$L^\infty$- and $L^2$-Error Estimates for a Finite Volume Approximation of Linear Advection

Authors:
Benoit Merlet
Affiliations:
-
Venue:
SIAM Journal on Numerical Analysis
Year:
2007

Citing 0
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Error estimate for the upwind finite volume method for the nonlinear scalar conservation law

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An optimal error estimate for upwind Finite Volume methods for nonlinear hyperbolic conservation laws

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Abstract

We study the convergence of the upwind finite volume scheme applied to the linear advection equation with a Lipschitz divergence-free speed in $\R^d$. We prove an $h^{1/2-\varepsilon}$-error estimate in the $L^\infty(\R^d\times [0,T])$-norm for Lipschitz initial data. The expected optimal result is an $h^{1/2}$-error estimate. In a second part, we also prove an $h^{1/2}$-error estimate in the $L^\infty(0,T;L^2(\R^d))$-norm for initial data in $H^1(\R^d)$.