Gregori Shishkin

High-order time-accuracy schemes for parabolic singular perturbation problems with convection

Abstract - The first boundary value problem for a singularly perturbed parabolic PDE with convection is considered on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh condensing in the boundary layer, which gives an e-uniformly convergent difference scheme. The order of convergence for such a scheme is exactly one and close to one up to a small logarithmic factor with respect to the time and space variables, respectively. In this paper we construct high-order time-accurate e-uniformly convergent schemes by a defect-correction technique. The efficiency of the new defect-correction scheme is confirmed by numerical experiments.

Difference Scheme of the Solution Decomposition Method for a Singularly Perturbed Parabolic Convection-Diffusion Equation

For a Dirichlet problem for an one-dimensional singularly perturbed parabolic convection-diffusion equation, a difference scheme of the solution decomposition method is constructed. This method involves a special decomposition based on the asymptotic construction technique in which the regular and singular components of the grid solution are solutions of grid subproblems solved on uniform grids, moreover, the coefficients of the grid equations do not depend on the singular component of the solution unlike the fitted operator method. The constructed scheme converges in the maximum norm \(\varepsilon \)-uniformly (i.e., independent of a perturbation parameter \(\varepsilon \), \(\varepsilon \in (0,1]\)) at the rate \(\mathcal{O}\left ({N}^{-1}\ln N + N_{0}^{-1}\right )\) the same as a scheme of the condensing grid method on a piecewise-uniform grid (here N and N 0 define the numbers of the nodes in the spatial and time meshes, respectively).

On Numerical Experiments with Central Difference Operators on Special Piecewise Uniform Meshes for Problems with Boundary Layers

Singularly perturbed second order elliptic equations with boundary layers are considered. Numerical methods composed of central difference operators on special piece-wise uniform meshes are constructed for the above problems. Numerical results are obtained which show that these methods give approximate solutions with error estimates that are independent of the singular perturbation parameter.

On the Design of Piecewise Uniform Meshes for Solving Advection-Dominated Transport Equations to a Prescribed Accuracy

The numerical performance of numerical methods specifically designed for singularly perturbed partial differential equations is examined. Numerical methods whose solutions have an accuracy independent of the small parameter are called e-uniform methods. In this paper, the advantages of using an e-uniform numerical method are discussed.

Numerical results for advection‐dominated heat transfer in a moving fluid with a non‐slip boundary condition

This paper is concerned with the laminar transfer of heat by forced convection where the velocity profile is taken to be parabolic. In the advection dominated case the problem is described mathematically by a singularly perturbed boundary value problem with a non‐slip condition. It has been established both theoretically and computationally that numerical methods composed of upwind finite difference operators on special piecewise uniform meshes have the property that they behave uniformly well, regardless of the magnitude of the ratio of the advection term to the diffusion term. A variety of choices of special piecewise uniform mesh is examined and it is shown computationally that these lead to numerical methods also sharing this property. These results validate a previous theoretical result which is quoted.