ds − εẑ (k−1)
m+1 + εẑ(k−1) m .
(2.11)
The integral term ∫ t m+1
t m
in equations (2.11) are approximated by a numerical quadrature. Especially, let
S be the integration matrix with its (m, k) element
where
S m,k = 1 h
∫ tm+1
t m
α k (s)ds, for m = 0, · · · , M − 1, k = 1, · · · M
α k (s) =
M∏
i≠k,i=1
is the Lagrangian basis function based on the node t k . Let
with ∑ M
j=1 Sm,j = 1. Then
S m ( ¯f) =
s − t k
t i − t k
(2.12)
M∑
S m,j f(y j , z j ), (2.13)
j=1
∫
hS m ( ¯f)
tm+1
− f(y(s), z(s))ds = O(h M+1 ),
t m
for any smooth function f. In other words, the quadrature formula given by hS m ( ¯f) approximates the
exact integration with (M + 1) order accuracy locally.
Remark 2.2. Considering the following change of variable s = t 0 + σh in the interval [t n , t n+1 ] we get
α k (t 0 + σh) = ∏ M
, i.e. there is only a dependence on M and not on h. Then the integral
i≠k σ−k
i−k
depends on M and not on h.
S m,k = 1 h
∫ tm+1
t m
α k (s)ds =
∫ m+1
m
α k (t 0 + σh)dσ (2.14)
2. Update the approximate solutions ¯ŷ (k) = ¯ŷ (k−1) + ¯ê (k−1) and ¯ẑ (k) = ¯ẑ (k−1) + ¯ˆd(k−1) .
Remark 2.3. Using these notations, we get from eqs. (2.9)
{
ŷ (k)
m+1 = ŷ(k) m + h∆f (k−1)
εẑ (k)
m+1 = εẑ(k) m
m+1 + hSm ( ¯ˆf (k−1) ),
+ h∆g (k−1)
m+1 + hSm (¯ĝ (k−1) )
(2.15)
where we use the S m notation introduced in (2.13).
Remark 2.4. Since we consider the nodes excluding the left most quadrature point t 0 , the order of approximation
for integration/interpolation will be one order lower than the usual one considered in [6, 5].
7

2.3 IDC Methods Based on General Implicit R-K
Below, we describe how we apply an s-stage implicit R-K methods in correction loops for IDC framework. For
the internal stages in the R-K method, we introduce the integration matrix and interpolation matrix as following
hS cmi,k =
∫ tm+c mih
t m
α k (s)ds, P cmi,k = α k (t m + c mi h), (2.16)
∀m = 0, · · · , M − 1, ∀k = 1, · · · M, ∀mi = 1, · · · s, where α j (s) as introduced in equation (2.12) is the
Lagrangian basis function based on the node t j . Let
Then
S cmi ( ¯f)
M∑
= S cmi,j f(y j , z j ), P cmi ( ¯f)
M∑
= P cmi,j f(y j , z j )
j=1
j=1
∫ tm+c ih
hS cmi ( ¯f) − f(y(s), z(s))ds = O(h M+1 )
t m
P cmi ( ¯f) − f(y(t m + c i h), z(t m + c i h)) = O(h M )
for any smooth function f. In other words, the quadrature formula given by hS cmi ( ¯f) approximates the exact
integration with (M +1) th order accuracy locally, while the interpolation formula given by P cmi ( ¯f) approximates
the exact solution at R-K internal stages with M th order accuracy.
To compute the numerical error approximating the error function e (k−1) (t m ), d (k−1) (t m ) with a general
implicit R-K method to (2.7),
with
(
(
ê (k−1)
m+1
ˆd (k−1)
m+1
(k−1)
∆ ˆK
ε∆
mi
(k−1) ˆL mi
)
)
=
.
=
=
(
ê (k−1)
m + h ∫ 1
0 δ(t m + τh)dτ
ˆd (k−1)
m + h/ε ∫ 1
0 ρ(t m + τh)dτ
(
(
)
f(Ŷ (k)
mi , Ẑ(k) mi ) − P cmi ( ¯ˆf (k−1) )
g(Ŷ (k)
mi , Ẑ(k) mi ) − P cmi (¯ĝ (k−1) )
+ h
)
(
s∑
b i
i=1
f(Ŷ (k)
mi , Ẑ(k) mi ) − f(P cmi (¯ŷ (k−1) ), P cmi (¯ẑ (k−1) ))
g(Ŷ (k)
mi , Ẑ(k) mi ) − g(P cmi (¯ŷ (k−1) ), P cmi (¯ẑ (k−1) ))
(k−1)
∆ ˆK
∆
mi
(k−1) ˆL mi
)
)
(2.17)
(2.18)
+ O(h M ), (2.19)
where the last equation above is due to the high order interpolation accuracy of P cmi . We note that equation
(2.18) is for numerical implementation, whereas equation (2.19) is preparation for our analysis. Here we put
with ( Ê(k−1) mi
ˆD (k−1)
mi
Ŷ (k)
mi
)
= P cmi (¯ŷ (k−1) ) + Ê(k−1) mi , Ẑ (k)
mi = P cmi (¯ẑ (k−1) ) +
=
(
ê (k−1)
m
ˆd (k−1)
m
We can rewrite the previous system (2.17) as
⎛
with
⎛
⎝ ŷ(k) m+1 − hSm,(k−1) ¯ˆf
ẑ (k)
m+1 − hSm,(k−1) ¯ĝ
⎛
⎝ Ŷ (k)
mi
− hS cmi,(k−1)
¯ˆf
Ẑ (k)
mi − hScmi,(k−1)
¯ĝ
⎝ Sm,(k−1) ¯ˆf
εS m,(k−1)
¯ĝ
+ h ∫ c mi
δ(t
0 m + τh)dτ
+ h/ε ∫ c mi
ρ(t
0 m + τh)dτ
⎞
⎠ =
⎞
= S m ( ¯ˆf (k−1) )
⎠ =
= S m (¯ĝ (k−1) )
⎞
(
(
⎠ ,
ŷ (k)
m
ẑ (k)
m
ŷ (k)
m
ẑ (k)
m
⎛
)
)
+ h
+ h
)
+ h
(
s∑
b i
i=1
(
i∑
a ij
j=1
⎝ Scmi,(k−1)
¯ˆf
εS cmi,(k−1)
¯ĝ
(
i∑
a ij
j=1
(k−1)
∆ ˆK
∆
mi
(k−1) ˆL mi
(k−1)
∆ ˆK mj
∆
ˆD
(k−1)
mi (2.20)
ˆL
(k−1)
mj
= S cmi ( ¯ˆf (k−1) )
= S cmi (¯ĝ (k−1) )
(k−1)
∆ ˆK mj
∆
ˆL
(k−1)
mj
)
)
)
. (2.21)
(2.22)
, (2.23)
⎞
⎠ . (2.24)
8

