Grigorii I. Shishkin

Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter @e, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates @e-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the @e-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

In this paper, parameter-uniform numerical methods for a class of singularly perturbed parabolic partial differential equations with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The solution is decomposed into a sum of regular and singular components. A numerical algorithm based on an upwind finite difference operator and an appropriate piecewise uniform mesh is constructed. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

Robust novel high‐order accurate numerical methods for singularly perturbed convection‐diffusion problems 1

Abstract For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy.

A class of singularly perturbed semilinear differential equations with interior layers

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

A class of singularly perturbed quasilinear differential equations with interior layers

A class of singularly perturbed quasilinear differential equations with discontinuous data is examined. In general, interior layers will appear in the solutions of problems from this class. A numerical method is constructed for this problem class, which involves an appropriate piecewise-uniform mesh. The method is shown to be a parameter-uniform numerical method with respect to the singular perturbation parameter. Numerical results are presented, which support the theoretical results.

High-order time-accurate parallel schemes for parabolic singularly perturbed problems with convection

Abstract The first boundary value problem for a singularly perturbed parabolic equation of convection-diffusion type on an interval is studied. For the approximation of the boundary value problem we use earlier developed finite difference schemes, ɛ-uniformly of a high order of accuracy with respect to time, based on defect correction. New in this paper is the introduction of a partitioning of the domain for these ɛ-uniform schemes. We determine the conditions under which the difference schemes, applied independently on subdomains may accelerate (ɛ-uniformly) the solution of the boundary value problem without losing the accuracy of the original schemes. Hence, the simultaneous solution on subdomains can in principle be used for parallelization of the computational method.

A robust method of improved order for convection-diffusion problems in a domain with characteristic boundaries

We consider a singularly perturbed two-dimensional convection-diffusion problem in a rectangle with flow parallel to the x-axis. On the inflow and outflow boundaries Dirichlet type conditions are imposed and on the characteristic boundaries regular Robin type conditions (including the possibility of Neumann conditions) are given. For small values of the parameter e, regular (strong) and parabolic (weak) layers appear. In this case the parabolic layers do not involve the first term of the asymptotic expansion. For the proposed problem, e-uniformly convergent methods are considered. Note, that for the upwind finite difference scheme the order of e-uniform convergence does not exceed one. Using the High Order Compact (HOC) standard technique, we construct a classical method on piecewise uniform Shishkin meshes having uniform convergence with order 3/2 in the maximum norm. Numerical results are presented and discussed.

Grid approximation of singularly perturbed parabolic reaction‐diffusion equations with piecewise smooth initial‐boundary conditions

Abstract A Dirichlet problem is considered for a singularly perturbed parabolic reaction–diffusion equation with piecewise smooth initial‐boundary conditions on a rectangular domain. The higher‐order derivative in the equation is multiplied by a parameter ϵ 2; ϵ ? (0, 1]. For small values of ϵ, a boundary and an interior layer arises, respectively, in a neighbourhood of the lateral part of the boundary and in a neighbourhood of the characteristic of the reduced equation passing through the point of nonsmoothness of the initial function. Using the method of special grids condensing either in a neighbourhood of the boundary layer or in neighbourhoods of the boundary and interior layers, special ϵ‐uniformly convergent difference schemes are constructed and investigated. It is shown that the convergence rate of the schemes crucially depends on the type of nonsmoothness in the initial–boundary conditions.

High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions

Abstract The boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter ε. The order of convergence for the known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct ε-uniformly convergent schemes of highorder time-accuracy. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. A new original technigue for experimental studying of convergence orders is developed for the cases where the orders of convergence in the x-direction and in the t-direction can be substantially different.

Conservative numerical method for a system of semilinear singularly perturbed parabolic reaction‐diffusion equations

Abstract On a vertical strip, a Dirichlet problem is considered for a system of two semilinear singularly perturbed parabolic reaction‐diffusion equations connected only by terms that do not involve derivatives. The highest‐order derivatives in the equations, having divergent form, are multiplied by the perturbation parameter e ; ϵ ∈ (0,1]. When ϵ → 0, the parabolic boundary layer appears in a neighbourhood of the strip boundary. Using the integro‐interpolational method, conservative nonlinear and linearized finite difference schemes are constructed on piecewise‐uniform meshes in the x 1 ‐axis (orthogonal to the boundary) whose solutions converge ϵ‐uniformly at the rate O (N1−2 ln2 N 1 + N 2 −2 + N 0 −1). Here N 1 + 1 and N 0 + 1 denote the number of nodes on the x 1‐axis and t‐axis, respectively, and N 2 + 1 is the number of nodes in the x 2‐axis on per unit length.