High-order finite-volume integration schemes for subsonic magnetohydrodynamics

TL;DR

This work develops a dimension-by-dimension, high-order finite-volume scheme for subsonic magnetohydrodynamics on Cartesian grids, incorporating constrained-transport to keep $\nabla\cdot\mathbf{b}=0$ and achieving up to 10th order accuracy with only one face reconstruction per cell. The solver combines a magnetic-field interpolation (Binterp) with WENO reconstructions, a flux module featuring a 1D Riemann solver and a CT-based 2D Riemann solver, and a passage-through-point-values framework (AtoP/PtoA) to preserve high-order accuracy in nonlinear fluxes. It also provides a consistent treatment of non-ideal terms (viscosity $\mu$, magnetic diffusivity $\eta$) and an internal-energy sink $S_e$, along with forcing in spectral space mapped to the grid, enabling driven turbulence at statistically stationary states. Numerical tests on a 3D MHD vortex and forced turbulence demonstrate reduced numerical dissipation at higher orders and show that high-order schemes can achieve comparable accuracy on coarser grids, with clearer inertial ranges and finer structures. The framework is flexible for including dissipative physics and forcing while preserving the intended accuracy, though stability considerations for non-smooth supersonic flows remain an area for potential stabilization or adaptive-order strategies.

Abstract

We present an efficient dimension-by-dimension finite-volume method which solves the adiabatic magnetohydrodynamics equations at high discretization order, using the constrained-transport approach on Cartesian grids. Results are presented up to tenth order of accuracy. This method requires only one reconstructed value per face for each computational cell. A passage through high-order point values leads to a modest growth of computational cost with increasing discretization order. At a given resolution, these high-order schemes present significantly less numerical dissipation than commonly employed lower-order approaches. Thus, results of comparable accuracy are achievable at a substantially coarser resolution, yielding overall performance gains. We also present a way to include physical dissipative terms: viscosity, magnetic diffusivity and cooling functions, respecting the finite-volume and constrained-transport frameworks.

Figures (15)

• Figure 1: Right-hand-side computation's workflow.
• Figure 2: Reconstruction module.
• Figure 3: Flux computation module.
• Figure 4: Illustration of the different terms appearing in the area-averages$\leftrightarrow$point value transformations. A term with a superscript of the form $\Sigma mn$ with $m,n\in[0,4]$ corresponds to the sum of the values in the cells where this superscript appears: it contains either four or eight terms.
• Figure 5: Constrained-transport module.