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Lyapunov Functional for Linear Error Estimates

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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

The main goal of the present Chapter is to emphasize the qualitative difference between Time-Splitting (TS, also called Fractional Step, FS) and Well-Balanced (WB) numerical schemes when it comes to computing the entropy solution [26] of a simple scalar, yet non-resonant, balance law:

$$ {\partial}_tu+{\partial}_xf(u)=k(x)g(u),\kern1.68em 0\le k\in {L}^1\cap {C}^0\left(\mathbb{R}\right) $$

A child of five would understand this. Send someone to fetch a child of five.

Groucho Marx

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Notes

  1. 1.

    Observe that it would already be problematic for an homogeneous scalar conservation law in which k ≡ 0 because its L 1 error is known to increase in time like \( \mathcal{O} \) (\( \sqrt{t}, \)) as explained in e.g. [36].

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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). Lyapunov Functional for Linear Error Estimates. 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_3

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