Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations
Provides rigorous numerical analysis for a class of complex PDEs, but the results are incremental as they extend existing finite volume techniques to a specific problem class.
The paper proves existence and uniqueness of entropy solutions for doubly nonlinear degenerate hyperbolic-parabolic equations and shows that discrete duality finite volume schemes converge strongly to these solutions in 2D and 3D, covering non-Lipschitz nonlinearities.
We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès \cite{DomOmnes}) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, Minty-Browder type arguments, and "hyperbolic" $L^\infty$ weak-$\star$ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna, ...). Our results cover the case of non-Lipschitz nonlinearities.