Globally Divergence-Free Discontinuous Galerkin Methods for Ideal Magnetohydrodynamic Equations

Abstract

Ideal magnetohydrodynamic (MHD) equations are widely used in many areas in physics and engineering, and these equations have a divergence-free constraint on the magnetic field. In this paper, we propose high order globally divergence-free numerical methods to solve the ideal MHD equations. The algorithms are based on discontinuous Galerkin methods in space. The induction equation is discretized separately to approximate the normal components of the magnetic field on elements interfaces, and to extract additional information about the magnetic field when higher order accuracy is desired. This is then followed by an element by element reconstruction to obtain the globally divergence-free magnetic field. In time, strong-stability-preserving Runge–Kutta methods are applied. In consideration of accuracy and stability of the methods, a careful investigation is carried out, both numerically and analytically, to study the choices of the numerical fluxes associated with the electric field at element interfaces and vertices. The resulting methods are local and the approximated magnetic fields are globally divergence-free. Numerical examples are presented to demonstrate the accuracy and robustness of the methods.