Semi-discrete Active Flux method for two-dimensional hyperbolic systems with Coriolis source terms
Abstract
We investigate the semi-discrete Active Flux numerical method on Cartesian grids applied to two-dimensional linear and nonlinear shallow water equations with Coriolis source terms. These hyperbolic systems admit non-trivial stationary solutions governed by a geostrophic equilibrium. In this setting, we analyze the stationarity-preserving properties of our semi-discrete Active Flux formulation. To do so for the linear system, we apply a discrete spatial Fourier transform to the semi-discrete method and analyze the resulting evolution matrix. We show that its discrete kernel is a non-trivial discretization of the geostrophic equilibrium, which implies that the method preserves a discrete geostrophic equilibrium state. This is an inherent feature of Active Flux and is achieved without any additional modifications. Numerical experiments confirm the method's ability to maintain geostrophic stationary states for the linear system. Additionally, we numerically investigate the performance of the Active Flux method for the nonlinear shallow water system.
Related
- Fully discrete energy stability for a one-dimensional hyperbolic relaxation model with complete topography terms
- Fully Discrete Active Flux Methods for the Euler Equations: Exploring Different Reconstructions and New Multi-Dimensional Evolution Operators
- Boundary stabilization of multi-dimensional first-order quasilinear hyperbolic systems
- Exponential Turnpike in Optimal Boundary Control of Multi-Dimensional First-Order Hyperbolic Systems
- Exact controllability and rapid stabilization of linear difference delay equations and one-dimensional hyperbolic systems
BibTeX
@misc{manoj2026semidiscrete,
title = {{Semi-discrete Active Flux method for two-dimensional hyperbolic systems with Coriolis source terms}},
author = {Nikhil Manoj and Wasilij Barsukow and Christian Klingenberg},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27555},
eprint = {2609.27555},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05794417v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Semi-discrete Active Flux method for two-dimensional hyperbolic systems with Coriolis source terms
AU - Nikhil Manoj
AU - Wasilij Barsukow
AU - Christian Klingenberg
PY - 2026
DA - 2026/09/23
PB - HAL
DO - 10.48550/arxiv.2609.27555
UR - https://hal.science/hal-05794417v1
AB - We investigate the semi-discrete Active Flux numerical method on Cartesian grids applied to two-dimensional linear and nonlinear shallow water equations with Coriolis source terms. These hyperbolic systems admit non-trivial stationary solutions governed by a geostrophic equilibrium. In this setting, we analyze the stationarity-preserving properties of our semi-discrete Active Flux formulation. To do so for the linear system, we apply a discrete spatial Fourier transform to the semi-discrete method and analyze the resulting evolution matrix. We show that its discrete kernel is a non-trivial discretization of the geostrophic equilibrium, which implies that the method preserves a discrete geostrophic equilibrium state. This is an inherent feature of Active Flux and is achieved without any additional modifications. Numerical experiments confirm the method's ability to maintain geostrophic stationary states for the linear system. Additionally, we numerically investigate the performance of the Active Flux method for the nonlinear shallow water system.
ER -