Remark 2.5. Assuming that A is invertible, following a similar procedure as in equation (1.12) and (1.13), we
get in the vectorial form from the second equation of (2.23)
ˆL
(k−1)
m1
with ∆ ¯ˆL(k−1) = (∆ , · · · , ∆
Plug this into the second equation of (2.22), we get
h∆ ¯ˆL(k−1) = A −1 ( ¯Ẑ (k) − ẑ (k)
m 1 − hS¯c (¯ĝ (k−1) )),
ˆL
(k−1)
ms ) T , 1 = (1, · · · , 1) T and ¯c = (c m1 , · · · , c ms ) are R-K internal stages.
ẑ (k)
m+1 = ẑ m
(k) + hS m (¯ĝ (k−1) ) + b T A −1 ( ¯Ẑ (k) − ẑ m (k) 1 − hS¯c (¯ĝ (k−1) )).
Especially for stiffly accurate R-K method with b T A −1 = e T s , we have
ẑ (k)
m+1 = R(∞)ẑ(k) m + b T A −1 ¯Ẑ(k) = Ẑ(k) ms, (2.25)
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
similar spirit to (1.17) for implicit R-K method.
2.4 ε-asymptotic expansion
In this paper we want to use the approach proposed in [8] considering the ε-expansion of the exact and numerical
solution to study the behavior of the local error for the IDC method. To prove the main results, ε-expansion
of the exact solutions and numerical solutions are performed; this is a preparation for local error estimates in
Section 4.
• The ε-expansion of the exact solution of problem (2.1) is the following,
( ) ( ∑ ∞ y(t)
=
ν=0 y )
∑ ν(t)ε ν
∞
z(t)
ν=0 z ν(t)ε ν
(2.26)
where y ν (t) and z ν (t) are ε-independent functions, which are solutions of a sequence of differential algebraic
equations of arbitrary index [8]. Evaluating (2.26) at t m this gives
( ) ( ∑ ∞ ym
=
ν=0 y )
∑
m,νε ν
∞
z m ν=0 z m,νε ν (2.27)
Inserting (2.26) into (2.1) and collecting terms of equal powers of ε yields
ε 0 :
{ y
′
0 = f(y 0 , z 0 )
0 = g(y 0 , z 0 )
(2.28)
ε 1 :
{ y
′
1 = f y (y 0 , z 0 )y 1 + f z (y 0 , z 0 )z 1
. = F1
z ′ 0 = g y (y 0 , z 0 )y 1 + g z (y 0 , z 0 )z 1
. = G1
(2.29)
· · · (2.30)
ε ν :
{ y
′
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
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
(2.31)
with initial values y ν (0), z ν (0) known from (2.26). We observe that system (2.28) under the condition
(1.3) is a differential algebraic system of index 1. According to [8], if we consider (2.28) and (2.29) together
we have a differential algebraic system of index 2. In general (2.28), (2.29) and (2.31) is a system of index
ν.
• The ε-expansion of the numerical solution at k th iteration is the following. The case of k = 0 is for
numerical solution at prediction.
( ) (
ŷ m
(k) ∑∞
)
ẑ m
(k) =
ν=0 ŷ(k) m,νε ν
∑ ∞
. (2.32)
ν=0 ẑ(k) m,νε ν
Backward Euler in prediction and correction steps of IDC.
By plugging the ε-expansion of numerical solution (2.32) into the numerical scheme (2.8)-(2.11), and
matching and collecting terms that are of equal powers of ɛ, one obtains the following.
9

– For the prediction step k = 0
where
⎧
⎨
⎩
ˆF (0)
m+1,1
Ĝ (0)
m+1,1
ε 0 :
ε 1 :
{
{
(
.
= f y (ŷ (0)
.
=
ŷ (0)
m+1,0 = ŷ(0) m,0 + hf(ŷ(0) m+1,0 , ẑ(0) m+1,0 )
0 = g(ŷ (0)
m+1,0 , ẑ(0) m+1,0 ). (2.33)
ŷ (0)
m+1,1 = ŷ(0) (0)
m,1 + hˆF m+1,1
ẑ (0)
m+1,0 = ẑ(0) m,0 + hĜ(0) m+1,1
m+1,0 , ẑ(0) m+1,0 )ŷ(0) m+1,1 + f z(ŷ (0)
m+1,0 , ẑ(0) m+1,0 )ẑ(0) m+1,1
(
g y (ŷ (0)
m+1,0 , ẑ(0) m+1,0 )ŷ(0) m+1,1 + g z(ŷ (0)
m+1,0 , ẑ(0) m+1,0 )ẑ(0) m+1,1
Equation (2.33)-(2.34) are consistent discretizations of equation (2.28)-(2.29).
– For the correction steps k ≥ 1,
⎧
⎪⎨
ε 0 :
⎪⎩
⎧
⎪⎨
ε 1 :
⎪⎩
ŷ (k)
m+1,0 = ŷ (k)
(k−1)
m,0 + h∆ ˆf m+1,0 + hSm ( ¯ˆf (k−1)
0 )
0 = h∆ĝ (k−1)
m+1,0 + hSm (¯ĝ (k−1)
0 )
ŷ (k)
m+1,1
ẑ (k)
m+1,0
= ŷ (k)
(k−1)
m,1 + h∆ˆF m+1,1 + hSm (¯ˆF(k−1) 1 )
= ẑ (k)
m,0 + h∆Ĝ(k−1) m+1,1 + hSm (k−1)
( ¯Ĝ 1 )
)
)
.
(2.34)
(2.35)
(2.36)
(2.37)
In equation (2.36)
and in equation (2.37)
{
(k−1)
∆ ˆf m+1,0 = f(ŷ (k)
m+1,0 , ẑ(k) m+1,0 ) − f(ŷ(k−1) m+1,0 , ẑ(k−1) m+1,0 )
∆ĝ (k−1)
m+1,0 = g(ŷ (k)
m+1,0 , ẑ(k) m+1,0 ) − g(ŷ(k−1) m+1,0 , ẑ(k−1) m+1,0 ) (2.38)
where
∆ˆF (k−1)
m+1,1 =
=
(k) ˆF m+1,1
(
f y (ŷ (k)
−
− ˆF
(k−1)
m+1,1
m+1,0 , ẑ(k) m+1,0 )ŷ(k) m+1,1 + f z(t m+1 , ŷ (k)
(
f y (ŷ (k−1)
m+1,0 , ẑ(k−1) m+1,0 )ŷ(k−1) m+1,1 + f z(ŷ (k−1)
m+1,0 , ẑ(k−1) m+1,0 )ẑ(k−1) m+1,1
= f y (y m+1,0 , z m+1,0 )ê (k−1)
m+1,1 + f z(y m+1,0 , z m+1,0 )
.
= f y ê (k−1)
m+1,1 + f (k−1)
z ˆd m+1,1 + O(hsk−1+1 ),
m+1,0 , ẑ(k) m+1,0 )ẑ(k) m+1,1
)
)
ˆd
(k−1)
m+1,1 + O(hs k−1+1 )
(2.39)
ˆF (k)
m,1 = f y(ŷ (k)
m,0 , ẑ(k) m,0 )ŷ(k) m,1 + f z(ŷ (k)
m,0 , ẑ(k) m,0 )ẑ(k) m,1 . (2.40)
Here we assume y m,0 − ŷ (k)
m,0 = O(hs k+1 ) locally for all k. For simplicity of notations we let
f y (y m+1,0 , z m+1,0 ) = f y , and similarly for f z . Similarly, we have
∆Ĝ(k−1) m+1,1 = g y ê (k−1)
m+1,1 + g (k−1)
z ˆd m+1,1 + O(hsk−1+1 ). (2.41)
Equation (2.36)-(2.37) are consistent discretizations of equation (2.28)-(2.29) respectively.
Implicit R-K in prediction and correction steps of IDC.
ˆK
(k−1)
mi
ˆL
(k−1)
mi
We formally expand the quantities ∆ , ∆
(2.22) with k ≥ 1 into power of ε with ε-independent coefficients
ŷ (k)
m
Ŷ (k)
mi
∆
ẑ (k)
m
= ŷ (k)
m,0 + εŷ(k) m,1 + ε2 ŷ (k)
= Ŷ (k)
mi,0
= ∆
ˆK
(k−1)
mi
m,2 + ...
+ εŶ
(k)
mi,1 + ε2 Ŷ (k)
ˆK
(k−1)
mi,0
mi,2 + ...
(k−1)
+ ε∆ ˆK + ε2 ∆
mi,1
= ẑ (k)
m,0 + εẑ(k) m,1 + ε2 ẑ (k)
m,2 + ...
Ẑ (k)
mi = Ẑ(k) mi,0 + εẐ(k) mi,1 + ε2 Ẑ (k)
∆ = ε −1 ∆
ˆL
(k−1)
mi
ˆL
(k−1)
mi,−1
from (2.18) and Ŷ (k)
mi , Ẑ(k) mi , ŷ(k) m+1 , ẑ(k) m+1 from (2.20)
ˆK
(k−1)
mi,2 + ...
mi,2 + ...
(k−1)
+ ε∆ ˆL
+ ∆ ˆL
(k−1)
mi,0
mi,1
+ ε2 (k−1)
∆ ˆL mi,2 + · · · . (2.42)
10

