This article reviews some of the salient features of the piecewiseuniform Shishkin mesh. The central analytical techniques involved in the associated numerical analysis are explained via a particular class of singularly perturbed differential equations. A detailed discussion of the Shishkin solution decomposition is included. The generality of the numerical approach introduced by Shishkin is highlighted. The impact of Shishkin’s ideas on the field of singularly perturbed differential is assessed in this selective review of his research output over the past thirty years.

/pdf/shishkin-meshes-in-the-numerical-solution-of-singularly-nfspqxxp54.pdf

Shishkin meshes in the numerical solution of singularly perturbed differential equations

In this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of x=0. First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is ϵ-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.

An ϵ-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection–diffusion problem

This article studies the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of this problem exhibits a boundary layer in the neighborhood of left boundary of the domain. We discretize the domain by piecewise-uniform Shishkin mesh in the spatial direction and uniform mesh in the temporal direction. The numerical scheme consists of the implicit-Euler scheme for the time derivative and upwind finite difference scheme for the spatial derivatives. By applying the Richardson extrapolation technique, we improve the order of accuracy of the numerical solution from $$O (N^{-1} \ln ^2 N + \varDelta t)$$
 to $$O (N^{-2} \ln ^2 N + \varDelta t^2)$$
 (measured in the discrete supremum norm), where $$N+1$$
 mesh points are used in spatial direction and $$\varDelta t$$
 is the step size in the temporal direction. Error estimates are derived. Some numerical experiments are carried out to validate the theoretical results.

Second-Order Uniformly Convergent Richardson Extrapolation Method for Singularly Perturbed Degenerate Parabolic PDEs

A parameter-uniform grid equidistribution method for singularly perturbed degenerate parabolic convection–diffusion problems

A singularly perturbed parabolic equation of convection-diffusion type with an interior layer in the initial condition is studied. The solution is decomposed into a discontinuous regular component, a con- tinuous outflow boundary layer component and a discontinuous interior layer component. A priori parameter-explicit bounds are derived on the derivatives of these three components. Based on these bounds, a parameter-uniform Shishkin mesh is constructed for this problem. Numerical analysis is presented for the associated numerical method, which concludes by showing that the numerical method is a parameter- uniform numerical method. Numerical results are presented to illus- trate the theoretical bounds on the error established in the paper.

A singularly perturbed convection-diffusion problem with a moving interior layer ∗

A singularly perturbed time-dependent convection-diffusion problem is examined on non-rectangular domains The nature of the boundary and interior layers that arise depends on the geometry of the domains For problems with different types of layers, various numerical methods are constructed to resolve the layers in the solutions and the numerical solutions are shown to converge independently of the singular perturbation parameter

Singularity Perturbed Parabolic Problems on Non-rectangular Domains

A system of three coupled, non-linear, singularly perturbed differential equations, which arise in the modelling of plasma sheaths, is studied. For one of the component equations theoretical results are given, which show that a numerical method based on a simple linearisation technique, a piecewise-uniform mesh of Shishkin type and a standard upwind finite difference operator generate parameter-uniform numerical approximations. Motivated by this, a numerical method for generating approximations to the solution of the full system of equations is constructed. Numerical results are presented, which indicate that this method is parameter uniform.

R. K. Dunne

Papers

Singularity Perturbed Parabolic Problems on Non-rectangular Domains

A numerical method for a singular perturbation problem arising in the modelling of plasma sheaths