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

Previously, Chang 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. 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.