Strong bounded variation estimates for the multi-dimensional finite volume approximation of scalar conservation laws and application to a tumour growth model

Abstract

A uniform bounded variation estimate for finite volume approximations of the nonlinear scalar conservation law ∂tα + div(uf(α)) = 0 in two and three spatial dimensions with an initial data of bounded variation is established. We assume that the divergence of the velocity div(u) is of bounded variation instead of the classical assumption that div(u) is zero. The finite volume schemes analysed in this article are set on nonuniform Cartesian grids. A uniform bounded variation estimate for finite volume solutions of the conservation law ∂tα + div(F(t,x,α)) = 0, where divxF ≠ 0 on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in Lp spaces, which is essential to prove the existence of a solution for a partial differential equation with nonlinear terms in α, when the uniqueness of the solution is not available. This application is demonstrated by establishing the existence of a weak solution for a model that describes the evolution of initial stages of breast cancer proposed by Franks et al. [J. Math. Biol. 47 (2003) 424–452]. The model consists of four coupled variables: tumour cell concentration, tumour cell velocity–pressure, and nutrient concentration, which are governed by a hyperbolic conservation law, viscous Stokes system, and Poisson equation, respectively. Results from numerical tests are provided and they complement theoretical findings.