Ruilin Fan | Searcharxiv

arXiv2026

A well-balanced path-conservative discontinuous Galerkin method with generalized hydrostatic reconstruction for non-conservative balance laws

We propose a high-order well-balanced path-conservative discontinuous Galerkin method for one-dimensional balance laws with non-conservative products. Generalized hydrostatic reconstruction at cell interfaces couples effective numerical fluxes with Dal Maso-LeFloch-Murat path integrals, while a cellwise reference-equilibrium decomposition cancels projection-induced residuals in the volume discretization. Under explicit assumptions on equilibrium recovery, admissibility, paths, and boundary data, the spatial discretization is path conservative, and prescribed rest and moving equilibria are exactly preserved by both the semi-discrete scheme and its fully discrete SSP-RK approximation. The framework is applied to the two-layer shallow water equations and a blood-flow model with spatially varying vessel properties. Numerical experiments confirm the designed order of accuracy and round-off-level preservation of smooth and discontinuous equilibria, demonstrate accurate resolution of small perturbations, and show that the interface treatment captures shocks and stationary material contacts without visible spurious oscillations.

math.NA↗

arXiv2025

Well-balanced path-conservative discontinuous Galerkin methods with equilibrium preserving space for shallow water linearized moment equations

This paper presents high-order, well-balanced, path-conservative discontinuous Galerkin (DG) methods for the shallow water linearized moment equations (SWLME), designed to preserve both still and moving water equilibrium states. Unlike the multi-layer shallow water equations, which model vertical velocity variations using multiple distinct layers, the SWLME employs a polynomial expansion of velocity profiles with up to $N$ moments. This approach enables a more detailed representation of vertical momentum transfer and complex velocity profiles while retaining hyperbolicity. However, the presence of non-conservative terms and complex steady-state structures introduces significant numerical challenges. Addressing these challenges, we develop path-conservative DG schemes grounded in the Dal Maso-LeFloch-Murat (DLM) theory for non-conservative products. Our method balances flux gradients, non-conservative terms, and source terms through equilibrium-preserving spaces. For the still water equilibrium, we reformulate the equations into a quasilinear form that eliminates source terms, inherently preserving steady states. For the moving water equilibrium, we extend the DG method by transforming conservative variables into equilibrium variables and employing linear segment paths. Theoretical analysis and numerical experiments demonstrate that the proposed methods achieve exact equilibrium preservation while maintaining high-order accuracy, even in scenarios with vertical velocity variations and complex topographies.

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