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Early Well-Balanced Derivations for Various Systems

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Computing Qualitatively Correct Approximations of Balance Laws

Part of the book series: SIMAI Springer Series ((SEMA SIMAI,volume 2))

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Abstract

Even more than the formers ones, the present chapter will emphasize that well-balanced methods consist essentially in recycling astutely homogeneous techniques in order to take advantage of their strong stability properties in the more delicate context of non-homogeneous systems: for instance, the well-known fact that the Godunov scheme has zero numerical viscosity at steady-state (the preservation of steady-state curves is actually a consequence of this feature). Obviously, there is a price to pay: if nearly all source terms can be reformulated painlessly as nonconservative products, the corresponding linearly degenerate fields can locally interfere with the already existing characteristic fields of the original equations, meaning that strict hyperbolicity doesn’t hold unconditionally. Such a pathological situation is referred to as non-linear resonance and led to numerous theoretical investigations. Kinetic formalism can sometimes save the day: in Chapter 6, we shall study a kinetic scheme which doesn’t suffer from any resonance issue. Hereafter, nonlinear resonance will be avoided, following the assumptions of the seminal paper [39].

Mathematicians are like Frenchmen: whatever is said to them, they translate into their own language, and forthwith it is something entirely different.

Goethe, Maximen und Reflektionen

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Notes

  1. 1.

    Rigorously, the average must be computed along the integral curve of the steady-state equation, but this is difficult to perform in practice.

  2. 2.

    Huang-Liu’s scheme suffers a little bit from its excessive numerical viscosity: indeed, at steadystate, one has \( {\mathbf{u}}_{j-\frac{1}{2}}^{+}={\mathbf{u}}_{j-\frac{1}{2}}^{-} \) for any j and the well-balanced property exists if \( 2{\mathbf{u}}_j^n={\mathbf{u}}_{j-\frac{1}{2}}^{+}+{\mathbf{u}}_{j+\frac{1}{2}}^{-} \) ,which holds only up to second-order terms in general (except for stationary curves linear in x).

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  1. Istituto per le Applicazioni del Calcolo “Mauro Picone”, CNR, Rome, Italy

    Laurent Gosse

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Gosse, L. (2013). Early Well-Balanced Derivations for Various Systems. In: Computing Qualitatively Correct Approximations of Balance Laws. SIMAI Springer Series, vol 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2892-0_4

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