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Journal Articles SIAM Journal on Scientific Computing Year : 2022

A very easy high-order well-balanced reconstruction for hyperbolic systems with source terms

Abstract

When adopting high-order finite-volume schemes based on MUSCL reconstruction techniques to approximate the weak solutions of hyperbolic systems with source terms, the preservation of the steady states turns out to be very challenging. Indeed, the designed reconstruction must preserve the steady states under consideration in order to get the required well-balancedness property. A priori, to capture such a steady state, one needs to solve some strongly nonlinear equations. Here, in order to preserve the required well-balancedness property to be satisfied by finite volume methods, we design a very easy correction. This correction can be applied to any scheme of order greater than or equal to 2, such as a MUSCL-type scheme, and ensures that this scheme exactly preserves the steady solutions. The main discrepancy with usual techniques lies in never having to invert the nonlinear function governing the steady solutions. Moreover, for under-determined steady solutions, several nonlinear functions must be considered simultaneously. Since the derived correction only considers the evaluation of the governing nonlinear functions, we are able to deal with under-determined stationary systems. Several numerical experiments illustrate the relevance of the proposed well-balanced correction.
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Dates and versions

hal-03271103 , version 1 (25-06-2021)
hal-03271103 , version 2 (04-07-2022)

Identifiers

Cite

Christophe Berthon, Solène Bulteau, Françoise Foucher, Meissa M'Baye, Victor Michel-Dansac. A very easy high-order well-balanced reconstruction for hyperbolic systems with source terms. SIAM Journal on Scientific Computing, 2022, 44 (4), pp.A2506-A2535. ⟨10.1137/21M1429230⟩. ⟨hal-03271103v2⟩
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