Stability estimates for singularly perturbed Fisher's equation using element-free Galerkin algorithm

Abstract

In the present article, a mesh-free technique has been presented to study the behavior of nonlinear singularly perturbed Fisher's problem, which exhibits the traveling wave propagation phenomenon. Some narrow regions adjacent to the left and right lateral boundary may possess rapid variations when the singular perturbation parameter $ \epsilon\rightarrow 0 $, which are not captured nicely by the traditional numerical schemes. In the current work, a robust numerical strategy is proposed, which comprises the implicit Crank-Nicolson scheme to discretize the time derivative term and the element-free Galerkin (EFG) scheme to discretize the spatial derivative terms with nodes densely distributed in the boundary layer regions. The stability of the semi-discrete scheme has been analyzed, and the rate of convergence is shown to be $ \mathcal{O}(\tau^{2}) $. The nonlinear nature of the considered problem has been tackled by employing the quasilinearization process, and its convergence rate has been discussed. Some numerical experiments have been performed to verify the computational uniformity and robustness of the suggested method, rate of convergence as well $ L_{\infty} $ errors have been presented, which depicts the effectiveness of the proposed method.

Keywords

• singularly perturbed problems
• nonlinear time-dependent fisher equation
• moving least-squares approximation
• implicit crank-nicolson technique
• element-free galerkin scheme