A matrix free implicit scheme for solution of resistive magneto-hydrodynamics equations on unstructured grids

The resistive magneto-hydrodynamics (MHD) governing equations represent eight conservation equations for the evolution of density, momentum, energy and induced magnetic fields in an electrically conducting fluid, typically a plasma. A matrix free implicit method is developed to solve the conservation equations within the framework of an unstructured grid finite volume formulation. The analytic form of the convective flux Jacobian is derived on a general unstructured mesh and used in a Lower-Upper Symmetric Gauss Seidel (LU-SGS) technique developed as part of the implicit scheme. A grid coloring technique is also developed to create data parallelism in the algorithm. The computational efficiency of the matrix free method is compared with two common approaches: a global matrix solve technique that uses the GMRES (Generalized minimum residual) algorithm and an explicit method. The matrix-free method is observed to be overall computationally faster than the global matrix solve method and demonstrates excellent parallel scaling on multiple cores. The computational effort and memory requirements for the matrix free approach is comparable to the explicit approach which in turn is much lower than the global solve implicit approach. Both the matrix free and global solve implicit techniques exhibit superior steady state convergence compared to the explicit method.