Pratibhamoy Das

Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems

This paper deals with the adaptive mesh generation for singularly perturbed nonlinear parameterized problems with a comparative research study on them. We propose an a posteriori error estimate for singularly perturbed parameterized problems by moving mesh methods with fixed number of mesh points. The well known a priori meshes are compared with the proposed one. The comparison results show that the proposed numerical method is highly effective for the generation of layer adapted a posteriori meshes. A numerical experiment of the error behavior on different meshes is carried out to highlight the comparison of the approximated solutions.

Optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction-diffusion boundary-value problems

In this article, we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction-diffusion problems having diffusion parameters with different magnitudes. The central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle. An a priori monitor function is obtained from the error estimate. A suitable a posteriori analogue of this monitor function is also derived for the mesh construction which will lead to an optimal second-order parameter uniform convergence. We present the results of numerical experiments for linear and semilinear reaction-diffusion systems to support the effectiveness of our preferred monitor function obtained from theoretical analysis.

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.

HIGHER-ORDER PARAMETER UNIFORM CONVERGENT SCHEMES FOR ROBIN TYPE REACTION-DIFFUSION PROBLEMS USING ADAPTIVELY GENERATED GRID

In this article, a singularly perturbed reaction-diffusion problem with Robin boundary conditions, is considered. In general, the solution of this problem possesses boundary layers at both the ends of the domain. To solve this problem, we propose a numerical scheme, involving the cubic spline scheme for boundary conditions and the classical central difference scheme for the differential equation (DE) at the interior points. The grid is generated by the equidistribution of a positive monitor function. It has been proved that classical forward–backward approximation for mixed type boundary conditions, gives first-order convergence, whereas our proposed cubic spline scheme provides second-order accuracy independent of the perturbation parameter. Numerical experiments have been provided to validate the theoretical results.

A higher order difference method for singularly perturbed parabolic partial differential equations

Abstract This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.

An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh

The present work considers a nonlinear system of singularly perturbed delay differential equation whose each component of the solution has multiple layers. Here, we provide an a posteriori based convergence analysis for the adaptation of these layer phenomena. We derive a parameter uniform a posteriori error estimate which will lead to a layer adaptive mesh by moving a fixed number of mesh points. It is theoretically shown that the layer adaptive solution on the a posteriori generated mesh will uniformly converge to the exact solution with optimal order accuracy where the optimality is measured with respect to the continuous problem discretization. The comparison results with the existing methods based on a priori meshes show that the proposed method on the a posteriori mesh is highly effective.