Virtual‎ ‎Element‎ ‎Method‎ ‎for‎ Numerical Simulation of Burgers-Fisher Equation on Convex‎ ‎and Non-Convex Meshes

Abstract

‎We present an enhanced approach to solving the combined non-linear time-dependent Burgers-Fisher equation‎, ‎which is widely used in mathematical biology and has a broad range of applications‎. ‎Our proposed method employs a modified version of the finite element method‎, ‎specifically the virtual element method‎, ‎which is a robust numerical approach‎. ‎We introduce a virtual process and an Euler-backward scheme for discretization in the spatial and time directions‎, ‎respectively‎. ‎Our numerical scheme achieves optimal error rates based on the degree of our virtual space‎, ‎ensuring high accuracy‎. ‎We evaluate the efficiency and flexibility of our approach by providing numerical results on both convex and non-convex polygonal meshes‎. ‎Our findings indicate that the proposed method is a promising tool for solving non-linear time-dependent equations in mathematical biology‎.

Keywords

• virtual element method
• burgers-fisher equation
• convex mesh
• non-convex mesh
• non-linearity