G. I. Shishkin

A numerical method for a system of singularly perturbed reaction-diffusion equations

A Dirichlet problem for a system of two coupled singularly perturbed reaction-diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results.

Difference methods for singular perturbation problems

Preface Part I: Grid Approximations of Singular Perturbation Partial Differential Equations Introduction Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Smooth Boundaries Boundary Value Problems for Elliptic Reaction-Diffusion Equations in Domains with Piecewise-Smooth Boundaries Generalizations for Elliptic Reaction-Diffusion Equations Parabolic Reaction-Diffusion Equations Elliptic Convection-Diffusion Equations Parabolic Convection-Diffusion Equations Part II: Advanced Trends in epsilon Uniformly Convergent Difference Methods Grid Approximations of Parabolic Reaction-Diffusion Equations with Three Perturbation Parameters Application of Widths for Construction of Difference Schemes for Problems with Moving Boundary Layers High-Order Accurate Numerical Methods for Singularly Perturbed Problems A Finite Difference Scheme on a priori Adapted Grids for a Singularly Perturbed Parabolic Convection-Diffusion Equation On Conditioning of Difference Schemes and Their Matrices for Singularly Perturbed Problems Approximation of Systems of Singularly Perturbed Elliptic Reaction-Diffusion Equations with Two Parameters Survey References

A parameter-uniform Schwarz method for a singularly perturbed reaction-diffusion problem with an interior layer

Abstract In this paper we consider numerical methods for a singularly perturbed reaction–diffusion problem with a discontinuous source term. We show that such a problem arises naturally in the context of models of simple semiconductor devices. We construct a numerical method consisting of a standard finite difference operator and a non-standard piecewise-uniform mesh. The mesh is fitted to the boundary and interior layers that occur in the solution of the problem. We show by extensive computations that, for this problem, this method is parameter-uniform in the maximum norm, in the sense that the numerical solutions converge in the maximum norm uniformly with respect to the singular perturbation parameter.

A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers

The problem of constructing a parameter-uniform numerical method for a singularly perturbed self-adjoint ordinary differential equation is considered. It is shown that a suitably designed discrete Schwarz method, based on a standard finite difference operator with a uniform mesh on each subdomain, gives numerical approximations which converge in the maximum norm to the exact solution, uniformly with respect to the singular perturbation parameter. This parameter-uniform convergence is shown to be essentially second order. That this new discrete Schwarz method is efficient in practice is demonstrated by numerical experiments.

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

Abstract We study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.

Grid approximation of singularly perturbed parabolic convection-diffusion equations subject to a piecewise smooth initial condition

AbstractA boundary value problem for a singularly perturbed parabolic convection-diffusion equation on an interval is considered. The higher order derivative in the equation is multiplied by a parameter ɛ that can take arbitrary values in the half-open interval (0, 1]. The first derivative of the initial function has a discontinuity of the first kind at the point x0. For small values of ɛ, a boundary layer with the typical width of ɛ appears in a neighborhood of the part of the boundary through which the convective flow leaves the domain; in a neighborhood of the characteristic of the reduced equation outgoing from the point (x0, 0), a transient (moving in time) layer with the typical width of ɛ1/2 appears. Using the method of special grids that condense in a neighborhood of the boundary layer and the method of additive separation of the singularity of the transient layer, special difference schemes are designed that make it possible to approximate the solution of the boundary value problem ɛ-uniformly on the entire set

A Richardson scheme of the decomposition method for solving singularly perturbed parabolic reaction-diffusion equation

\bar G

The convergence of classical schwarz methods applied to convection-diffusion problems with regular boundary layers

, approximate the diffusion flow (i.e., the product ɛ(ϱ/ϱx)u(x, t)) on the set