Divergence-Preserving MHD Solutions Using a Noniterative Hyperbolic Conservation Law Scheme

Abstract

The divergence constraint, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\nabla$</tex-math> </inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\cdot$</tex-math> </inline-formula> B <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$=$</tex-math> </inline-formula> 0, presents a central challenging numerical feature of electromagnetic and magnetohydrodynamic solvers. In general, this constraint is not strictly enforced in hyperbolic solvers and tends to accumulate error over the duration of a simulation, polluting the solution with unphysical behavior. In this article, we present a numerical scheme that constructs a system of hyperbolic conservation laws that already satisfy the divergence constraint, starting from the potential form of Maxwell’s equations and basic derivative relationships. The new system consists only of the first-order spatial differentiation, in contrast to the potential equations which are second order; this allows them to be directly integrated into the existing hyperbolic solvers with no additional implementation changes needed. Furthermore, since the system of conservation laws constructed automatically satisfies the divergence criteria, no subiteration is required to resolve the divergence-free solution. We present the results of integrating this new conservation law approach in an existing finite volume plasmadynamic solver that discretizes the coupled Maxwell and Navier–Stokes equations. Numerical tests indicate that the new approach can strongly resolve physical solutions. The divergence error is also measured and quantified for these problems, showing that the new system suppresses divergence error by many orders of magnitude.