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Lax theorem and finite volume schemes

Author: Bruno Despres
Journal: Math. Comp. 73 (2004), 1203-1234
MSC (2000): Primary 65M12; Secondary 65M15, 65M60
Published electronically: November 5, 2003
MathSciNet review: 2047085
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Abstract: This work addresses a theory of convergence for finite volume methods applied to linear equations. A non-consistent model problem posed in an abstract Banach space is proved to be convergent. Then various examples show that the functional framework is non-empty. Convergence with a rate $h^{\frac12}$ of all TVD schemes for linear advection in 1D is an application of the general result. Using duality techniques and assuming enough regularity of the solution, convergence of the upwind finite volume scheme for linear advection on a 2D triangular mesh is proved in $L^\alpha$, $2\leq \alpha\leq +\infty$: provided the solution is in $W^{1,\infty}$, it proves a rate of convergence $h^{\frac14 -\varepsilon}$ in $L^\infty$.

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Additional Information

Bruno Despres
Affiliation: Commissariat à l’Energie Atomique, 91680, Bruyères le Chatel, France
Address at time of publication: Laboratoire d’analyse numérique, 175 rue du Chevaleret, Université de Paris VI, 75013 Paris, France

Keywords: Finite volume schemes, linear advection
Received by editor(s): November 28, 2001
Received by editor(s) in revised form: January 10, 2003
Published electronically: November 5, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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