A Two-Dimensional Third-Order CESE Scheme for Ideal MHD Equations

In this paper,we construct a two-dimensional third-order space-time conservation element and solution element(CESE)method and apply it to the magnetohydrodynamics(MHD)equations.This third-order CESE method preserves all the favorable attributes of the original second-order CESEmethod,such as:(i)flux conservation in space and time without using an approximated Riemann solver,(ii)genuine multi-dimensional algorithm without dimensional splitting,(iii)the use of the most compact mesh stencil,involving only the immediate neighboring cells surrounding the cell where the solution at a new time step is sought,and(iv)an explicit,unified space-time integration procedure without using a quadrature integration procedure.In order to verify the accuracy and efficiency of the scheme,several 2D MHD test problems are presented.The result of MHD smooth wave problem shows third-order convergence of the scheme.The results of the other MHD test problems show that the method can enhance the solution quality by comparing with the original second-order CESE scheme.