A fully semi-Lagrangian technique for viscous and dispersive conservation laws

We consider a scheme of Semi-Lagrangian type for the approximation of conservation laws with convex or concave flux. Following a previously proposed approach, the scheme treats in explicit form the diffusive/dispersive terms, although it still requires an implicit phase to determine the correct characteristic. We prove consistency and one-sided Lipschitz stability in the viscous case, and show that, despite being nonconservative, the scheme has good performances and negligible conservation error with large Courant numbers as far as the diffusion scales are resolved. Then, the approach is generalized to conservation laws with dispersive terms (in particular, the Korteweg–de Vries equation). We also validate the scheme with a set of numerical examples.