Localized method of approximate particular solutions with Cole–Hopf transformation for multi-dimensional Burgers equations

The Burgers equations depict propagating wave with quadratic nonlinearity, it can be used to describe nonlinear wave propagation and shock wave, where the nonlinear characteristics cause difficulties for numerical analysis. Although the solution approximation can be executed through iterative methods, direct methods with finite sequence of operation in time can solve the nonlinearity more efficiently. The resolution for nonlinearity of Burgers equations can be resolved by the Cole–Hopf transformation. This article applies the Cole–Hopf transformation to transform the system of Burgers equations into a partial differential equation satisfying the diffusion equation, and uses a combination of finite difference and the localized method of approximate particular solution (FD-LMAPS) for temporal and spatial discretization, respectively. The Burgers equations with behaviors of propagating wave, diffusive N-wave or within multi-dimensional irregular domain have been verified in this paper. Effectiveness of the FD-LMAPS has also been further examined in some experiments, and all the numerical solutions prove that the FD-LMAPS is a promising numerical tool for solving the multi-dimensional Burgers equations.