High-Order CESE Methods for Solving Hyperbolic PDEs (Preprint)

Abstract : In the present paper, we extend Chang's high-order method for system of linear and non-linear 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 non-linear 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. 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., < or = 1, as compared to the original second-order method, and (iii) shock capturing capability without using an approximate Riemann solver.