A robust layer adapted difference method for singularly perturbed two-parameter parabolic problems

Abstract

A finite difference method for a time-dependent singularly perturbed convection-diffusion-reaction problem involving two small parameters in one space dimension is considered. We use the classical implicit Euler method for time discretization and upwind scheme on the Shishkin-Bakhvalov mesh for spatial discretization. The method is analysed for convergence and is shown to be uniform with respect to both the perturbation parameters. The use of the Shishkin-Bakhvalov mesh gives first-order convergence unlike the Shishkin mesh where convergence is deteriorated due to the presence of a logarithmic factor. Numerical results are presented to validate the theoretical estimates obtained.

Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

“…First‐order accurate methods for smooth data are also observed in other studies . Analysis on different meshes like Shishkin‐Bakhvalov meshes can be also seen in Jha and Kadalbajoo . The a posteriori based convergence analysis for singularly perturbed parameterized problems is also carried out to get optimal order parameter uniform accuracy in Das, which avoids the requirement of a priori derivative bound.…”

“…Theorem 1. The continuous solution y(x, t) of (1) to (2) and the numerical solution Y(x i , t j ) of (27) to (29) satisfy the following error estimate:…”

“…, if 2 ≥ , ie, the difference scheme (27) to (29) converges with almost first-order accuracy in space and time.…”

“…[23][24][25][26] Analysis on different meshes like Shishkin-Bakhvalov meshes can be also seen in Jha and Kadalbajoo. 27 The a posteriori based convergence analysis for singularly perturbed parameterized problems is also carried out to get optimal order parameter uniform accuracy in Das, 28 which avoids the requirement of a priori derivative bound. An equidistribution based new adaptive mesh is proposed in Das and Mehrmann 29 for parabolic problems and in Das and Vigo-Aguiar 30 for systems of reaction-diffusion problems with smooth data.…”

Singularly Perturbed System of Parabolic Equations in the Critical Case

“…In [5] - [6], various problems for split systems of two equations of parabolic type were studied, and asymptotics of the boundary layer type were constructed. The problems of differential equations of parabolic type with a small parameter were studied in [7]- [9]. In [10]- [13], numerical methods for solving singularly perturbed problems are studied.…”