A Two-Dimensional Fourth-Order CESE Method for the Euler Equations on Triangular Unstructured Meshes

Abstract

Previously, Chang 1 reported a new high-order Conservation Element Solution Element (CESE) method for solving nonlinear, scalar, hyperbolic partial differential equations in one dimensional space. Bilyeu et al. 2 have extended Chang’s scheme for solving a onedimensional, coupled equations with an arbitrary order of accuracy. In the present paper, the one-dimensional, high-order CESE method is extended for two-dimensional unstructured meshes. A formulation is presented for solving the coupled equations with the fourthorder in accuracy. The new high-order CESE methods share many favorable attributes of the original second-order CESE method, including: (i) the use of 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 � 1, and (iii) superb shock capturing capability without using an approximate Riemann solver. To demonstrate the formulation, four test cases are reported, including (i) solution of a two-dimensional, scalar convection equation, (ii) the solution of the linear acoustic equations, (iii) the solution of the Euler equations for waves of small amplitudes, and (iv) the solution of the Euler equations for expanding shock waves. In all calculations, unstructured triangular meshes are used. In the first three cases, convergence tests show the fourth-order accuracy of the solutions. In the last case, numerical results of the fourth-order scheme are superior than that obtained by the second-order CESE method. In this paper, we extend the one-dimensional, high-order CESE method for solving two-dimensional coupled, nonlinear hyperbolic Partial Differential Equations (PDEs) by using unstructured meshes. The new formulation is currently fourth-order in accuracy, but it can be easily extended to a higher-order scheme by including more terms in the Taylor series expansion. The tenet of the CESE method is treating space and time in a unified manner in integrating the model equations for flux conservation. In the CESE method, the space-time domain is divided into non-overlapping Conservation Elements (CEs), over which the spacetime integration is performed to enforce flux conservation. The integration is facilitated by using Solution Elements (SEs), which in general do not coincide with the CEs. The unknowns are discretized inside each SE by using a Taylor series expansion. Aided by the prescribed discretization schemes for the unknowns inside each SE, space-time flux over each surface of a CE can be calculated and the overall flux conservation over each CE can be enforced. The result is an explicit formulation for updating the unknowns for time-marching calculation. Special features of the CESE method include: (i) The unknowns are discretized by using the Taylor series expansion in both space and time. The order of the Taylor series is also the order of accuracy of the method. (ii) The method has the most compact mesh stencil, which involves only the immediate neighboring mesh points of the cell where the solution is sought. (iii) The method is an explicit scheme in time-marching calculation. The stability criterion is CFL � 1. (iv) No approximate Riemann solver is used and the scheme is simple and efficient. (v) The CESE method includes a suite of numerical algorithms,