Pei Fu
School of Mathematical Sciences at University of Science and Technology of China, Hefei, China
In this talk, we overview our main works on the structure preserving discontinuous Galerkin (DG) methods for magnetohydrodynamic (MHD) equations and PDEs with high order spatial derivatives. For the ideal MHD equations, the main difficulties are to solve nonlinear systems and preserve the divergence-free constraint.
We propose one globally divergence-free DG method to solve the ideal MHD equations. We utilized DG methods to solve nonlinear systems for fluid variables and discretize the induction equation separately to approximate the normal components of the magnetic field on element interfaces. Then, we reconstruct the globally divergence-free magnetic field in an element-by-element manner. The resulting methods are local and the approximated magnetic fields are globally divergence-free.
For PDEs with high order spatial derivatives, we formulate and analyze the DG methods with optimal accuracy in one dimension. We first rewrite each PDE into its first order form and then apply a general DG formulation. Then, we design one group of numerical fluxes as linear combinations of average values of fluxes, and jumps of the solution as well as the auxiliary variables at cell interfaces. The main focus of this part is to identify a sub-family of the numerical fluxes by choosing the coefficients in the linear combinations, so the solution and some auxiliary variables of the proposed DG methods are optimally accurate. Regarding error estimates, we develop some special projection operators, tailored for each choice of numerical fluxes in the sub-family, to eliminate those terms at cell interfaces that would otherwise contribute to the sub-optimality of the error estimates.