Inserting (2.42) into (2.18) we obtain the following
ε 0 :
ε 1 :
ε ν :
Similarly, we have
(k−1)
∆ ˆK mi,0 = f(Ŷ (k)
(k−1)
∆ ˆK mi,1 =
(k−1)
∆ ˆK mi,ν =
(
f y (Ŷ (k)
−
(k−1)
∆ ˆL mi,−1 = g(Ŷ (k)
(k−1)
∆ ˆL mi,0 =
mi,0 , Ẑ(k) mi,0 ) − f(P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 )) + O(h M ) (2.43)
)
mi,0 , Ẑ(k) (k)
mi,0 )Ŷ mi,1 + f z(Ŷ (k)
mi,0 , Ẑ(k) mi,0 )Ẑ(k) mi,1
(
f y (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ))P cmi (¯ŷ (k−1)
1 )
)
+f z (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ))P cmi (¯ẑ (k−1)
1 )
(
f y (Ŷ (k)
mi,0 , Ẑ(k) (k)
mi,0 )Ŷ mi,ν + f z(Ŷ (k)
mi,0 , Ẑ(k) mi,0 )Ẑ(k) mi,ν
(
− f y (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ))P cmi (¯ŷ ν (k−1) )
)
+f z (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ))P cmi (¯ẑ (k−1)
ν )
+ψ ν (Ŷ (k)
mi,0 , Ẑ(k) (k)
mi,0 , ..., Ŷ mi,ν−1 , Ẑ(k) mi,ν−1 )
+ O(h M ) (2.44)
)
+ψ ν (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ), ..., P cmi (¯ŷ (k−1)
ν−1 ), P cmi (¯ẑ (k−1)
ν−1 ))
+O(h M ). (2.45)
(
g y (Ŷ (k)
−
mi,0 , Ẑ(k) mi,0 ) − g(P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 )) + O(h M ) (2.46)
)
mi,0 , Ẑ(k) (k)
mi,0 )Ŷ mi,1 + g z(Ŷ (k)
mi,0 , Ẑ(k) mi,0 )Ẑ(k) mi,1
(
g y (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ))P cmi (ȳ (k−1)
1 )
)
+g z (P cmi (¯ŷ (k−1)
0 ), P cmi (¯ẑ (k−1)
0 ))P cmi (¯z (k−1)
1 )
+ O(h M ).
Because of the linearity of relations (2.22) and (2.23), we have to order ε ν with ν = −1 in vectorial form
and for ν ≥ 0,
where
with S m,(k−1)
¯ˆF ν
⎛
(k−1)
hA∆ ˆL ¯m,−1 + hS−→c (¯ĝ) = 0,
⎝ ŷ(k) m+1,ν − hSm,(k−1) ¯ˆF ν
ẑ (k)
m+1,ν − hSm,(k−1) ¯Ĝ ν
⎛
⎝ Ŷ (k)
mi,ν − hScmi,(k−1)
¯ˆF ν
Ẑ (k)
mi,ν − hScmi,(k−1)
¯Ĝ ν
⎛
⎞
⎠ =
⎞
⎠ =
⎝ Sm,(k−1) ¯ˆF ν
εS m,(k−1)
¯Ĝ ν
(
⎞
(
⎠ =
hb T ∆L (k−1)
¯m,−1 + hSm (¯ĝ) = 0, (2.47)
ŷ (k)
m,ν
ẑ (k)
m,ν
ŷ (k)
m,ν
ẑ (k)
m,ν
(
)
)
+ h
+ h
(
s∑
b i
i=1
(
i∑
a ij
j=1
S m (¯ˆF(k−1) ν )
S m (
¯Ĝ
(k−1)
ν )
)
(k−1)
∆ ˆK mi,ν
∆
ˆL
(k−1)
mi,ν
(k−1)
∆ ˆK mj,ν
∆
ˆL
(k−1)
mj,ν
)
)
(2.48)
. (2.49)
= S m (¯ˆF(k−1) ν ) and S m,(k−1) = S ¯Ĝ m (k−1)
( ¯Ĝ ν ). Similarly for S cmi,(k−1) and S cmi,(k−1) .
ν
¯ˆF ν
¯Ĝ ν
(2.50)
• Let the ε-expansion of error function e (k) (t), d (k) (t) at the k th iteration be the following.
( ) (
e (k) ∑∞
) ( ∑∞
)
m
d (k) =
ν=0 e(k) m,νε ν
∑ ∞
=
ν=0 (y m,ν − ŷ m,ν)ε (k) ν
m
ν=0 d(k) m,νε ν ∑ ∞
ν=0 (z m,ν − ẑ m,ν)ε (k) . (2.51)
ν
Let the ε-expansion of numerical approximations of error functions ê (k) (t), ˆd (k) (t) at the k th iteration be
the following. ( ) (
ê (k) ∑∞ ) ( ∑∞ )
m
ˆd (k)
l=0
=
ê(k) m,l
∑ εl
l=0
∞ (k) =
(ŷ(k+1) m,l
− ŷ (k)
m,l
∑ )εl
∞
m
l=0
ˆd
m,l εl l=0 (ẑ(k+1) m,l
− ẑ (k) . (2.52)
m,l )εl
Combining (2.51) and (2.52), we observe with k, ν ≥ 0, m = 0, · · · M
e (k)
m,ν = ê (k)
m,ν + e (k+1)
m,ν , d (k)
m,ν =
ˆd
(k)
m,ν + d (k+1)
m,ν . (2.53)
11

