BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...
G.I. SHISHKIN: Grid Approximation of Parabolic Equations with Nonsmooth Initial Condition in the Presence of Boundary Layers of Different Types ✬ ✫ Grid Approximation of Parabolic Equations with Nonsmooth Initial Condition in the Presence of Boundary Layers of Different Types ∗ Grigory I. Shishkin Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Yekaterinburg 620219, Russia shishkin@imm.uran.ru A Dirichlet problem for a singularly perturbed parabolic equation with the perturbation vector-parameter ε, ε = (ε1, ε2), is considered on a semiaxis. The highest derivative of the equation and also the first derivative with respect to x contain respectively the parameters ε1 and ε2, which take arbitrary values in the half-open interval (0, 1] and the segment [−1, 1]. Depending on the parameter ε2, the type of the equation may be reaction-diffusion or convectiondiffusion. The first order derivative of the initial function has a discontinuity of the first kind at the point x0. For small values of the parameter ε1, a boundary layer appears in a neighbourhood of the lateral part of the domain boundary. Depending on the ratio between the parameters ε1 and ε2, these layers may be regular, parabolic or hyperbolic (characteristic scales of these boundary layers also depend on the ratio between ε1 and ε2). In a neighbourhood of the set S γ , that is, the characteristic of the reduced equation outgoing from the point (x0, 0), the parabolic transient layer arises. Using the method of piecewise uniform meshes condensing in a neighbourhood of the layer, we construct a special difference scheme that converges ε-uniformly. Numerical methods for problems with different types of boundary layers for elliptic convectiondiffusion equations in the case of sufficiently smooth boundary data are studied, e.g., in [1]. References [1] G.I. Shishkin, “Grid approximation of a singularly perturbed elliptic equations with convective terms in the presence of various boundary layers”, Comput. Maths. Math. Phys., 45 (1), 104–119 (2005). ∗ This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578, 04–01–89007–NWO a), by the Dutch Research Organisation NWO under grant No. 047.016.008 and by the Boole Centre for Research in Informatics, National University of Ireland, Cork. Speaker: SHISHKIN G.I. 48 BAIL 2006 1 ✩ ✪
L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for a Quasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of the Third-Kind Boundary Condition ✬ ✫ A Difference Scheme of Improved Accuracy for a Quasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of the Third Kind Boundary Condition ∗ Lidia P. Shishkina and Grigory I. Shishkin Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Yekaterinburg 620219, Russia Lida@convex.ru and shishkin@imm.uran.ru A boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion equation on a strip is considered. The third kind boundary condition admitting both Dirichlet and Neumann conditions is given on the domain boundary. For small values of the perturbation parameter ε, a boundary layer arises in a neighbourhood of the outflow part of the boundary. For such a problem, the base (nonlinear) difference scheme constructed by classical approximations of the problem on piecewise uniform meshes condensing in the layer converges ε-uniformly with an order of accuracy not higher than 1. Our aim is for this boundary value problem to construct grid approximations that converge ε-uniformly with an order of convergence close to two. Using the�Richardson technique, we construct a (nonlinear) scheme that converges ε-uniformly at the rate O N −2 1 ln 2 N1+N −2 � 2 , where N1+1 is the number of nodes in the mesh with respect to x1 and N2+1 is the number of mesh points on a unit interval along the x2-axis. Based on the nonlinear Richardson scheme, a linearized iterative scheme is constructed where the nonlinear term is computed using the unknown function taken at the previous iteration. The solution of this iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of a geometry progression ε-uniformly with respect to the number of iterations. Thus, the number of iterations required for solving the problem (as well as the accuracy of the resulting solution) is independent of the parameter ε. The use of lower and upper solutions of the linearized iterative Richardson scheme as a