Stable higher-order finite-difference schemes for extremely low-beta plasmas

Develop a more stable higher-order finite-difference scheme for the 10-moment magnetohydrodynamics equations that does not crash in extremely low-beta systems, including cases where the filtering parameter is varied over the range 0.01 to 10.0.

Background

The paper reports that its fourth- and sixth-order implementations crash in the extremely low-beta blast-wave problem, even after varying the nonlinear-filtering parameter c_diff from 0.01 to 10.0. The second-order scheme remains stable, but the authors identify the lack of a robust higher-order method as an unresolved task for simulations requiring improved spatial accuracy.

References

Nonetheless, for strong shocks or high-Mach-number flows, this scheme can become unstable even while time steps obey the CFL condition; indeed, there is no known method for guaranteeing stability in multidimensional schemes with accuracies greater than first order.

— Magnetising the quokka code with fast, second-order accurate, error-correcting schemes for magnetohydrodynamics on GPUs  (2609.10816 - Kriel et al., 9 Sep 2026) in Section 2, subsection “Time-stepping with flux correction” (Section 2.5)

The first is that simulations crash in extremely low beta systems when higher-order schemes are used. We performed simulations of blast-wave problem with $c_\mathrm{diff}$ set to various values ranging from $0.01$ to $10.0$, but the simulations still crash. Developing a more stable scheme for higher-order finite differences is one of the remaining tasks.

— A generalized energy-consistent finite difference scheme for 10-moment magnetohydrodynamics  (2608.25441 - Akutagawa et al., 26 Aug 2026) in Section 6, Conclusion