3 Main results and numerical evidence
In this section, we present the main theoretical results in the form of theorems, and provide numerical evidence
supporting the main theorems. We will provide a rigorous mathematical proof in the next section.
3.1 Main results
The aim of this section is to present convergence results of IDC framework based on backward Euler method
and general implicit R-K method when applied to (1.2).
Theorem 3.1. Consider the stiff system (1.2) with the condition (1.3) holds with initial values y(0), z(0)
admitting a smooth solution. Consider the IDC method constructed with M uniformly distributed quadrature
nodes excluding the left-most point, and backward Euler method for the prediction and correction loops k =
1, · · · K. Then for any fixed constant c > 0 the global error after K correction loops satisfies the following
estimates
e (K)
n
d (K)
n
= y n (K) − y(t n ) = O(H min{K+1,M} ) + O(εH) + O(ε 2 ) + O(ε 3 /H)
= z n (K) − z(t n ) = O(H min{K+1,M} ) + O(εH) + O(ε 2 ) + O(ε 3 /H),
for ε ≤ cH, where H = Mh is one IDC time step as in equation (2.2).
H ≤ H 0 and nH ≤ Const.
(3.1)
The estimates hold uniformly for
Theorem 3.2. Consider the stiff system (1.2) with the condition (1.3) holds with initial values y(0), z(0)
admitting a smooth solution. Consider the IDC method constructed with M uniformly distributed quadrature
nodes excluding the left-most point, an implicit stiffly accurate R-K method of order p (0) , stage order q (0) with
(q (0) < p (0) ) for the prediction step. Apply implicit R-K methods of different classical orders (p (1) , p (2) , . . . , p (K) )
in the correction loops k = 1, · · · K. Assume that each of these implicit R-K methods in the correction loops are
stiffly accurate. Then for any fixed constant c > 0 the global error after K correction loops satisfies the following
estimates
e (K)
n
d (K)
n
= 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)
= 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)
for ε ≤ cH, 1 ≤ ν ≤ q (0) + 1, where s K = ∑ K
k=0 p(k) , and H = Mh is one IDC time step as in equation (2.2).
The estimates hold uniformly for H ≤ H 0 and nH ≤ Const.
Remark 3.3. We note that the estimates (3.1) and (3.2) can be rewritten as
(3.2)
e (K)
n = e (K)
n,0 + εe(K) n,1 + · · · + εν e (K)
n,ν + O(ε ν+1 /H)
d (K)
n = d (K)
n,0 + εd(K) n,1 + · · · + εν d (K)
n,ν + O(ε ν+1 /H)
(3.3)
It represents the global error functions e (K)
n
and d (K)
n
at the K-th correction as an ε-expansion of e (K)
n,ν , ν =
0, 1, · · · , which are the global errors of the IDC implicit R-K method applied to the differential algebraic systems
systems (2.28), (2.29) and (2.31) of different indexs. Then the terms O(· · · ) in (3.1) and (3.2) are the estimates
of such global errors. An estimate of the remainder is given by O(ε ν+1 /H). We will justify these estimates in
the next section.
3.2 Numerical evidence
We present numerical evidence of Theorem 3.1 and Theorem 3.2. Below, we consider the IDC method embedded
with the following implicit methods.
• The first order backward Euler method (BE). The order p = 1 and the stage order q = 1.
• The second order stiffly accurate DIRK method (DIRK2-SA) with the Butcher tableau
γ γ 0
1 1 − γ γ
1 − γ γ
(3.4)
where γ = 1 − √ 2
2
. This method is stiffly accurate with the order p = 2 and the stage order q = 1.
12

• The second order not stiffly accurate midpoint method (DIRK2-NSA)
1/2 1/2
1
(3.5)
This method is not stiffly accurate with the order p = 2 and the stage order q = 1.
• The third order Radau IIA method (Radau) with the Butcher tableau
1/3 5/12 −1/12
1 3/4 1/4
3/4 1/4
(3.6)
This method is stiffly accurate with the order p = 3 and the stage order q = 2.
The indicated order of convergence by the Theorems for the y- z- component in the singular perturbation system
(1.2) are summarized in Table 3.1. Below is a discussion of the Table.
• When the first order backward Euler method is used in both prediction and k correction steps in an
IDC framework with M quadrature points (IDC-BE-M-k), the order of convergence will increase by 1 for
index 1 problem leading to a term of H min(M,k+1) ; the order of convergence for index 2 problem will be
dominated by the stage order of the prediction, leading to a term of εH.
• When the second order stiffly accurate DIRK method is used in both prediction and k correction steps
in an IDC framework with M quadrature points (IDC-DIRK2-SA-M-k), the order of convergence will
increase by 2 for index 1 problem leading to a term of H min(M,2(k+1)) ; the order of convergence for index
2 problem will be dominated by the stage order of the prediction, leading to a term of εH.
• An important ingredient, suggested by the analysis is the property of stiffly accuracy for implicit R-
K method. Such a choice provides a significant benefit for the convergence of the numerical solution,
without which would lead to the divergence of numerical solution. For example, if we consider using the
second order non stiffly accurate DIRK method in both the prediction and k correction steps of an IDC
framework with M quadrature points (IDC-DIRK2-NSA-M-k), divergence results are expected. Note that
in Sect. 4.2, a satisfactory theoretical explanation of this fact is given.
• When the third order stiffly accurate Radau IIA method (with stage order q = 2) is used in prediction
step and the first order backward Euler method is used in k correction steps in an IDC framework with M
quadrature points (IDC-Radau-BE-M-k), the order of convergence will increase by 1 for index 1 problem
leading to a term of H min(M,3+k) ; the order of convergence for index 2 problem will be dominated by the
stage order of the prediction leading to a term of εH 2 .
Table 3.1: Global error predicted by Theorem 3.1 and Theorem 3.2 with H ≫ ε. Note that ‘SA’/‘NSA’ means
stiffly accurate/not stiffly accurate.
Method y−comp z−comp
IDC-BE-M-k H min(M,k+1) + εH H min(M,k+1) + εH
IDC-DIRK2-SA-M-k H min(M,2(k+1)) + εH H min(M,2(k+1)) + εH
IDC-DIRK2-NSA-M-k diverge diverge
IDC-Radau-BE-M-k H min(M,3+k) + εH 2 H min(M,3+k) + εH 2
For numerical verification, we first consider a scalar example [8]
εz ′ = −z + cos(t) (3.7)
with the analytical solution
z(t) =
cos(t) + ε sin(t)
1 + ε 2 + Cexp(−t/ε),
13

where C = z(0) − 1 is determined by the initial condition. For a well-prepared initial condition, let C = 0. This
is a good example to investigate the order of convergence for the ε 1 term in equation (1.22), as the error for
ε 0 is 0. Indeed, for stiff parameter ε = 10 −6 only a region of first order convergence is observed for the stiffly
accurate backward Euler method where the global and local error given for the z-component in Theorem 1.5 is
O(εH). Figure 3.1 gives the one step error (local error) and global error of backward Euler method; expected
O(εH) is observed. We also test the IDC embedded with second order but not stiffly accurate midpoint method
with three quadrature nodes in Figure 3.2. Divergence is observed when time step is large compared to ε if an
IDC-correction is performed.
Figure 3.1: Scalar example. Local, i.e. one step error (left plot) and global error at T = 0.5 (right plot) of
backward Euler method. O(ɛH) is observed in both plots.
Then we consider the van der Pol equation [8] with the well-prepared initial data up to O(ε 3 )
{ {
y ′ = z
y(0) = 2
εz ′ = (1 − y 2 )z − y , z(0) = − 2 3 + 10
81 ε − 292
(3.8)
2187 ε2
• The numerical results of the IDC methods embedded with the first order stiffly accurate backward Euler
method in both prediction and two correction steps are presented in the upper row of Figure 3.3. The
order of convergence for ε 0 term would increase with the correction loops. The ε 1 term of error behaves
like O(εH) for both y- z-components.
• The numerical results of IDC methods embedded with the second order stiffly accurate DIRK method
in both prediction and one correction step are presented in the middle row of Figure 3.3. The order of
convergence for ε 0 term would increase with second order with the correction loop. The ε 1 term of error
behaves like O(εH) for both y- z-components.
• The numerical results of IDC methods embedded with the third order stiffly accurate Radau IIA method
in the prediction step and the first order backward Euler method in two correction steps are presented in
Figure 3.2: Scalar example. Global error (T = 0.1) of the IDC-second order DIRK method that is not stiffly
accurate with three quadrature points and one correction step. ε = 10 −4 .
14

