A Lax-Wendroff type theorem of doubly conservative schemes for degenerate convection-diffusion equations

It is well known that a numerical scheme for solving a hyperbolic conservation law, when convergent, may not always converge to a weak solution. The remedy is the famous Lax-Wendroff theorem, stating that a conservative numerical scheme, when convergent, always converges to a weak solution of the conservation law. In this paper, we address the same issue for numerically solving degenerate non-linear diffusion or convection-diffusion equations. We start with an example showing that a conservative scheme, in the sense of that for conservation laws, may still converge to a function which is not a weak solution of the degenerate non-linear diffusion equation. We then introduce a stronger form of conservative schemes, which we term as doubly-conservative (DoC) schemes, that would allow the proof of a Lax-Wendroff type theorem, namely if a DoC scheme converges, then it will converge to the weak solution of the degenerate non-linear diffusion or convection-diffusion equation. The concept of DoC schemes is applicable to high order schemes, on one-dimensional non-uniform meshes as well as on two-dimensional Cartesian meshes and unstructured triangular meshes. Finally, we design a new DoC local discontinuous Galerkin scheme that remains semi-discrete stable even when the diffusion coefficient degenerates. Numerical experiments demonstrate optimal convergence rates of this new scheme and validate our theoretical results.