A High-order Implicit Finite Element Method for Integrating the Extended-magneto-hydrodynamic Equations in Two Dimensions
Authors: S.C. Jardin, J. Breslau, and N. Ferraro
Date of PPPL Report: January 2007
Submitted to: Journal of Computational Physics.
We describe a new method for solving the time-dependent extended-magnetohydrodynamic (X-MHD) equations in two dimensions that has significant advantages over other methods. The stream-function/potential representation of the velocity and magnetic field vectors, while fully general, allows accurate description of nearly incompressible fluid motions and manifestly satisfies the divergence condition on the magnetic field. Through analytic manipulation, the split semi-implicit method breaks the full matrix time advance into four sequential time advances, each involving smaller matrices. The use of a high order triangular element with continuous first derivatives (C1 continuity) allows the Galerkin method to be applied without introduction of new auxiliary variables (such as the vorticity or the current density). These features, along with the manifestly compact nature of the fully node-based C1 finite elements, lead to minimum size matrices for an unconditionally stable method with order of accuracy h4. The resulting matrices are compatible with direct factorization using SuperLU_dist. We demonstrate the accuracy of the method by presenting examples of twofluid linear wave propagation, two-fluid linear eigenmodes of a tilting cylinder, and of a challenging nonlinear problem in two-fluid magnetic reconnection.