the bottown row of Figure 3.3. The order of convergence for ε 0 term would increase with first order with
the correction loop and is observed to be O(H 5 ). The ε 1 term of error behaves like O(εH 2 ) for both y-
z-components.
Numerical observations in Figure 3.3 are consistent with Theorem 3.1 and 3.2 and Table 3.1. Especially,
it is observed that the IDC method embedded with implicit R-K methods exhibits order reduction both in the
differential and algebraic component. They produce an estimate for the y and z component of the following
form
e (k)
n = y n − y (k) (t n ) = O(H s k
) + εO(H q(0) ) + ε 2 O(H q(0) −1 ) + · · · ,
d (k)
(3.9)
n = z n − z (k) (t n ) = O(H s k
) + εO(H q(0) ) + ε 2 O(H q(0) −1 ) + · · · ,
after k correction steps. For example, in Figure 3.3, we observe a behavior like e (k)
n = O(H 3 ) + O(εH) + O(ε 2 )
where the term O(ε 2 s
) can be neglected since ε ≪ H. Furthermore, if the step size H > ε k −q (0) , O(H s k
)
is dominant; otherwise the term O(εH q(0) ) is observed. A singularity may appear in the neighborhood of
1
s
H ≈ ε k −q (0)
where we have a cancellation of error terms between O(H s k
) and εO(H q(0) ) with error constants
of an opposite sign, see for example Figure 3.3.
4 Proofs of main results
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
the proof for Theorem 3.1 first to demonstrate the basic ingredients of the proof. The proof is then generalized
for Theorem 3.2. Our error estimate is based on the ε-expansion outlined in Section 2.4.
4.1 Error estimates for Theorem 3.1.
We perform local error estimate for Theorem 3.1 by four Lemmas. We again note that since h = H M
, we use
O(h p ) and O(H p ) interchangeably below in our proof. We then prove the global error estimate in Proposition 4.8
based on the four Lemmas.
Lemma 4.1. (Prediction step, ε 0 ) Suppose that the reduce system (1.4) satisfies (1.3) and that the initial
values are consistent. Consider the backward Euler method for the prediction step (2.33). Then the numerical
solutions have the following local error estimate at each interior node of IDC t m with m = 0, ..., M
and
ŷ (0)
m,0 − y m,0 = O(h 2 ), ẑ (0)
m,0 − z m,0 = O(h 2 ),
g(ŷ (0)
m,0 , ẑ(0) m,0 ) = 0.
Proof. 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)) =
0 and z 0 (t) = G(y 0 (t)). For the numerical solution, we have (2.33), indicating a similar behavior of the numerical
solution
0 = g(ŷ (0)
m,0 , ẑ(0) m,0 ), (4.1)
with m = 0, ..., M. Now by (4.1), we get ẑ (0)
m,0 = G(ŷ(0) m,0 ). This implies that ŷ(0) m,0 represents the numerical
solution of the ordinary differential equation y 0(t) ′ = ˆf(y 0 (t)) with ˆf = . f(y 0 (t), G(y 0 (t))). Then from the
backward Euler method we have for the local truncation error |ŷ (0)
m,0 − y m,0| ≤ C m h 2 with m = 0, ..., M and for
some constant C m independent of H. Therefore, ŷ (0)
m,0 − y m,0 = O(h 2 ). By ẑ (0)
m,0 = G(ŷ(0) m,0 ) and the Lipschitz
condition of G, it follows that ẑ (0)
m,0 − z m,0 = O(h 2 ) with m = 0, ..., M.
Lemma 4.2. (Correction steps: ε 0 ) Under the same assumptions of Lemma 4.1 we consider the backward Euler
method for the correction steps (2.9)-(2.11). Assume that the numerical solutions after k − 1 correction loops
have the local error estimate with m = 0, ..., M
e (k−1)
m,0 = y m,0 − ŷ (k−1)
m,0 = O(h min(k+1,M+1) ), d (k−1)
m,0 = z m,0 − ẑ (k−1)
m,0 = O(h min(k+1,M+1) ), (4.2)
1
15

,
,
,
Figure 3.3: Van der Pol equation. Global error (T = 0.5) of the IDC-BE method with M = 3 quadrature
points and two correction steps (upper row); and of the IDC method with stiffly accurate DIRK with M = 4
quadrature points and one correction step (middle row); and of the IDC method with the third order stiffly
accurate Radau IIA method for prediction and first order backward Euler for two correction steps and with
M = 6 quadrature points (bottom row). ε = 10 −6 .
16

and
g(ŷ (k−1)
m,0 , ẑ (k−1)
m,0 ) = 0.
Then the numerical solutions after k correction loops have the local error estimate at the interior nodes of IDC
with m = 0, ..., M
and
e (k)
m,0 = y m,0 − ŷ (k)
m,0 = O(hmin(k+2,M+1) ), d (k)
m,0 = z m,0 − ẑ (k)
m,0 = O(hmin(k+2,M+1) ), (4.3)
g(ŷ (k)
m,0 , ẑ(k) m,0 ) = 0. (4.4)
Proof. Without loss of generosity and for simplicity, we let k = 1 and assume a fixed M ≥ 1. Consider equation
(2.36) with k = 1 be the numerical scheme for the first correction. From the prediction step (2.33) we have
g(ŷ (0)
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.
From the invertibility condition of the function g z in equation (1.3), we get
ŷ (1)
m+1,0 = ŷ(1)
(1)
(0)
m,0 + h( ˆf(ŷ m+1,0 ) − ˆf(ŷ m+1,0 )) + hSm ( ¯ˆf (0)
0 ), (4.5)
(1)
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)
0 ) = Sm (0) (0)
( ˆf(¯ŷ 0 , G(¯ŷ 0 ))). The scheme
(4.5) of updating ŷ (1)
m,0 can be interpreted as the the applying a correction step in the IDC framework to the
non-stiff ordinary differential equation (1.6). Therefore applying classical results as [17] of IDC frameworks using
backward Euler method applied to a classical ordinary differential equation we can have for the local truncation
error after one correction |y m,0 − ŷ (1)
m,0 | ≤ C mh 3 for some constant C m independent of H and with h ≤ h 0 .
Therefore y m,0 − ŷ (1)
m,0 = O(h3 ). By ẑ (1)
m,0 = G(ŷ(1) m,0 ) and Lipschitz condition of G, we get z m,0 − ẑ (1)
m,0 = O(h3 ),
∀m = 1, · · · M. The estimate for general k > 1 can be proved by mathematical induction in a similar fashion.
Lemma 4.3. (Prediction step, ε 1 ) Assume the condition (1.3) holds and initial values of the differential algebraic
system (2.28) -(2.29) are consistent, then the local error estimate at the interior nodes of IDC of the backward
Euler method (2.33)-(2.35) for the prediction step, holds
ŷ (0)
m,1 − y m,1 = O(h 2 ), ẑ (0)
m,1 − z m,1 = O(h). (4.6)
Proof. The proof is a special case of Theorem 3.4 in Chap. VI with ν = 1 and Lemma 4.4 in Chap. VII of [8].
Lemma 4.4. (Correction steps, ε 1 ) Under the same assumptions of Lemma 4.3 we consider the backward Euler
method for the correction steps (2.9)-(2.11). Assume that the numerical solutions at the interior nodes of IDC
after k correction loops have the local error estimate (4.2) for ε 0 term. Assume after k − 1 correction loops have
the local error estimate
e (k−1)
m,1 = y m,1 − ŷ (k−1)
m,1 = O(h 2 ), d (k−1)
m,1 = z m,1 − ẑ (k−1)
m,1 = O(h), (4.7)
holds for m = 1, · · · M. Then the numerical solutions after k correction loops have the local error at the interior
nodes of IDC
e (k)
m,1 = y m,1 − ŷ (k)
m,1 = O(h2 ), d (k)
m,1 = z m,1 − ẑ (k)
m,1 = O(h), (4.8)
holds for m = 1, · · · M.
Proof. Without loss of generosity and for simplicity, we let k = 1 and assume M ≥ 1 fixed. We prove (4.8) by
mathematical induction w.r.t. m. Especially, we know e (k)
m,1 = d(k) m,1 = 0, with m = 0. We assume (4.8) is valid
for 0, · · · , m. We will prove that (4.8) is valid for m + 1. We consider the ε-expansion of the exact solution.
Integration (2.29) over [t m , t m+1 ] gives
{
ε 1 y m+1,1 = y m,1 + ∫ t m+1
t
:
m
F 1 (τ)dτ,
z m+1,0 = z m,0 + ∫ t m+1
(4.9)
t m
G 1 (τ)dτ,
with F 1 and G 1 defined in (2.29). We consider now
e (1)
m+1,1 = y m+1,1 − ŷ (1)
m+1,1 ,
d(1) m+1,1 = z m+1,1 − ẑ (1)
m+1,1 (4.10)
17

