Discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations

TL;DR

This paper develops and analyzes discrete duality finite volume schemes for doubly nonlinear degenerate hyperbolic-parabolic equations, proving convergence to entropy solutions in multiple dimensions with non-Lipschitz nonlinearities.

Contribution

The paper introduces a novel discrete duality finite volume scheme for complex nonlinear PDEs and proves its convergence to entropy solutions, extending applicability to non-Lipschitz cases.

Findings

Established existence and uniqueness of entropy solutions.

Proved convergence of discrete schemes to the continuous solution.

Derived discrete duality formulas and entropy dissipation inequalities.

Abstract

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 Omn\`es \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.