Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

TL;DR

The paper develops invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics model in Lagrangian coordinates, leveraging group-classification results for bottom topographies. It constructs three-layer schemes in Lagrangian coordinates and two-layer schemes in mass Lagrangian coordinates that preserve discrete conservation laws and key symmetries, and it analyzes their invariance properties. Numerical implementations using Newton linearization and tridiagonal solvers demonstrate energy preservation (with negligible loss) and reveal that magnetic fields accelerate compression waves and influence column collapse, depending on bottom topography. The work provides a practical framework for symmetry-preserving discretizations of SMHD in 1D, with explicit handling of bottom topographies and rigorous discussion of invariance, conservation, and numerical stability. Overall, the results offer a robust approach for accurate, energy-consistent simulations of SMHD on uniform 1D meshes in Lagrangian-type coordinates.

Abstract

Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

Figures (6)

• Figure 1: Dam break problem for the horizontal bottom. 'SW' (black) is the solution for the standard shallow water equations ($\alpha=0$), 'Gilman' is the solution for the case $\alpha^2 > 0$ (SMHD). Initial profile is given by the dotted line, and the magnetic field gradient $B$ is denoted by the dashed line.
• Figure 2: Dam break problem for the horizontal bottom. Trajectories $x(t,s)$ for the case $\alpha > 0$ (gray) and $\alpha = 0$ (black). The characteristics of the flow are outlined with thick dashed lines.
• Figure 3: Dam break problem for the parabolic bottom.
• Figure 4: Dam break problem for the logarithmic bottom.
• Figure 5: Collapse of the liquid column over the inclined bottom.