the differences with the exact solutions and numerical ones.
From (2.39) and (2.41) we have
⎧
⎨
Subtract equation (2.37) from equation (4.9) gives
ε 1 :
⎧
⎨
⎩
⎩
∆ˆF (0)
m+1,1 = ( f y ê (0)
m+1,1 + f (0)
z ˆd m+1,1 ) + O(h2 )
∆Ĝ(0) m+1,1 = ( g y ê (0)
m+1,1 + g (0)
z ˆd m+1,1 ) + O(h2 ).
e (1)
m+1,1 = e(1)
(0)
m,1 − h∆ˆF m+1,1 − hSm (¯ˆF(0) 1 ) + ∫ t m+1
t m
F 1 (τ)dτ
d (1)
m+1,0 = d(1) m,0 − h∆Ĝ(0) m+1,1 − hSm (0)
( ¯Ĝ 1 ) + ∫ t m+1
t m
G 1 (τ)dτ.
(4.11)
(4.12)
On the right-hand side of the equations in (4.12) we add and subtract the following quantities: hS m (¯F 1 )
and hS m (Ḡ1), these are the integrals of the M-th degree interpolating polynomials on (t m , F 1 (t m )) M m=1 and
(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
t m
F 1 (τ)dτ−
hS m (¯F 1 ) = O(h M+1 ). By the assumption of local error estimate of (4.2) and (4.7), S m (¯F 1 ) − S m (¯ˆF1 ) and
S m (Ḡ1) − S m ( ¯Ĝ 1 ) are accurate to the order O(h). Thus,
⎧
(
)
⎪⎨ e (1)
m+1,1 = e (1)
m,1 − h f y ê (0)
m+1,1 + f (0)
z ˆd m+1,1 + O(h 2 ),
(
)
(4.13)
⎪⎩ d (1)
m+1,0 = d (1)
m,0 − h g y ê (0)
m+1,1 + g (0)
z ˆd m+1,1 + O(h 2 )
From (2.53) and (4.2), we have
⎧
⎨
⎩
ê (0)
m,1 = ŷ(1) m,1 − ŷ(0) m,1 = e(0) m,1 − e(1) m,1 = −e(1) m,1 + O(h2 ),
ˆd (0)
m,1 = ẑ(1) m,1 − ẑ(0) m,1 = d(0) m,1 − d(1) m,1 = −d(1) m,1 + O(h),
and put it into equation (4.13) gives,
⎧
(
)
⎪⎨ e (1)
m+1,1 = e(1) m,1 + h f y e (1)
m+1,1 + f zd (1)
m+1,1 + O(h 2 ),
(
)
⎪⎩ d (1)
m+1,0 = d(1) m,0 + h g y e (1)
m+1,1 + g zd (1)
m+1,1 + O(h 2 )
(4.14)
Now using the estimate (4.2) about d (1)
m,0 , by the second equation in (4.14) we obtain
d (1)
m+1,1 = −g−1 z g y e (1)
m+1,1 + O(h) (4.15)
with the invertibility of g z . Inserting this into the first equation in (4.14) gives
e (1)
m+1,1 = (1 − h(f y − f z g −1
z g y )) −1 e (1)
m,1 + O(h2 ) (4.16)
Finally e (1)
m+1,1 = O(h2 ) follows from (4.16), and d (1)
m+1,1 = O(h) follows from (4.15). We note that the proof of
the general k is similar.
Remark 4.5. In [4], the IDC with explicit R-K in the prediction and correction loops is incorporated as
a high-order explicit R-K method. Similarly, the IDC-BE can be viewed as an implicit R-K method with
a corresponding Butcher tableau. Below, we present the Butcher tableau for the IDC-BE with one loop of
correction. The Butcher tableau can be generated by a similar fashion if there are more than one correction
loops. The Butcher tableau takes the form
⃗c T Z
⃗c P T
⃗ b
T
1
⃗ b
T
2
(4.17)
18

where ⃗c = 1 M [1, · · · , M]T , Z is a M × M matrix of zeros, T and P are M × M matrices, with
⎡
⎤
1 0 0 . . . 0
T = 1 1 1 0 . . . 0
⎢
M ⎣
.
. . ..
⎥
. . ⎦ .
1 1 1 . . . 1
⎡
P = ⎢
⎣
( ˜S 11 − 1 M ) ⎤
˜S12 . . . ˜S1,M−1 ˜S1,M
( ˜S 21 − 1 M ) ( ˜S 22 − 1 M ) . . . ˜S2,M−1 ˜S2,M
. ⎥
.
. .. .
. ⎦ ,
( ˜S M,1 − 1 M ) ( ˜S M,2 − 1 M ) . . . ( ˜S M,M−1 − 1 M ) ( ˜S M,M − 1 M )
where the term ˜S ij = ∫ t i
t 0
α j (s)ds with α j (s) as defined in equation (2.12). The vectors
⃗ b
T
1 =
((
˜S M,1 − 1 M
) (
, ˜S M,2 − 1 ) (
, · · · , ˜S M,M − 1 ))
M
M ) , ⃗ b
T
2 = 1 (1, 1, · · · , 1).
M
Now by the Butcher table constructed, the following Proposition follows.
Proposition 4.6. The IDC-BE is an implicit stiffly accurate RK method with the matrix
( ) T Z
A =
P T
in (1.10) invertible.
Remark 4.7. In the estimates above, we show that there is no improvement in the order of convergence for
approximating y 1 and z 1 in IDC corrections. This is consistent with our numerical evidence presented in the
previous section. The reason is that both the local and global error for ẑ (k)
1 approximating z 1 in the prediction
and correction steps is of first order. This sets the bottleneck for order increase for the term O(h 2 ) in equation
(4.13).
The Proposition below brings the local estimate from four Lemmas above to a global error estimate for terms
in equation (3.3).
Proposition 4.8. Under the same assumption of Theorem 3.1, then for any fixed constant c > 0, the following
global error estimate for equation (3.3) holds with
e (K)
n,0 = O(Hmin(k+1,M) ), d (K)
n,0 = O(Hmin(k+1,M) )
e (K)
n,1
= O(H), d(K) n,1 = O(H),
for ε ≤ cH and ν ≤ 2. The estimates hold uniformly for H ≤ H 0 and nH ≤ Const.
Proof. From Lemma 4.2, we have g(ŷ (k)
m,0 , ẑ(k) m,0 ) = 0, ∀m. Hence, the IDC-BE method is a state space form
method (for Definition see Chap. VI in [8]). In fact, from the above Proposition, it is also a stiffly accurate
method. By the mathematical induction with respect to k, Lemma 4.1 and 4.2 give the local estimate
y 0 (t 1 ) − ŷ (k)
M,0 = O(Hmin(k+2,M+1) ). (4.18)
With the help of Theorem 3.4 in Chap. II [7] in estimating global error from local error, we have the global
error estimate
e (K)
n,0 = y 0(nH) − ŷ (k)
n,0 = O(Hmin(k+1,M) ).
As IDC-BE method is a stiffly accurate method, z = G(y) for both exact solution and numerical solution. By
the Lipschitz condition of G, we have the global error estimate
d (K)
n,0 = z 0(nH) − ẑ (k)
n,0 = O(Hmin(k+1,M) )
19

