Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Locally Divergence-Free Local Characteristic Decomposition Based Path-Conservative Central-Upwind Scheme for Ideal Magnetohydrodynamics

Shaoshuai Chu, Alexander Kurganov, Maria Lukacova-Medvidova, Mingye Na

TL;DR

The paper addresses the challenge of accurately solving ideal MHD while preserving a divergence-free magnetic field. It introduces LCD-PCCU, combining local characteristic decomposition with a path-conservative central-upwind flux in the Godunov-Powell augmented system to reduce dissipation. The method augments the system with derivatives of the magnetic field to enable LCD-driven reconstruction and divergence control, and constructs 2-D fluxes using eigenstructure. Numerical experiments on standard MHD benchmarks show improved resolution and robustness over the previous PCCU method, confirming practical benefits for high-fidelity MHD simulations.

Abstract

We introduce a locally divergence-free local characteristic decomposition based path-conservative central-upwind (LCD-PCCU) scheme for ideal magnetohydrodynamics (MHD) equations. The proposed method is a low-dissipation extension of the recently proposed locally divergence-free PCCU scheme. To reduce the numerical dissipation, we incorporate the LCD into the PCCU framework. The resulting LCD-PCCU method enhances the resolution of numerical solutions as demonstrated through a series of benchmark tests.

A Locally Divergence-Free Local Characteristic Decomposition Based Path-Conservative Central-Upwind Scheme for Ideal Magnetohydrodynamics

TL;DR

The paper addresses the challenge of accurately solving ideal MHD while preserving a divergence-free magnetic field. It introduces LCD-PCCU, combining local characteristic decomposition with a path-conservative central-upwind flux in the Godunov-Powell augmented system to reduce dissipation. The method augments the system with derivatives of the magnetic field to enable LCD-driven reconstruction and divergence control, and constructs 2-D fluxes using eigenstructure. Numerical experiments on standard MHD benchmarks show improved resolution and robustness over the previous PCCU method, confirming practical benefits for high-fidelity MHD simulations.

Abstract

We introduce a locally divergence-free local characteristic decomposition based path-conservative central-upwind (LCD-PCCU) scheme for ideal magnetohydrodynamics (MHD) equations. The proposed method is a low-dissipation extension of the recently proposed locally divergence-free PCCU scheme. To reduce the numerical dissipation, we incorporate the LCD into the PCCU framework. The resulting LCD-PCCU method enhances the resolution of numerical solutions as demonstrated through a series of benchmark tests.

Paper Structure

This paper contains 7 sections, 52 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Ideal MHD Equations
  3. 2-D Flux Globalization Based LCD-PCCU Scheme
  4. Numerical Examples
  5. Conclusions
  6. Eigendecomposition for Conservative Variables
  7. Eigendecomposition for Primitive Variables

Figures (9)

  • Figure 4.1: Example 1: $\rho$, $b_1$, and $b_2$ computed by the LCD-PCCU and PCCU schemes (top row) and zooms for $\rho$ and $b_2$ at $x\in[-0.2,0.35]$, $[-0.2,0]$, and $[0.2,0.8]$ (bottom row).
  • Figure 4.2: Example 2: Time evolution of the $L^1$- and $L^\infty$-norms of $(\bm\nabla\cdot\bm b)_{j,k}$ computed by the Uncorrected LCD-PCCU scheme on a uniform $320\times320$ mesh.
  • Figure 4.3: Example 3: Density $\rho$ computed by the LCD-PCCU (left) and PCCU (right) schemes.
  • Figure 4.4: Example 3: 1-D slices along the line $y=\pi$ of the solutions from Figure \ref{['fig31']} together with the reference solution.
  • Figure 4.5: Example 3: Time evolution of the $L^1$- and $L^\infty$-norms of $(\bm\nabla\cdot\bm b)_{j,k}$ computed by the Uncorrected LCD-PCCU scheme.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 3.1