Two-dimensional simulation of the damped Kuramoto–Sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization

Abstract

We apply a numerical scheme based on a meshless method in space and an explicit exponential Runge-Kutta in time for the solution of the damped Kuramoto–Sivashinsky equation in two-dimensional spaces. The proposed meshless method is radial basis function-generated finite difference, which approximates the derivatives of the unknown function with respect to the spatial variables by a linear combination of the function values at given points in the domain and weights. Also, in this approach there is no need a mesh or triangulation for approximation. For each point, the weights are computed separately in its local sub-domain by solving a small radial basis function interpolant. Besides, a numerical algorithm based on singular value decomposition of the local radial basis function interpolation matrix [59] is applied to find the suitable shape parameter for each interpolation problem. We also consider an explicit time discretization based on exponential Runge–Kutta scheme such that its stability region is bigger than the classical form of Runge-Kutta method. Some numerical simulations are provided on the square, circular and annular domains to show the capability of the numerical scheme proposed here.