Applications of the local RBF collocation method and the fictitious time integration method for Burgers'equations

Abstract

In this paper, a meshfree numerical scheme, which is a combination of the local RBF (radial basis function) collocation method (LRBFCM) and the fictitious time integration method (FTIM), is proposed to efficiently and accurately analyze the two-dimensional Burgers’ equations. The Burgers’ equations, which can be regarded as the simplified form of the Navier-Stokes equations, are a system of time-and space-dependent nonlinear partial differential equations, so it is very difficult and important to be accurately solved. The LRBFCM, one of newly-developed meshfree methods, is adopted for spatial discretization and, at the meantime, the implicit Euler method is used for temporal discretization. Then, the system of nonlinear algebraic equations at every time step will be efficiently resolved by the FTIM, which can avoid the calculation of the Jacobian matrix. The numerical results and comparisons of one numerical example will be provided to validate the accuracy and the efficiency of the proposed meshfree numerical scheme.