High-Order CESE Methods for the Euler Equations

Recently, Chang1 reported a new class of high-order CESE methods for solving nonlinear hyperbolic partial differential equations. A series of high-order algorithms have been developed based on a systematic, recursive formulation that achieves fourth-, sixth-, and eighth-order accuracy. The new high-order CESE method shares many favorable attributes of the original second- order CESE method, including: (i) compact mesh stencil involving only the immediate mesh nodes surrounding the node where the solution is sought, (ii) the CFL stability constraint remains to be the same, i.e., less or equal to 1, as compared to the original second-order method, and (iii) superb shock capturing capability without using an approximate Riemann solver. The new algorithm has been demonstrated by solving Burger's equation. In the present paper, we extend Chang's high-order method for system of linear and nonlinear hyperbolic partial differential equations. A general formulation is presented for solving the coupled equations with arbitrarily high-order accuracy. To demonstrate the formulation, several linear and nonlinear cases are reported. First, we solve a convection equation with source term and the linear acoustics equations. We then solve the Euler equations for acoustic waves, a blast wave, and Shu and Osher's test case for acoustic waves interacting with a shock. Numerical results show higher-order convergence by continuous mesh refinement. Presented at the AIAA Aerospace Sciences Meeting (49th) held in Orlanda, FL on 4-7 Jan 2011. Published in Proceedings of the AIAA Aerospace Sciences Meeting (49th), p1-14, Jan 2011.