# Preprint 2012-010

# L¹ Error Estimates for Difference Approximations of Degenerate Convection-Diffusion Equations

## Kenneth H. Karlsen, Nils Henrik Risebro and Erlend B. Storrøsten

**Abstract:**
We analyze a class of semi-discrete monotone difference
schemes for degenerate convection-diffusion equations
in one spatial dimension.
These nonlinear equations are well-posed
within a class of (discontinuous) entropy solutions.
We prove that the $L^1$ error between the approximate
solutions and the unique entropy solution is
$\mathcal{O}(∆x^{1/3})$,
where $∆x$ denotes the spatial discretization parameter.
This result should be compared with the classical
$\mathcal{O}(∆x^{1/2})$
result for conservation laws [20],
and a very recent error estimate of $\mathcal{O}(∆x^{1/11})$
for degenerate convection-diffusion equations [18].

**References**

[18] K. H. Karlsen, U. Koley, and N. H. Risebro. An error estimate for the finite difference approximation to degenerate convection-diffusion equations.

*Numer. Math.*, 2012.

[20] N. N. Kuznetsov. The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation.

*U.S.S.R. Computational Math. and Math. Phys.*,

**16**(6):105–119, 1976.