A globally divergence-free entropy stable nodal DG method for conservative ideal MHD equations

TL;DR

The paper introduces a globally divergence-free entropy-stable nodal DG method (ES-GDF) for the conservative 2D ideal MHD equations, enabling direct discretization of the original conservation form while preserving $\nabla\cdot\mathbf B=0$ through interface constraint updates and a least-squares interior reconstruction. By combining Gauss–Lobatto SBP operators, an entropy-conservative flux inside elements, and an entropy-stable flux at interfaces, the scheme achieves discrete entropy stability under the divergence-free constraint; a novel limiter and an energy-correction step ensure robust shock-capturing without sacrificing the divergence-free property. A semi-discrete entropy analysis and a fully discrete ES-GDF framework (with SSP-RK4 time stepping) are developed, including a LS-based magnetic-field reconstruction to maintain divergence-freeness and a targeted limiter to control oscillations near shocks. Numerical experiments across smooth and discontinuous MHD benchmarks demonstrate high-order accuracy, effective entropy dissipation, and tight control of divergence, validating the method's robustness and potential for scalable, physically faithful simulations.

Abstract

We propose an arbitrarily high-order globally divergence-free entropy stable nodal discontinuous Galerkin (DG) method to directly solve the conservative form of the ideal MHD equations using appropriate quadrature rules. The method ensures a globally divergence-free magnetic field by updating it at interfaces with a constraint-preserving formulation [5] and employing a novel least-squares reconstruction technique. Leveraging this property, the semi-discrete nodal DG scheme is proven to be entropy stable. To handle the problems with strong shocks, we introduce a novel limiting strategy that suppresses unphysical oscillations while preserving the globally divergence-free property. Numerical experiments verify the accuracy and efficacy of our method.

Figures (16)

• Figure 1: The notations of the reconstruction problem.
• Figure 1: Example 5.1: Smooth MHD vortex. Time evolution of the absolute values of relative deviation in the conservative variables on different meshes for $k=2$.
• Figure 1: The notations around a vertex $\Lambda$.
• Figure 1: The structure of nonzero elements in $\mathbf G^{-1}$.
• Figure 2: Example 5.2: The rotated shock tube. The numerical solution at $T = 0.2/\sqrt{5}$ on $N_x\times N_y = 512\times 2$ meshes. No limiter is applied.

Theorems & Definitions (11)

• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Theorem 3.6
• Remark 3.7
• Lemma 4.1
• Remark 4.2