An error estimate for the finite difference approximation to degenerate convection–diffusion equations

Abstract

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection–diffusion equations in one space dimension, and prove an L 1 error estimate. Precisely, we show that the \({L^1_{\rm{loc}}}\) difference between the approximate solution and the unique entropy solution converges at a rate \({\mathcal{O}(\Delta x^{1/11})}\) , where \({\Delta x}\) is the spatial mesh size. If the diffusion is linear, we get the convergence rate \({\mathcal{O}(\Delta x^{1/2})}\) , the point being that the \({\mathcal{O}}\) is independent of the size of the diffusion.