A Very Easy High-Order Well-Balanced Reconstruction for Hyperbolic Systems with Source Terms

C. Berthon, Solène Bulteau, F. Foucher, Meissa M'Baye, Victor Michel-Dansac
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引用次数: 2

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, we design a very easy correction to high-order finite volume methods. 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 avoiding the inversion of the nonlinear function that governs 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, as well as its main limitation, namely the fact that it may fail at being both well-balanced and more than second-order accurate for a specific class of initial conditions.
具有源项的双曲型系统的一种非常简单的高阶良好平衡重构
当采用基于MUSCL重构技术的高阶有限体积格式逼近带源项的双曲型系统弱解时,稳态的保持是非常具有挑战性的。实际上,所设计的重构必须保持所考虑的稳态,以获得所需的正平衡性。先验地,为了获得这样一个稳定状态,我们需要解一些强非线性方程。在这里,我们设计了一个非常容易的高阶有限体积方法的修正。这种修正可以应用于任何大于或等于2阶的格式,如muscl型格式,并确保该格式准确地保留稳定解。该方法与常用方法的主要区别在于避免了控制稳态解的非线性函数的反转。此外,对于欠定稳态解,必须同时考虑多个非线性函数。由于导出的修正只考虑控制非线性函数的评估,因此我们能够处理欠定的平稳系统。几个数值实验说明了所提出的良好平衡校正的相关性,以及它的主要局限性,即它可能无法在良好平衡和超过二阶精度的特定类别的初始条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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