A well-balanced path-conservative discontinuous Galerkin method with generalized hydrostatic reconstruction for non-conservative balance laws
Abstract
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.
Related
- High-order in-cell discontinuous reconstruction path-conservative methods for conservative systems in primitive variables
- Entropy-Stable and Well-Balanced Discontinuous Galerkin Methods for the Compressible Euler Equations in Vector-Invariant Form
- GPU Performance of an Entropy-Stable Discontinuous Galerkin Euler Solver with Non-Conservative Terms
- A Fully Discrete, Mass-Conservative Adaptive Discontinuous Spectral-Element Method for the One-Dimensional Richards Equation in Mixed Form
- Limit Cycles in a Discontinuous Generalized Liénard Systems with Bivariate Perturbation
BibTeX
@misc{fan2026wellbalanced,
title = {{A well-balanced path-conservative discontinuous Galerkin method with generalized hydrostatic reconstruction for non-conservative balance laws}},
author = {Ruilin Fan and Yinhua Xia and Yan Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41484},
eprint = {2610.41484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41484},
}
RIS
TY - UNPB
TI - A well-balanced path-conservative discontinuous Galerkin method with generalized hydrostatic reconstruction for non-conservative balance laws
AU - Ruilin Fan
AU - Yinhua Xia
AU - Yan Xu
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41484
UR - https://arxiv.org/abs/2610.41484
AB - 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.
ER -