A robust structure-preserving surface reconstruction scheme for two-layer shallow water equations based on a relaxation model and an extension on adaptive moving triangles

We present a structure-preserving surface reconstruction scheme for the relaxation two-layer shallow water equation, inspired by the approach in [Computers & Fluids 272 (2024) 106193], along with its two-dimensional extension on adaptive moving triangular meshes. The original two-layer shallow water equation is conditionally hyperbolic, which presents challenges in designing shock-capturing numerical schemes. To address this, we propose a relaxation two-layer shallow water equation that is hyperbolic. However, this relaxation equation still contains nonconservative products associated with layer heights and bottom topography, which cannot be defined in the distributional sense. Utilizing the surface reconstruction method, we define Riemann states linked to the layer heights and bottom topography. This approach smooths the solution, facilitating the discretization of the nonconservative product across cell boundaries. We introduce a structure-preserving parameter crucial for demonstrating convergence, maintaining stationary steady states, and ensuring the positivity-preserving property. We establish a Lax-Wendroff type convergence theorem for the structure-preserving surface reconstruction scheme applied to the relaxation two-layer shallow water equation. To validate our approach, we conduct several classical numerical experiments for both one-dimensional and two-dimensional cases. Notably, we numerically confirm that the structure-preserving surface reconstruction scheme for the one-dimensional relaxation two-layer shallow water equation exhibits $K$-convergence based on the Cesàro average, particularly in the context of the well-known Kelvin-Helmholtz instabilities. Finally, we show several numerical results of the two-dimensional two-layer shallow water equations on adaptive moving triangles to verify the theoretical results.