High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics

TL;DR

This work develops high-order entropy-stable finite-difference schemes for the shallow water magnetohydrodynamics (SWMHD) equations with bottom topography by incorporating the Janhunen source term. It starts from a second-order entropy-conservative scheme that preserves lake-at-rest and then builds high-order EC schemes and corresponding ES schemes by introducing carefully matched high-order fluxes and source-term discretizations, augmented with WENO/ENO-based dissipation and strong-stability preserving time integrators. A positivity-preserving mechanism is provided, ensuring nonnegative water height under a CFL constraint. The authors extend the framework to 2D, and present extensive numerical tests—1D and 2D—confirming accuracy, well-balancedness, entropy stability, and robustness in the presence of discontinuities and complex bottom topography, showing the practical viability of the approach for SWMHD simulations.

Abstract

This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations.They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second-order EC schemes. Next, by using the second-order EC schemes as building block, high-order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high-order accurate well-balanced semi-discrete ES schemes %satisfying the entropy inequality are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order well-balanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiter. Finally, extensive numerical tests are conducted to validate the accuracy, the well-balanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes.

Figures (10)

• Figure 5.1: Example \ref{['ex:1DWB']}: The symbols "$\triangle$" and "$\circ$" denote the numerical solutions at $t=10$ obtained by using the EC and the ES schemes with $N_x=40$, respectively.
• Figure 5.2: Example \ref{['ex:1Dsteady']}: Left: The bottom topography and the reference solutions obtained by using the ES scheme with $N_x=1000$. Right: The enlarged view of the numerical solutions obtained by using the ES schemes with $N_x=50$ ("$\triangledown$") and $N_x=100$ ("$\circ$"), respectively.
• Figure 5.3: Example \ref{['ex:1Dperturb']}: The surface level $h+b$ and the discharge $hv_1$ obtained by using the ES scheme with $N_x=200$. The solid lines denote the reference solutions obtained by using the ES schemes with $N_x=3000$.
• Figure 5.4: Same as Figure \ref{['fig:1DSW_perturb']} except for $hB_1=1$.
• Figure 5.5: Example \ref{['ex:RP1']}: The solutions at $t=0.4$ ("$\circ$") are obtained by the ES schemes with $N_x=100$. The solid lines denote the reference solutions obtained by using the Lax-Friedrichs scheme with a fine mesh of $N_x=20000$.

Theorems & Definitions (23)

• definition 1: Entropy function
• definition 2: Entropy solution
• definition 3: Entropy conservative scheme
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4