A Well-Balanced Space-Time ALE Compact Gas-Kinetic Scheme for the Shallow Water Equations on Unstructured Meshes

TL;DR

This work develops a high-order space-time ALE compact gas-kinetic scheme for the shallow water equations on moving unstructured meshes. It rigorously preserves the geometric conservation law (GCL) and the well-balanced lake-at-rest state by incorporating mesh motion directly into fluxes and source terms within a unified space-time framework, updating on the moving mesh without remapping. A fourth-order compact spatial reconstruction coupled with a one-stage second-order time update (extendable to $S2O4$ for higher temporal accuracy) yields high-fidelity, non-oscillatory solutions. Bottom topography evolves consistently with mesh motion, and cell-averaged gradients are updated via Gauss's theorem to maintain a cohesive ALE scheme. Numerical tests across 1D/2D dam-breaks and perturbations validate the scheme’s accuracy, stability, and efficiency on moving meshes, highlighting its potential for adaptive, high-resolution geophysical flow simulations.

Abstract

This study presents a high-order, space-time coupled arbitrary Lagrangian Eulerian (ALE) compact gas-kinetic scheme (GKS) for the shallow water equations on moving unstructured meshes. The proposed method preserves both the geometric conservation law (GCL) and the well-balanced property. Mesh motion effects are directly incorporated by formulating numerical fluxes that account for the spatial temporal nonuniformity of the flow field and the swept area of moving cell interfaces. This allows temporal updates to be performed on the physical moving mesh, avoiding data remapping. The compact GKS provides time accurate evolution of flow variables and fluxes, enabling the scheme to achieve second-order temporal accuracy within a single stage. To consistently treat bottom topography on moving meshes, an evolution equation for the topography is established and discretized using a compatible space-time scheme, in which the fluxes induced by mesh motion are computed accurately. Mathematical proofs demonstrating the GCL preserving and well-balanced properties of the proposed ALE formulation are also provided. For improved accuracy and robustness, a nonlinear fourth-order compact reconstruction technique is employed. A comprehensive set of numerical experiments verifies the scheme's theoretical properties and demonstrates its accuracy, stability, and effectiveness in simulating complex shallow-water flow problems.

Figures (15)

• Figure 1: Schematic of the moving triangular mesh cell at times $t^n$ (black) and $t^{n+1}$ (red).
• Figure 2: Schematic of the motion of edge $AB$ of a mesh cell. The motion is determined by the nodal velocities $V_1$ and $V_2$. Within a time step, the motion velocities are constant, and $A'B'$ remains a straight line.
• Figure 3: A schematic of reconstruction stencil of compact GKS. The flow variables and their gradients are known in each cell.
• Figure 4: A schematic of mesh node motion. Adaptive mesh moving is achieved through node motion.
• Figure 5: Verification of the GCL of the compact GKS. Left: Comparison of the moving mesh at its initial state ($t=0$, blue) and a deformed state ($t=5.5$, red). Right: Time evolution of the numerical errors for water height and momentum.