Stable meshless discretization of two-dimensional Fisher-type equations by local multiquadric method over non-rectangular domains

Abstract

Reaction-diffusion equations play important role in problems related to population dynamics, developmental biology, and phase-transition. For such equations, we propose a strong-form local (multiquadric) RBF method that gives sparse well-conditioned differentiation matrices with reduced computational cost and memory storage; thus, avoids solving dense ill-conditioned system matrices, an inherited drawback of strong-form global RBF methods if compared to the limitations of mesh-based methods. After spatial discretization of the time-dependent PDE problem by sparse differentiation matrices, the resultant system of ODEs can be stably integrated in time via a high-order and high-quality ODE solver. The proposed method is tested on two-dimensional Fisher-type equations for its geometric flexibility, accuracy, and efficiency. Unlike the mesh-based methods, the proposed local method works for arbitrary scattered data points and is equally effective for problems over non-rectangular domains. Some recommendations are also made for further efficient implementation of the proposed local multiquadric method.