Boundary and interior layer phenomena in coupled multiscale parabolic convection–diffusion interface problems: efficient numerical resolution and analysis

Abstract

Purpose

This paper aims to study boundary and interior layer phenomena in coupled multiscale parabolic convection–diffusion interface problems and to present their efficient numerical resolution and analysis.

Design/methodology/approach

This study includes cases in which the diffusion parameters are small, distinct and can differ in order of magnitude. The source term is considered to be discontinuous. The asymptotic behavior of the solution is examined. The layer structure is analyzed, leading to the development of a variant of layer-resolving Shishkin mesh. For efficient numerical resolution, two methods are developed by combining additive schemes on a uniform mesh to discretize in time and an upwind difference scheme away from the line of discontinuity and a specific upwind difference scheme along the line of discontinuity, defined on a variant of layer resolving Shishkin mesh, to discretize in space. The analysis of the numerical resolution is discussed using the barrier function approach. Numerical simulations provide a verification of the theory and efficiency of the approach.

Findings

The discontinuity in the source term, along with the inclusion of small and distinct diffusion parameters, results in multiple overlapping and interacting boundary and interior layers. The work demonstrates that the present approach is robust in resolving boundary and interior layers. From a computational cost perspective, the numerical resolution presented in the paper is more efficient than conventional approaches.

Originality/value

Efficient numerical resolution and analysis of boundary and interior layer phenomena in coupled multiscale parabolic convection–diffusion interface problems are provided. The discretization of the coupled system in the approach incorporates a distinctive feature, wherein the components of the approximate solution are decoupled at each time level, resulting in tridiagonal linear systems to be solved, in contrast to large banded linear systems with conventional approaches.