Well-balanced semi-implicit relaxed finite volume schemes for hyperbolic balance laws
Abstract
The aim of this work is to design well-balanced semi-implicit relaxed finite volume methods for one-dimensional systems of hyperbolic balance laws with several time scales, in which the fast waves impose a severe time step restriction of fully explicit time discretizations. The strategy is based on the semi-implicit relaxation approach in which the flux is split into a slow and a fast part and the latter is linearised by means of Jin-Xin relaxation, such that the implicit contributions reduce to linear decoupled systems. The main difficulty with respect to well-balancing is the generation of implicit viscosity in these schemes which does not vanish on non-constant steady states. This is circumvented by rewriting the scheme in terms of fluctuations with respect to local steady states, and the resulting implicit operator is driven by equilibrium-corrected jumps that vanish identically at equilibrium. A second order extension of the first order well-balanced relaxed scheme is given in the framework of implicit-explicit Runge-Kutta schemes. Therein, to achieve the desired accuracy, the projection to the equilibrium manifold on each stage is exploited. The well-balanced properties of the first-and second-order schemes are proven and numerical results for the shallow water system and the Ripa model are given to assess the performance of the novel well-balanced schemes.
Related
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BibTeX
@misc{gomezbueno2026wellbalanced,
title = {{Well-balanced semi-implicit relaxed finite volume schemes for hyperbolic balance laws}},
author = {Irene Gómez-Bueno and Manuel J Castro and Andrea Thomann},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05835613v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-balanced semi-implicit relaxed finite volume schemes for hyperbolic balance laws
AU - Irene Gómez-Bueno
AU - Manuel J Castro
AU - Andrea Thomann
PY - 2026
DA - 2026/10/22
PB - HAL
UR - https://hal.science/hal-05835613v1
AB - The aim of this work is to design well-balanced semi-implicit relaxed finite volume methods for one-dimensional systems of hyperbolic balance laws with several time scales, in which the fast waves impose a severe time step restriction of fully explicit time discretizations. The strategy is based on the semi-implicit relaxation approach in which the flux is split into a slow and a fast part and the latter is linearised by means of Jin-Xin relaxation, such that the implicit contributions reduce to linear decoupled systems. The main difficulty with respect to well-balancing is the generation of implicit viscosity in these schemes which does not vanish on non-constant steady states. This is circumvented by rewriting the scheme in terms of fluctuations with respect to local steady states, and the resulting implicit operator is driven by equilibrium-corrected jumps that vanish identically at equilibrium. A second order extension of the first order well-balanced relaxed scheme is given in the framework of implicit-explicit Runge-Kutta schemes. Therein, to achieve the desired accuracy, the projection to the equilibrium manifold on each stage is exploited. The well-balanced properties of the first-and second-order schemes are proven and numerical results for the shallow water system and the Ripa model are given to assess the performance of the novel well-balanced schemes.
ER -