An adaptive mesh generation and higher-order difference approximation for the system of singularly perturbed reaction–diffusion problems

Abstract

The paper presents a hybrid difference method to solve a coupled system of singularly perturbed reaction–diffusion problems over an adaptive mesh. The solution to the problem manifests a distinctive multi-scale nature, characterized by localized narrow regions where the solution undergoes exponential changes. The mesh-generating procedure relies upon equidistributing a non-negative monitor function. The method utilizes a suitable combination of the cubic and exponential spline difference schemes over a layer adaptive mesh. The proposed method is unconditionally stable and uniformly convergent. The convergence obtained is optimal, being free from logarithmic factors. Moreover, it is free from directional bias. Numerical experiments have been performed and presented for two model problems. The results obtained are in agreement with the theoretical estimates.

Keywords

• System of reaction–diffusion problems
• Singular perturbation
• Multiscale phenomena
• Mesh generation
• Grid equidistribution
• Spline difference method