An efficient localized meshless collocation method for the two-dimensional Burgers-type equation arising in fluid turbulent flows

Abstract

This paper focusses on the numerical technique based on a localized meshless collocation method for approximating the Burgers-type equation in two dimensions. The method uses two main stages to approximate the unknown solution. First, the time derivative is discretized by the backward Euler method and a semi-discrete approach is obtained. Meanwhile, the unconditional stability and convergence of the semi-discrete scheme are proved by virtue of the discrete energy method in an appropriate Sobolev space. Second, a local radial basis function partition of unity (LRBF-PU) method is applied to approximate the spatial direction and a fully-discrete scheme is established. The LRBF-PU is based on subdividing the original domain to several subdomains and uses the radial basis function interpolation on each local subdomain. Compared with the global collocation technique, the main advantages of this technique are that it can reduce the computational cost and obtain a well linear system. Three illustrative examples are presented to demonstrate the accuracy and efficiency of the proposed method.