Locally Divergence-Free Spectral-DG Methods for Ideal Magnetohydrodynamic Equations on Cylindrical Coordinates

Abstract

In this paper, we propose a class of high order locally divergence-free spectraldiscontinuous Galerkin (DG) methods for three dimensional (3D) ideal magnetohydrodynamic (MHD) equations on cylindrical geometry. Under the conventional cylindrical coordinates (r, ϕ, z), we adopt the Fourier spectral method in the ϕ-direction and discontinuous Galerkin (DG) approximation in the (r, z) plane, motivated by the structure of the particular physical flows of magnetically confined plasma. By a careful design of the DG approximation function space, our spectral-DG methods are divergence-free inside each element for the magnetic field. Numerical examples with third order strong-stability-preserving Runge-Kutta methods are provided to demonstrate the efficiency and performance of our proposed methods.

Conservative and Dissipative Local Discontinuous Galerkin Methods for Korteweg-de Vries Type Equations

“…Their work was motivated by the successful numerical experiments of Bassi and Rebay [1] for compressible Navier-Stokes equations. The LDG methods can be applied in many equations, such as KdV type equations [32,26,27,28,15,39], Camassa-Holm equations [24,35], Degasperis-Procesi equation [31], Schrödinger equations [25,23], and more nonlinear equations or system [33,24,30,34,17].…”