Preprint 2000-035

Convergence of Finite Difference Schemes for Viscous and Inviscid Conservation Laws with Rough Coefficients

Abstract: We consider the initial value problem for degenerate viscous and inviscid scalar conservation laws where the flux function depends on the spatial location through a ``rough'' coefficient function $k(x)$. We show that the Engquist-Osher (and hence all monotone) finite difference approximations converge to the unique entropy solution of the governing equation if, among other demands, $k'$ is in $BV$, thereby providing alternative (new) existence proofs for entropy solutions of degenerate convection-diffusion equations as well as new convergence results for their finite difference approximations. In the inviscid case, we also provide a rate of convergence. Our convergence proofs are based on deriving a series of a priori estimates and using a general $L^p$ compactness criterion.

Paper:

Available as PDF (364 Kbytes).

Author(s):

Kenneth Hvistendahl Karlsen, <kennethk@mi.uib.no>

Nils Henrik Risebro, <nilshr@math.uio.no>

Publishing information:

Preprint, Department of Mathematics, University of Bergen.

Comments:

Updated version, February 22 2001.

Submitted by:

<kennethk@mi.uib.no> September 7 2000.