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.
Highlights
IntroductionConsider the following scalar hyperbolic conservation law in R2 with a homogeneous source term and an initial data of bounded variation (BV):
Consider the following scalar hyperbolic conservation law in R2 with a homogeneous source term and an initial data of bounded variation (BV):∂tα + div (uf (α)) = 0 in ΩT and }︂ α(0, ·) = α0 in Ω, (1.1)where α is the unknown, α0 : Ω → R is known a priori function of BV, u = (u, v) is the advecting velocity, ΩT := (0, T ) × Ω, Ω := I × J, I := (a, b) ⊂ R and J := (c, d) ⊂ R are intervals
Finite volume methods are extensively used to discretise and compute numerical solutions to (1.1) since such schemes respect the conservation of mass property associated with the underlying partial differential equation (PDE)
Summary
Consider the following scalar hyperbolic conservation law in R2 with a homogeneous source term and an initial data of bounded variation (BV):. Where α is the unknown, α0 : Ω → R is known a priori function of BV, u = (u, v) is the advecting velocity, ΩT := (0, T ) × Ω, Ω := I × J, I := (a, b) ⊂ R and J := (c, d) ⊂ R are intervals. Nonlinear flux, finite volume schemes, bounded variation, Cartesian grids, convergence analysis, breast cancer model.
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