Lemma 4.4 gives the local estimate
y 1 (t 1 ) − ŷ (k)
M,1 = O(H2 ). (4.19)
From the Remark 1.4 for stiffly accurate method, and with the help of Theorem 4.5, 4.6 in Chap. VII.4 of [8],
we have the global error estimate for y and z
e (K)
n,1 = y 1(nH) − ŷ (k)
n,1 = O(H), d(K) n,1 = z 1(nH) − ẑ (k)
n,1 = O(H).
4.2 Error estimates for Theorem 3.2
In this subsection, we extend the above result to the general case of using implicit R-K methods in the IDC
framework. We remark that the crucial assumption in Theorem 3.2 is that the implicit RK method is stiffly
accurate. In the case that this property is not satisfied, the method becomes unstable and the numerical solutions
diverge, see Figure 3.2. In order to justify this, from the invertibility of matrix A and by the first formula of
(2.47) we get
(k)
∆ ˆL m,−1 = −A−1 S −→c (¯ĝ (k) ), (4.20)
substituting which into the second formula of (2.47) yields
−b T A −1 S¯c (¯ĝ (k−1) ) + S m (¯ĝ (k−1) ) = 0. (4.21)
Proposition 4.9. Equation (4.21) is automatically satisfied, if the implicit RK method is stiffly accurate in the
IDC framework.
Proof. An implicit RK method is stiffly accurate if
with e s = (0, · · · , 0, 1) T . From (4.21) we get
b T A −1 = e T s , (4.22)
−e T s S¯c (¯ĝ (k−1) ) + S m (¯ĝ (k−1) ) = 0. (4.23)
Since the last row of the spectral integration matrix is s m,k = ∫ t m+c sh
α k (τ)dτ by (4.22) we get c s = 1 and then
∫ tm+c sh
t m
α k (τ)dτ = ∫ t m+1
t m
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
satisfied.
In fact, similar to Prop. 4.6, we have the following Proposition for the IDC embedded with stiffly accurate
implicit R-K methods. The proof is omitted for brevity.
Proposition 4.10. The IDC method embedded with stiffly accurate implicit RK methods is an implicit stiffly
accurate R-K method with a corresponding Butcher Tableau that has the matrix A in (1.10) invertible.
We prove the error estimate in Theorem 3.2 by two Lemmas. Lemma 4.11 is about the local truncation
error estimate in the case ε 0 , i.e., R-K methods applied to the reduced system (1.4) in the correction steps.
Lemma 4.14 is about the local truncation error estimate in the case ε ν with ν ≥ 1.
Lemma 4.11. (The case of ε 0 .) Consider the same assumptions as in Theorem 3.2. Consider the limiting case
of ε = 0. The numerical solutions after k correction loops have the local error estimate at the interior nodes of
IDC with m = 0, · · · M
t m
e (k)
m,0 = O(hmin(s k+1,M+1) ),
d (k)
m,0 = O(hmin(s k+1,M+1) ).
(4.24)
Proof. As done in Lemma 4.4, without loss of generosity and for simplicity, we let k = 1 and assume M ≥ 1.
The proof for the general k can be proved by mathematical induction. We will omit the superscript (k) for
different R-K methods when there is no confusion.
Since the R-K method for the prediction is stiffly accurate, by Remark 1.2, we have R(∞) = 0 and b T A −1 =
e T s . This implies by (1.17) and (1.16) ẑ (0)
m+1,0 = Ẑ(0) ms,0
for all i and, in particular, Ẑ(0) ms,0
= G(Ŷ
(0)
ms,0
and ŷ(0) m+1,0 = Ŷ (0)
ms,0
). Then this gives ẑ(0)
20
m+1,0 = G(ŷ(0) m+1,0 ).
. By (1.15) we get Ẑ(0)
mi,0
= G(Ŷ
(0)
mi,0 )

Now for the correction step k = 1, by the stiffly accurate property of the R-K method applied in the first
(0)
correction step and ¯ĝ 0 = (g(ŷ (0)
1,0 , ẑ(0) 1,0 ), · · · g(ŷ(0)
M,0 , ẑ(0) M,0 )) = ⃗0 from the prediction step, it follows
{
ŷ (1)
m+1,0 = ŷ(1) m,0 + hSm ( ¯ˆf (0)
0 ) + h ∑ s
i=1 b (0)
i∆ ˆK mi,0
g(ŷ (1)
m+1,0 , ẑ(1) m+1,0 ) = 0 (4.25)
The internal stages are given by
{
(1) Ŷ mi,0 = ŷ(1) m,0 + hScmi ( ¯ˆf (0)
0 ) + h ∑ i
j=1 a ij∆
ˆK
(0)
mj,0
g(Ŷ (1)
mi,0 , Ẑ(1) mi,0 ) = 0 (4.26)
Now, from the invertibility of function g z , by (4.26) we get Ẑ(1) mi,0
R-K method reads
Ŷ (1)
mi,0 = ŷ(1) m,0 + h ∑ s
j=1 a (0)
ij∆ ˆK mj,0 + hScmi ( ¯ˆf (0)
ŷ (1)
m+1,0 = ŷ(1) m,0 + h ∑ s
i=1 b i∆
(1)
= G(Ŷ mi,0 ) and ẑ(1) m+1,0 = G(ŷ(1) m+1,0 ). Thus the
0 )
ˆK
(0)
mi,0 + hSm ( ¯ˆf (0)
0 ) (4.27)
and ¯ˆf (0)
0 = (f(ŷ (0)
0,0 , G(ŷ(0) 0,0 )), · · · f(ŷ(0)
M,0 , G(ŷ(0) M,0
)). The scheme (4.27) of updating ŷ(1) m,0 can be interpreted as the
applying a correction step in the IDC framework to the ordinary differential equation (1.6). Therefore applying
classical results of local truncation error as in [4, 5] of IDC frameworks using R-K method applied to a classical
ordinary differential equation, we obtain the local error estimate for m = 0, · · · M
with s 2 = p (0) + p (1) , and
e (1)
m,0 = O(hmin(s2+1,M+1) ), (4.28)
E (1)
m,0 = y 0(t m + c i h) − Ŷ (1)
mi,0 = O(hmin(s1+q(1) +1,M+1) ). (4.29)
By ẑ (1)
m,0 = G(ŷ(1)
(1)
m,0 ), Ẑ(1) mi,0 = G(Ŷ mi,0 ) and using the Lipschitz condition of G, we get
and
d (1)
m,0 = z m,0 − ẑ (1)
m,0 = O(hmin(s2+1,M+1) ).
D (1)
mi,0 = z 0(t m + c i h) − Ẑ(1) mi,0 = O(hmin(s1+q(1) +1,M+1) ),
where q (1) is the stage order for the R-K method applied to the first correction loop. We note that the proof of
the general k is similar.
Remark 4.12. The estimate (4.28) is from [5] via estimating the smoothness of the rescaled error function.
The estimate (4.29) follows a similar fashion.
Remark 4.13. With the estimates in the above Lemma, and from equation (2.44)
where ∆K (k−1)
mi,1
(k−1)
∆ ˆK mi,1 = f y (y mi,0 , z mi,0 )Ê(k−1) mi,1
+ f (k−1)
z(y mi,0 , z mi,0 ) ˆD mi,1 + O(hs k−1+1 ).
.
= ∆K (k−1)
mi,1 + O(hs k−1+1 ) (4.30)
.
= f y (y mi,0 , z mi,0 )Ê(k−1) mi,1 + f z (y mi,0 , z mi,0 )
(k−1) ˆD mi,1
(k)
. Here, we replace Ŷ mi,0 and P cmi (¯ŷ (k−1)
0 ) by
y mi,0 with an error of 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.
Similarly,
where ∆L (k−1)
mi,0
(k−1)
∆ ˆL mi,0 = g y (y mi,0 , z mi,0 )Ê(k−1) mi,1
+ g (k−1)
z(y mi,0 , z mi,0 ) ˆD mi,1 + O(hs k−1+1 ).
.
= ∆L (k−1)
mi,0 + O(hs k−1+1 ), (4.31)
.
= g y (y mi,0 , z mi,0 )Ê(k−1) mi,1 + g (k−1)
z(y mi,0 , z mi,0 ) ˆD mi,1 .
21

