A Constrained-Transport E-CUSP scheme for Ideal Magnetohydrodynamic Equations Yiqing Shen ∗ 1 , Gecheng Zha †1 , and Manuel A. Huerta ‡ 2 1 Dept. of Mechanical and Aerospace Engineering, 2 Dept. of Physics,, University of Miami, Coral Gables,, Florida 33124, Abstract The constrained transport algorithm combined with an E-CUSP scheme is developed to solve the ideal magnetohydrodynamic equations. The algorithm can preserve the divergence-free condition for the magnetic field and maintain the advantages of simplicity and low diffusion of the E-CUSP scheme. The numerical results demonstrate the robustness and efficiency of this new algorithm. 1 Introduction Many hypersonic aerodynamics and astrophysics problems need to solve the solutions of ideal magneto- hydrodynamics(MHD) equations. Since the ideal MHD equations have a wave-like structure analogous to that of the hydrodynamics equations, various numerical schemes for hydrodynamics equations have been extended to solve the MHD equations in the past two decades. The approximate Riemann solvers, which are based on eigenvalue and eigenvector analysis, are widely used for high speed flow as well as for high speed MHD applications. Beginning with the work of Brio and Wu[1], the numerical methods for MHD equations based on approximate Riemann solvers have been extensively studied and developed. For example, Roe’s Riemann solvers are developed by Brio and Wu[1], Dai and Woodward[2], Zachary and Collelaz[3], Roe and Balsara[4], and Cargo and Gallice[5]. HLL(Harten-Lax-van Leer)-type schemes are developed by Janhunen[6] and Honkkila and Janhunen[7], Gurski[8], Li[9], Miyoshi and Kusano[10], Balsara et al.[11]. Flux vector splitting methods are developed by MacCormack[12], Jiang and Wu[13]. The equations of magnetohydrodynamics are not homogeneous of degree one with respect to the state vector and hence can not directly perform flux vector splitting. To overcome this difficulty MacCormack introduces an extra variable ˜ a in Ref. [12]. The flux splitting schemes based on eigenvalues and eigenvec- tors system are generally very complicated. In our study, we noticed that, in the eigensystem of Roe and Balsara[4], the eigenvalues of the Alfven waves do not affect the flux. In other words, any values can be used for the eigenvalues of the Alfven waves and the flux will be the same. This makes the flux splitting based on Roe’s approximate Riemann solver uncertain. The low disspation high order filter schemes developed by Yee and Sjogreen[14] for MHD systems involve a dissipative portion of higher order Lax-Friedrichs scheme or an approximate Riemann solver. Moreover, Balbas[15] developed a central differencing scheme based on the evolution of cell averages over staggered grids. Gaitonde[16] developed a compact difference method for MHD with a local filter switching procedure to change the higher order filter to a second order filter locally for shock capturing. The central differencing ∗ AIAA Member, current address: LHD, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR. China, yqshen@imech.ac.cn † Associate Professor, AIAA Senior Member, gzha@miami.edu ‡ Professor 1 42nd AIAA Plasmadynamics and Lasers Conference<br>in conjunction with the<br>18th Internati 27 - 30 June 2011, Honolulu, Hawaii AIAA 2011-3745 Copyright © 2011 by all the authors of this paper. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.