Local flux high-order well-balanced WENO finite-difference methods for hyperbolic systems with source terms
Abstract
We present a class of high-order well-balanced WENO finite-difference schemes for hyperbolic systems with source terms based on the local flux approach. The governing equations are reformulated in terms of a generalized flux that incorporates the source terms and nonconservative products, so that its variation vanishes at stationary states within the WENO reconstruction stencils. A path-conservative formulation with Roe linearization is employed to approximate the flux/nonconservative contribution and source contributions. The proposed approach does not require solving the ODE system associated with the equilibria or explicitly reconstructing equilibrium solutions. Upwind flux splitting and characteristic-wise WENO reconstruction are used to achieve high-order accuracy and nonoscillatory shock-capturing properties. Numerical experiments for Burgers' equations with source terms, the shallow water equations, and the two-layer shallow water equations demonstrate accurate preservation of moving equilibria, high-order convergence, resolution of small perturbations of steady states, and robust performance for Riemann problems.
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BibTeX
@misc{ren2026local,
title = {{Local flux high-order well-balanced WENO finite-difference methods for hyperbolic systems with source terms}},
author = {Baifen Ren and Maria Kazolea and Carlos Parés-Madronal},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://inria.hal.science/hal-05788823v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local flux high-order well-balanced WENO finite-difference methods for hyperbolic systems with source terms
AU - Baifen Ren
AU - Maria Kazolea
AU - Carlos Parés-Madronal
PY - 2026
DA - 2026/10/09
PB - HAL
UR - https://inria.hal.science/hal-05788823v1
AB - We present a class of high-order well-balanced WENO finite-difference schemes for hyperbolic systems with source terms based on the local flux approach. The governing equations are reformulated in terms of a generalized flux that incorporates the source terms and nonconservative products, so that its variation vanishes at stationary states within the WENO reconstruction stencils. A path-conservative formulation with Roe linearization is employed to approximate the flux/nonconservative contribution and source contributions. The proposed approach does not require solving the ODE system associated with the equilibria or explicitly reconstructing equilibrium solutions. Upwind flux splitting and characteristic-wise WENO reconstruction are used to achieve high-order accuracy and nonoscillatory shock-capturing properties. Numerical experiments for Burgers' equations with source terms, the shallow water equations, and the two-layer shallow water equations demonstrate accurate preservation of moving equilibria, high-order convergence, resolution of small perturbations of steady states, and robust performance for Riemann problems.
ER -