L.P. Shishkina

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 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’.

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.

Approximation of the solution and its derivative for the singularly perturbed Black-Scholes equation with nonsmooth initial data

A problem for the black-Scholes equation that arises in financial mathematics is reduced, by a transformation of variables, to the Cauchy problem for a singularly perturbed parabolic equation with the variables x, t and a perturbation parameter ɛ, ɛ ∈ (0, 1]. This problem has several singularities such as the unbounded domain, the piecewise smooth initial function (its first-order derivative in x has a discontinuity of the first kind at the point x = 0), an interior (moving in time) layer generated by the piecewise smooth initial function for small values of the parameter ɛ, etc. In this paper, a grid approximation of the solution and its first-order derivative is studied in a finite domain including the interior layer. On a uniform mesh, using the method of additive splitting of a singularity of the interior layer type, a special difference scheme is constructed that allows us to ɛ-uniformly approximate both the solution to the boundary value problem and its first-order derivative in x with convergence orders close to 1 and 0.5, respectively. The efficiency of the constructed scheme is illustrated by numerical experiments.

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

For the one-dimensional singularly perturbed parabolic reaction-diffusion equation with a perturbation parameter ɛ, where ɛ ∈ (0, 1], the grid approximation of the Dirichlet problem on a rectangular domain in the (x, t)-plane is examined. For small ɛ, a parabolic boundary layer emerges in a neighborhood of the lateral part of the boundary of this domain. A new approach to the construction of ɛ-uniformly converging difference schemes of higher accuracy is developed for initial boundary value problems. The asymptotic construction technique is used to design the base decomposition scheme within which the regular and singular components of the grid solution are solutions to grid subproblems defined on uniform grids. The base scheme converges ɛ-uniformly in the maximum norm at the rate of O(N−2ln2N + N0−1), where N + 1 and N0 + 1 are the numbers of nodes in the space and time meshes, respectively. An application of the Richardson extrapolation technique to the base scheme yields a higher order scheme called the Richardson decomposition scheme. This higher order scheme convergesɛ-uniformly at the rate of O(N−4ln4N + N0−2). For fixed values of the parameter, the convergence rate is O(N−4 + N0−2).

A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N1−2ln2N1 + N2−2), where N1 + 1 and N2 + 1 are the number of grid nodes along the x1-axis and per unit interval of the x2-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.

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.

Robust numerical method for a system of singularly perturbed parabolic reaction‐diffusion equations on a rectangle

Abstract A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction‐diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ϵ tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obtained. Using the condensing mesh technique and the classical finite difference approximations of the boundary value problem under consideration, a difference scheme is constructed that converges ϵ‐uniformly at the rate O ‘N−2 ln2 N + N0 −1) , where N = mins Ns, s = 1, 2, Ns + 1 and N0 + 1 are the numbers of mesh points on the axis xs and on the axis t, respectively.