stopping criterion allows us during the computational process to define a current iteration under which the same ε-uniform accuracy of the solution is achieved as for the nonlinear Richardson scheme. To construct the improved scheme, the technique developed in [1]–[4] for a Dirichlet problem is applied. References [1] G.I. Shishkin, “The method of increasing the accuracy of solutions of difference schemes for parabolic equations with a small parameter at the highest derivative”, USSR Comput. Maths. Math. Phys., 24 (6), 150–157 (1984). [2] G.I. Shishkin, ”Finite-difference approximations of singularly perturbed elliptic equations”, Comp. Math. Math. Phys., 38 (12), 1909–1921 (1998). [3] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singularly perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030– 1039 (2005). ∗ This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578, 04–01–89007–NWO a) and by the Dutch Research Organisation NWO under grant No. 047.016.008. Speaker: SHISHKINA, L.P. 49 BAIL 2006 1 ✩ ✪
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G.I. SHISHKIN: Grid Approximation <strong>of</strong> Parabolic Equations with Nonsmooth Initial<br />
Condition in the Presence <strong>of</strong> Bo<strong>und</strong>ary Layers <strong>of</strong> Different Types<br />
✬<br />
✫<br />
Grid Approximation <strong>of</strong> Parabolic Equations with Nonsmooth Initial<br />
Condition in the Presence <strong>of</strong> Bo<strong>und</strong>ary Layers <strong>of</strong> Different Types ∗<br />
Grigory I. Shishkin<br />
<strong>Institut</strong>e <strong>of</strong> Mathematics and Mechanics,<br />
Ural Branch <strong>of</strong> Russian Academy <strong>of</strong> Sciences,<br />
Yekaterinburg 620219, Russia<br />
shishkin@imm.uran.ru<br />
A Dirichlet problem for a singularly perturbed parabolic equation with the perturbation<br />
vector-parameter ε, ε = (ε1, ε2), is considered on a semiaxis. The highest derivative <strong>of</strong> the<br />
equation and also the first derivative with respect to x contain respectively the parameters<br />
ε1 and ε2, which take arbitrary values in the half-open interval (0, 1] and the segment [−1, 1].<br />
Depending on the parameter ε2, the type <strong>of</strong> the equation may be reaction-diffusion or convectiondiffusion.<br />
The first order derivative <strong>of</strong> the initial function has a discontinuity <strong>of</strong> the first kind at<br />
the point x0.<br />
For small values <strong>of</strong> the parameter ε1, a bo<strong>und</strong>ary layer appears in a neighbourhood <strong>of</strong> the<br />
lateral part <strong>of</strong> the domain bo<strong>und</strong>ary. Depending on the ratio between the parameters ε1 and<br />
ε2, these layers may be regular, parabolic or hyperbolic (characteristic scales <strong>of</strong> these bo<strong>und</strong>ary<br />
layers also depend on the ratio between ε1 and ε2).<br />
In a neighbourhood <strong>of</strong> the set S γ , that is, the characteristic <strong>of</strong> the reduced equation outgoing<br />
from the point (x0, 0), the parabolic transient layer arises.<br />
Using the method <strong>of</strong> piecewise uniform meshes condensing in a neighbourhood <strong>of</strong> the layer,<br />
we construct a special difference scheme that converges ε-uniformly.<br />
Numerical methods for problems with different types <strong>of</strong> bo<strong>und</strong>ary layers for elliptic convectiondiffusion<br />
equations in the case <strong>of</strong> sufficiently smooth bo<strong>und</strong>ary data are studied, e.g., in [1].<br />
References<br />
[1] G.I. Shishkin, “Grid approximation <strong>of</strong> a singularly perturbed elliptic equations with convective<br />
terms in the presence <strong>of</strong> various bo<strong>und</strong>ary layers”, Comput. Maths. Math. Phys.,<br />
45 (1), 104–119 (2005).<br />
∗ This research was supported in part by the Russian Fo<strong>und</strong>ation for Basic Research (grants No. 04-01-00578,<br />
04–01–89007–NWO a), by the Dutch Research Organisation NWO <strong>und</strong>er grant No. 047.016.008 and by the Boole<br />
Centre for Research in Informatics, National University <strong>of</strong> Ireland, Cork.<br />
Speaker: SHISHKIN G.I. 48 <strong>BAIL</strong> <strong>2006</strong><br />
1<br />
✩<br />
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