Stability and existence of solutions of time-implicit finite volume schemes for viscous nonlinear conservation laws

Abstract

We introduce a time-implicit Voronoi-box-based finite volume discretization for the initial-boundary value problem of a scalar nonlinear viscous conservation law in a heterogeneous one-, two- or three-dimensional domain. Using notations from the theory of explicit finite volume methods for hyperbolic problems and results from the Perron–Frobenius theory of nonnegative matrices, we establish various existence, stability and uniqueness results for the discrete problem. The class of schemes introduced covers hyperbolic problems as well as nonlinear diffusion problems. To clarify our results, we provide numerical examples, and we show the practical relevance of our considerations with a groundwater flow example.