# Discrete fully well balanced WENO finite difference schemes via a global-flux quadrature method

Maria Kazolea, Carlos Parés-Madronal, Mario Ricchiuto

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Maria Kazolea, Carlos Parés-Madronal, Mario Ricchiuto. Discrete fully well balanced WENO finite difference schemes via a global-flux quadrature method. ECCOMAS 2024 - 9th European Congress on Computational Methods in Applied Sciences and Engineering, Jun 2024, Lisbon, Portugal. 10.1016/j.jcp.2023.112673. hal-04883587

HAL Id: hal-04883587

https://inria.hal.science/hal-04883587v1

Submitted on 13 Jan 2025

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# Discrete fully well balanced WENO finite difference schemes via a global-flux quadrature method

Maria Kazolea<fnref n="1" />, Carlos Parés-Madronal<fnref n="2" /> and Mario Ricchiuto<fnref n="1" />

**Key Words:**

In this work high order discrete well-balanced methods for one dimensional hyperbolic systems of balance laws are proposed. Inspired by the approximate well-balanced schemes introduced e.g in [1] and [3] we aim at constructing a method whose discrete steady states are solutions of arbitrary very high order ODE integrators. However, as in [4] we embed this property directly in the scheme, so that the ODE integrator is never actually applied to solve the local Cauchy problem. To achieve this, we work in the WENO finite difference setting already explored in [5]. As in [2] we apply the WENO reconstruction to a global flux assembled nodewise as the sum of the physical flux and a source primitive. The novel idea is to compute the source primitive by means of very high order multi-step ODE methods applied on the finite difference grid. We obtain in this way a local well balanced splitting of a source integral with weights depending on the ODE integrator. As in [4] this approach is referred to as global flux quadrature of the source. The discrete solutions of our schemes are the solutions of the underlying ODE integrator by construction. In practice, we combine WENO flux reconstructions of orders from 3 to 7, with Adams-Multon or Adams-Bashford methods of orders up to 8. The results confirm that the steady accuracy of the scheme is independent on the order of the WENO approximation, and only depends on the consistency of the ODE method. For our approach moreover the two are independent, which is a very interesting improvement compared to [4]. Moreover, as in the last reference our approach does not require to explicitly solve the Cauchy problem, which differentiates our approach from the one of [1,3]. Applications to scalar balance laws and to the shallow water equations confirm all the desired properties.

**REFERENCES**

[1] M.J. Castro, C. Parés-Madronal, Well-Balanced High-Order Finite Volume methods for systems of balance laws, J.Sci.Comp. 82(2), 2020

[2] M. Ciallella, D. Torlo, M. Ricchiuto, Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibrium Preservation, J.Sci.Comp. 96, 2023

[3] I. Gómez-Bueno, M.J. Castro, C. Parés-Madronal, G. Russo, Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws, Mathematics 9(15), 2021

[4] Y. Mantri, P. Oeffner, M. Ricchiuto, Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: Global flux quadrature and cell entropy correction, J.Comput.Phys. 2023, 10.1016/j.jcp.2023.112673

[5] C. Parés and C. Parés-Pulido, Well-balanced high-order finite difference methods for systems of balance laws, 425: 109880, 2021.

<footnote n="1">¹ Team CARDAMOM, INRIA, U. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, France</footnote>

<footnote n="2">² Departamento de Análisis Matemático, Estadística e I.O. y Matemática Aplicada. Facultad de Ciencias, Campus de Teatinos. Universidad de Málaga, 29071 Málaga, Spain</footnote>