Liwei Xu, Department of Mathematical Sciences, Rensselaer Polytechnic Institute (RPI), Troy, NY, U.S.A
601 Pao Yue-Kong Library
Maintaining the divergence-free constraint on the magnetic field and preserving the positivity of density and pressure are two challenges in numerical simulation for ideal magnetohydrodynamic (MHD) equations. In this talk, we mainly discuss the design of exactly divergence-free central discontinuous Galerkin (DG) schemes solving ideal MHD equations. We first consider the second and third order divergence-free schemes. Exactly divergence-free magnetic fields are achieved by first approximating the normal component of the magnetic field through discretizing the magnetic induction equations on the mesh skeleton, namely, the element interfaces. Then it is followed by an element-by-element divergence-free reconstruction with the matching accuracy. The extension of this technique to design arbitrary order schemes is not trivial. We next discuss how this strategy, combined with extra consideration on the magnetic induction equations, can be extended to design divergence-free schemes of arbitrary order of accuracy. Essential analysis on these divergence-free schemes is presented. Numerical results are presented to demonstrate the accuracy and the robustness of the schemes. This is a joint work with Prof. Fengyan Li at RPI.
 Li, F., Xu, L. and Yakovlev, S., Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, Journal of Computational Physics, 230 (2011), 4828-4847
 Li, F. and Xu, L., Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, Journal of Computational Physics, 231(2012), 2655-2675
 Yakovlev, S., Xu, L. and Li, F. Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations, to appear, Journal of Computational Sciences, 2012
 Cheng, Y., Qiu, J., Li, F. and Xu, L., Positivity-preserving DG and central DG methods for ideal MHD equations, to be submitted, 2012