Lemma 4.14. (The case of 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
where Ē(0) 1 = (E (0)
m1,1 , · · · , E(0) ms,1
Ē (0)
1 = e (1)
(0)
m,11 − hA∆ ¯K 1 + O(h q(0) +1 )
¯D (0)
0 = d (1)
(0)
m,01 − hA∆ ¯L 0 + O(h q(0) +1 )
(4.42)
(0)
), ¯D 0 = (D (0)
m1,0 , · · · , D(0) ms,0 ), where s is the number of internal stages in a R-K
method and 1 = (1, 1, · · · , 1) T is a vector of size s. Then from the second equation in (4.42) and using (4.24),
we get
A(g y (t m + c i h)E (1)
mi,1 + g z(t m + c i h)D (1)
mi,1 ) = O(hq(0) ) (4.43)
where we replace Ê(0) mi,1 by E(1) mi,1 with +2 O(hq(0) ) error, and replace
(4.41). Thus, from the invertibility of A
(0) ˆD mi,1 by D(1) mi,1 with O(hq(0) ) error due to
D (1)
mi,1 = −(g−1 z g y )(t m + c i h)E (1)
mi,1 + O(hq(0) ), (4.44)
for all mi. Plug the above equation (4.44) into the first equation of (4.42)
∆K (0)
mi,1 = (f y − f z g −1
z g y )(t m + c i h)E (1)
mi,1 + O(hq(0) ). (4.45)
Next, we prove the local error e (1)
m,1 = +1 O(hq(0) ) by mathematical induction w.r.t. m. Especially, we would like
to show that e (1)
m+1,1 = +1 O(hq(0) ), if we assume the local error e (1)
l,1 = +1 O(hq(0) ), ∀l ≤ m. To show this, we
plug equation (4.45) into the first equation of the vectorial form (4.42) and obtain E (1)
mi,1 = +1 ). Thus,
O(hq(0)
from (4.44),
D (1)
mi,1 = ). (4.46)
O(hq(0)
From (4.45), ∆K (0)
mi,1 = O(hq(0) ). Plug this estimate into the first equation of (4.39), we obtain the desired
estimate of
e (1)
m+1,1 = O(hq(0) +1 ). (4.47)
Now in order to prove the estimate d (1)
m,1 = ), we start to considering equation (2.25). Since the R-K
O(hq(0)
method is stiffly accurate, from Remark 2.5, we have ẑ (1)
m+1,1 = Ẑ(1) ms,1 . Hence
z 1 (t m+1 ) − ẑ (1)
m+1,1 = d(1) m+1 = D(1)
(4.46)
ms,1 = O(h q(0) ), m = 0, · · · M − 1 (4.48)
The above proof can be generalized for the IDC method with different RK methods applied to k correction
loops. The local error estimate at the interior nodes of the IDC method with m = 0, · · · M is
e (k)
m,1 = O(hq(0) +1 ), d (k)
m,1 = O(hq(0) ).
We have thus proved the case ν = 1. The general estimates for ν > 1 (4.32) can be obtained in a similar
fashion to the case of ν = 1, as in the Theorem 3.4 in Chap.VI of [8], then we have for the local errors
e (k)
m,ν = O(h q(0) +2−ν ), d (k)
m,ν = O(h q(0) +1−ν ).
Remark 4.15. We remark that we can not improve the estimate of the global error for the y-component as
done in Theorem 3.4 in [8] for high-indices. Indeed the reason of this lost of accuracy for the y-component is
represented of the evaluation of the integrals by (2.50). These integrals contain the estimate of the algebraic
variable z obtained in the prediction step that reduces the order of the differential variable y, then a definition
of a new variable as done in Theorem 3.4 in [8] can not produce any benefit to the y-component. This can be
seen in the evaluation from (4.37) to (4.39) due to (4.38). We note that a similar conclusion for the remainder
can be drawn.
Similar to Proposition 4.8, we have the following Proposition for global error estimates of IDC implicit R-K
method. The proof follows from Lemma 4.11 and 4.14, in a similar spirit to that of Proposition 4.8. We omit
the proof for brevity.
Proposition 4.16. Under the same assumption of Theorem 3.2, then for any fixed constant c, the following
global error estimate for equation (3.3) holds with
e (K)
n,0 = O(Hmin(s k,M) ), d (K)
n,0 = O(Hmin(s k,M) )
e (K)
n,ν = O(H q(0) +1−ν ), d (K)
n,1 = O(Hq(0) +1−ν ),
for ε ≤ cH and 1 ≤ ν ≤ q (0) + 1. The estimates hold uniformly for H ≤ H 0 and nH ≤ Const.
23

4.3 Proof of main Theorems and estimation of the remainder
The proof of Theorem 3.1 and 3.2 follow directly from Prop. 4.8, 4.16 and the estimate about the remainder in
Prop. 4.17 below. We justify that the term O(ε ν+1 /H) representing the remainder in the expansion (3.3).
Proposition 4.17. Under the same hypothesis as those of theorem 3.2 for any fixed constant c > 0, the global
error satisfies for ε ≤ cH
q (0) ∑+1
e (k)
n = e (k)
n,ν + O(ε ν+1 /H),
ν=0
q (0) ∑+1
d (k)
n = d (k)
n,ν + O(ε ν+1 /H). (4.49)
ν=0
Here e (k)
n,ν = y n,ν − ŷ n,ν, (k) d (k)
n,ν = z n,ν − ẑ n,ν (k) are the global errors of the IDC method applied to (2.28), (2.29) and
(2.31). The estimates (4.49) hold uniformly for H ≤ H 0 and nH ≤ Const.
Proof. In order to prove the theorem we consider the truncated series for e (k)
n
∑
q (0) +1
e (k)
n = e (k)
n,ν,
ν=0
Using (2.51), then the statement (4.49) is equivalent to prove
e (k)
n
∑
q (0) +1
d (k)
n =
ν=0
and d (k)
n
as
d (k)
n,ν (4.50)
− e (k)
n = O(εν+1 /H), d (k)
n − d (k)
n = O(εν+1 /H). (4.51)
In this situation, the same conclusions of Theorem 3.8 in Chap. VI of [8] hold. The use of stiffly accurate implicit
Runge-Kutta methods with matrix A invertible from Prop. 4.10 guarantees the hypothesis of Theorem 3.6 in
Chap. I of [8]. It is worth commenting that the estimate for the y component in (4.51) is not optimal as in
Theorem 3.8 in [8]. We obtain this estimate by considering remark 4.15, and by the estimates in Prop. 4.16.
5 Conclusion
Global errors are studied for an IDC framework with uniform distribution of quadrature points excluding the
leftmost point, embedded with high order implicit R-K method for a class of singular perturbation problems.
Two Theorems on the estimate of global error in the form of an ɛ expansion are presented and proved. The
asymptotic analysis enables us to understand the phenomenon of order reduction for IDC methods when applied
to stiff problems. Numerical results on van der Pol equations are presented to reveal the convergence results.
In the future, our goal is to extend a similar analysis in order to study the global error and stability property
of IDC framework embedded with high order implicit-explicit (IMEX) R-K methods.
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