Parameter-uniform finite difference method for singularly perturbed parabolic problem with two small parameters

TL;DR: A layer resolving numerical scheme for solving the numerical solution of the singularly perturbed parabolic convection-diffusion problem exhibiting interior layer due to the convection coefficient is introduced in this paper .

TL;DR: In this paper , a numerical treatment via a difference scheme constructed by the Crank-Nicolson scheme for the time derivative and cubic spline in tension for the spatial derivatives on a layer resolving nonuniform Bakhvalov-type mesh for a singularly perturbed unsteady-state initial-boundary-value problem with two small parameters is presented.

TL;DR: In this paper , the numerical solution of parabolic convection-diffusion problems involving two small positive parameters and arising in modeling of hydrodynamics is dealt with, and the backward Euler method for time stepping and fitted trigonometric-spline scheme for spatial discretization are considered on uniform meshes.

TL;DR: Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems as mentioned in this paper, and the global errors in the numerical approximations are measured in the pointwise maximum norm.

TL;DR: In this paper, the authors propose a method to solve the problem of the "missing link" problem in the context of medical decision-making, and the final publication is available at Springer via http://dx.doi.

TL;DR: The comparison results show that the proposed numerical method is highly effective for the generation of layer adapted a posteriori meshes in the adaptive mesh generation for singularly perturbed nonlinear parameterized problems.

Abstract: In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.