Schemes Convergent ε-Uniformly for Parabolic Singularly Perturbed Problems with a Degenerating Convective Term and a Discontinuous Source

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed

ON CONDITIONING OF A SCHWARZ METHOD FOR SINGULARLY PERTURBED CONVECTION–DIFFUSION EQUATIONS IN THE CASE OF DISTURBANCES IN THE DATA OF THE BOUNDARY VALUE PROBLEM 1

Abstract. In this paper we consider a Dirichlet problem for singularly perturbed ordinary differential equations with convection terms and a small perturbation parameter ε. To solve the problem numerically we use an ε-uniformly convergent difference scheme (on special piecewise-uniform meshes) and a decomposition of this scheme based on a Schwarz technique with overlapping subdomains. The step-size of such special meshes is extremely small in a neighbourhood of the layer and changes sharply on its boundary, which can generally lead to a loss of conditioning of the above schemes. We study the influence of perturbations in the data of the boundary value problem on disturbances of numerical solutions. We derive estimates for the disturbances of numerical solutions (in the maximum norm) depending on a subdomain in which the disturbance of the data appears. When the right-hand side of the discrete equations is considered in a “natural” norm, i.e., in the maximum norm with a special weight multiplier (that is ε ln N, for ε = O(ln −1 N), in a neighbourhood of the boundary layer, where N defines the number of mesh points), the finite difference schemes under consideration are well conditioned ε-uniformly. In addition, for the Schwarz method a special restriction is imposed on the width of the overlapping region. Note that for these special schemes an ε-uniform estimate for the condition number is the same as that for schemes on uniform meshes in the case of regular boundary value problems. We give conditions under which the solution of the iterative scheme based on the overlapping Schwarz method is convergent ε-uniformly to the solution of the Dirichlet problem as the number of mesh points and the number of iterations increase

High-Order Time-Accurate Schemes For Singularly Perturbed Parabolic Convection-Diffusion Problems With Robin Boundary Conditions

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 #. In contrast to the Dirichlet boundary-value problem, for the problem under consideration the errors of the well-known classical methods, generally speaking, grow without bound as # N -1 where N defines the number of mesh points with respect to x. The order of convergence for the known #-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct #-uniformly convergent schemes of highorder time-accuracy. The e#ciency 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 di#erent

COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.3(2003), No.3, pp.387–404 c ○ 2003 Editorial Board of the Journal ”Computational Methods in Applied Mathematics ” Joint Co. Ltd. NOVEL DEFECT-CORRECTION HIGH-ORDER, IN SPACE AND TIME, ACCURATE SCHEMES FOR

Dedicated to John J.H.Miller on the occasion of his 65th birthday. Abstract — New high-order accurate finite difference schemes based on defect correction are considered for an initial boundary-value problem on an interval for singularly perturbed parabolic PDEs with convection; the highest space derivative in the equation is multiplied by the perturbation parameter ε, ε ∈ (0, 1]. Solutions of the well-known classical numerical schemes for such problems do not converge ε-uniformly (the errors of such schemes depend on the value of the parameter ε and are comparable with the solution itself for small values of ε). The convergence order of the existing ε-uniformly convergent schemes does not exceed 1 in space and time. In this paper, using a defect correction technique, we construct a special difference scheme that converges ε-uniformly with the second (up to a logarithmic factor) order of accuracy with respect to x and with the second order of accuracy and higher with respect to t. The conditions are given which ensure the ε-uniform convergence of the defect-correction schemes with a rate of O(N −k ln k N + K −k0), k = 1, 2, k0 = 1, 2, 3, where N + 1 and K + 1 denote the number of the mesh points in x and t, respectively. Theoretical results and the efficiency of the newly constructed schemes are confirmed by numerical experiments