A Comparison Study of Three CESE Schemes in MHD Simulation

Abstract

The space-time conservation element and solution element (CESE) scheme is a new second order numerical scheme based on the concept of space-time conservation integration. In order to further overcome excessive numerical damping due to small Courant-Friedrichs-Lewy (CFL) number and to obtain a high quality solution, a Courant number insensitive (CNIS) scheme and a high-order scheme have been proposed by Chang et al. for fluid mechanics problems recently. In this study, to explore the potential capability of applications of the CNIS CESE scheme and the high-order CESE scheme to magnetohydrodynamics (MHD) equations, several benchmark MHD problems are calculated in one and two dimensions: (i) Brio and Wu's shock tube, (ii) Dai and Woodward's case, (iii) the Orszag-Tang vortex problem, (iv) the Riemann problem. The numerical results just prove that the CNIS scheme is more accurate and can keep the divergence free condition of the magnetic field, even if the CFL number is ≪ 1. Meanwhile, the tests show that the high order CESE scheme possesses the ability to solve MHD problems but is sensitive to the Courant number.

An improved two-dimensional unstructured CE/SE scheme for capturing shock waves

“…During the past two decades, the CE/SE method became successful in several fields such as MHD, [7,8] detonation, [9−15] and spall fracture. [16] Chang's original CE/SE scheme is based on triangular or tetrahedral grids.…”

“…To simplify the CE/SE method while calculating at structured grids, an improved CE/SE method and its high-order accurate scheme, which is based on a redefinition of smaller CE and SE, are proposed and investigated. [8][9][10][11][12][13][14]16,20,21] Their scheme has smaller spatial steps. The derivatives are calculated based on two groups of equations instead of Chang's four groups.…”