A High-Order Compact Finite Volume Method For Nonlinear Wave Equations

A high-order compact finite volume scheme is presented for the solving the Burgers equation,the generalized Burger’s-Huxley equation and the generalized Burger’s-Fisher equation. The exact solution of the these equation is usually difficult to solve. Consequently, to solve its numerical solution is an important method for solving these equations at present. The aim of this paper is to reconstruct a high-order compact finite volume scheme to approximate the exact of these equations. Firstly, the three equations can be written into united form, which include the convection term, the diffusion term and the source item. The fourth-order compact finite volume scheme and the fourth-order compact upwind finite volume scheme are used to discrete the convection term, the diffusion term and the source item, which combine with the Simpson formula and (Local)Lax-Friedrich flux. The temporal discretization is done by a 4th order Runge-Kutta scheme(RK4) in order to guarantee the precision. Secondly, the truncation error of the numerical scheme are performed by using the Fourier analysis, and the linear stability analysis of the numerical scheme was carried out. Finally, we select some numerical examples about Burgers equation, the generalized Burger’s-Huxley equation and the generalized Burger’s-Fisher equation, respectively. Numerical results demonstrate that the present scheme possesses high precision and the robustness.