Higher-order magnetohydrodynamic numerics

TL;DR

The chapter develops and validates a higher-order finite-volume framework for ideal MHD that leverages constrained transport to preserve magnetic solenoidality while achieving fourth-order accuracy. It combines CWENO4 reconstructions, area-to-point value transformations, a multidimensional Riemann solver for electric fluxes, and SSPRK time integration to reduce numerical dissipation and capture fine-scale dynamics, including turbulence, with fewer grid points. The authors demonstrate robust performance across smooth and shock-dominated tests, introducing a-priori and a-posteriori fallbacks (flatteners) to maintain physical states near strong discontinuities. The work highlights the practical benefits and challenges of upgrading MHD codes to higher-order schemes, emphasizing the continued importance of CT and carefully designed reconstruction and flux strategies for reliable, efficient simulations. Overall, the framework provides a concrete path to higher-fidelity MHD simulations that can resolve complex multiscale phenomena more efficiently than traditional second-order approaches.

Abstract

In this chapter, we aim at presenting the basic techniques necessary to go beyond the widely accepted paradigm of second-order numerics. We specifically focus on finite-volume schemes for hyperbolic conservation laws occuring in fluid approximations such as the equations of ideal magnetohydrodynamics or the Euler equations of gas dynamics. For the sake of clarity, a simple fourth-order ideal magnetohydrodynamic (MHD) solver which allows to simulate strongly shocked systems serves as an instructive example. Issues that only or mainly arise in the world of higher-order numerics are given specific focus. Alternative algorithms as well as refinements and improvements are dicussed and are referenced to in the literature. As an example of application, some results on decaying compressible turbulence are presented.

Figures (8)

• Figure 1: Definition of the magnetic quantities and the fluxes in the cell $\Omega_{i,j,k}$. Only the flux in the ${\bf {x}}$-direction is shown, the ones in the other directions are defined analogously.
• Figure 2: Simple polynomial reconstruction over a discontinuity. The dashed line is the function to be reconstructed and the solid line is the unique polynomial of degree 4 which cell averages verify \ref{['eq:poptrel']}. The grid-size $\Delta x$ is unity.
• Figure 3: Illustration of the computation of $(E^{L}_z)_{i+1/2,j+1/2,k}$ (black circle in subgraph $(c)$): $(a)$ CWENO line-average reconstruction of the area-averaged electric field, $(b)$ CWENO line-average reconstruction of the area-averaged magnetic field and taking the mean of the two possibilities for the electric field in each quadrant, $(c)$ Deducing the electric flux thanks to the multidimensional Riemann solver.
• Figure 4: Circularly polarized Alfvén wave: cuts of $v_z$ along the main diagonal after 100 periods, (top), left: at resolution $64^2$, right: at resolution $128^2$. The CWENO4 solution (in red) is extremely close to the reference solution at $t=0$ (in black), whereas the TVD2 solution (in blue) presents some significant amplitude and shape error. Bottom: convergence of errors, EOC and numerical dissipation for different schemes after one period.
• Figure 5: 3D MHD vortex problem: slices of the magnetic pressure at $z\approx 0.04$, resolution $128^3$ after one period (top), left: CWENO4, right: TVD2. Bottom: convergence of errors, EOC and numerical dissipation for different schemes after one period.