𝐿¹ error estimates for difference approximations of degenerate convection-diffusion equations

Abstract

We analyze monotone finite difference schemes for strongly 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 L^1 error between the approximate and exact solutions is O ( Δ x 1 / 3 ) \mathcal {O}(\Delta x^{1/3}) , where Δ x \Delta x is the spatial grid parameter. This result should be compared with the classical O ( Δ x 1 / 2 ) \mathcal {O}(\Delta x^{1/2}) error estimate for conservation laws (Kuznecov, 1976), and a recent estimate of O ( Δ x 1 / 11 ) \mathcal {O}(\Delta x^{1/11}) for degenerate convection-diffusion equations (Karlsen, Koley, Risebro 2012).