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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.

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