Generalized finite difference method for solving two-dimensional Burgers’ equations

Abstract

In this paper, the two-dimensional Burgers’ equations are numerically analyzed by a meshfree numerical scheme, which is a combination of the implicit Euler method, the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). Since both of the convective and the diffusive terms simultaneously appear in the time-dependent quasi-linear Burgers’ equations, it is necessary and very difficult to develop a reliable numerical scheme to solve it. The GFDM, which can truly get rid of time-consuming mesh generation and numerical quadrature, and the implicit Euler method are used for spatial and temporal discretization, respectively. Then, the resultant system of nonlinear algebraic equations for every time step is resolved by the newly-developed FTIM. Since, in comparing with the Newton's method, the calculation of the inverse of Jacobian matrix can be avoided in the FTIM, to adopt the FTIM for solving the system of nonlinear algebraic equations is very efficient and has great potential for large-scale engineering problems. Some numerical results and comparisons are provided to validate the accuracy and the simplicity of the proposed meshfree scheme.