Stability and convergence of radial basis function finite difference method for the numerical solution of the reaction–diffusion equations

Abstract

Stability, convergence and application of radial basis function finite difference (RBF-FD) scheme is studied for solving the reaction–diffusion equations (RDEs). We show that the explicit RBF-FD method is stable, and stability condition depends on the shape parameter of related radial basis function.The generalized multiquadric (GMQ) is applied as radial basis function and weight coefficients are explicitly presented for equispaced node distribution. Also, two methods are presented to compute the optimal shape parameter. The combination of these methods with the GMQ-FD method will produce two efficient algorithms for numerical solution of RDEs: the variable GMQ-FD (VGMQ-FD) and the constant GMQ-FD (CGMQ-FD). We test the scheme on traveling wave and compare its accuracy with the conventional finite difference method (FDM).