High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well-balancedness and positivity
Zhihao Zhang, Huazhong Tang, Kailiang Wu
TL;DR
This work develops high-order, well-balanced (WB) and positivity-preserving (PP) finite-volume schemes for the shallow water equations (SWEs) on adaptive moving structured meshes. It introduces a time-dependent curvilinear coordinate framework and decomposes the WB property into flux-source balance and mesh movement balance, achieved via hydrostatic-reconstruction fluxes, careful source-term discretization, and discretization of geometric conservation laws (GCLs). A PP analysis shows WB schemes remain PP under a sufficient limiter-driven condition, even with mesh metrics and movement, and bottom-topography updates are incorporated to maintain WB during mesh adjustments. The schemes are implemented on adaptive moving meshes, employing a monitor-function-driven Euler–Lagrange mesh adaptation functional, and validated through extensive 1D and 2D tests demonstrating fifth-order accuracy, WB, PP, and computational efficiency advantages over fixed-grid counterparts. The results indicate robust performance for perturbed lake-at-rest states, moving vortices, and complex topography, with confirmed maintenance of $h\ge 0$ and $J>0$ under the prescribed CFL constraints.
Abstract
This paper develops high-order accurate, well-balanced (WB), and positivity-preserving (PP) finite volume schemes for shallow water equations on adaptive moving structured meshes. The mesh movement poses new challenges in maintaining the WB property, which not only depends on the balance between flux gradients and source terms but is also affected by the mesh movement. To address these complexities, the WB property in curvilinear coordinates is decomposed into flux source balance and mesh movement balance. The flux source balance is achieved by suitable decomposition of the source terms, the numerical fluxes based on hydrostatic reconstruction, and appropriate discretization of the geometric conservation laws (GCLs). Concurrently, the mesh movement balance is maintained by integrating additional schemes to update the bottom topography during mesh adjustments. The proposed schemes are rigorously proven to maintain the WB property by using the discrete GCLs and these two balances. We provide rigorous analyses of the PP property under a sufficient condition enforced by a PP limiter. Due to the involvement of mesh metrics and movement, the analyses are nontrivial, while some standard techniques, such as splitting high-order schemes into convex combinations of formally first-order PP schemes, are not directly applicable. Various numerical examples validate the high-order accuracy, high efficiency, WB, and PP properties of the proposed schemes.