A class of nonlinear degenerate mixed-type hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions in two and three space dimensions is considered. This class of equations combines a number of difficulties such as nonlinear convection, doubly nonlinear diffusion, strong degeneracy and shocks which necessitate the use of suitable framework of discontinuous entropy solutions.
The main goal of the presented work is to propose and analyse a specific class of finite volume schemes for this class of mixed-type equations. It appears to be the first time that convergent numerical schemes are constructed for this class of equations and in the generality considered in this work. The authors first establish the existence and uniqueness of entropy solutions working with weak solutions and imposing additional entropy inequalities in the spirit of Kruzhov. The authors then construct and analyse discrete duality finite volume (DDFV) schemes in two and three dimensions and derive a series of discrete duality formulas and entropy dissipation inequalities. The existence of solutions to the discrete problem is established, and is proved that sequences of approximate solutions generate by the DDFV schemes converge strongly to the (unique) entropy solution of the continuous problem. Strong convergence is obtained of both convective and diffusive fluxes. Related consistency estimates and properties are also given.
It should be noted that the paper contains extensive references on the subject which include references to some well-known finite volume schemes for nonlinear convection-diffusion and their relation to the proposed schemes.