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

Chirappurathu Remesan, Gopikrishnan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 4, pp. 1405-1437

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($$(t,x,α)) = 0, where div$$ ≠ 0 on nonuniform Cartesian grids is also proved. Such an estimate provides compactness for finite volume approximations in L$$ 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.

Classification : 65M08, 65M12, 35L65
Keywords: Scalar conservation laws, nonlinear flux, finite volume schemes, bounded variation, Cartesian grids, convergence analysis, breast cancer model

[1] D. Boffi, F. Brezzi and F. Michel, Mixed Finite Element Methods and Applications. In: Vol. 44 of Series in Computational Mathematics. Springer (2013). | MR | Zbl

[2] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models. In: Vol. 183 of Applied Mathematical Sciences (Switzerland). Springer (2013). | MR | Zbl | DOI

[3] H. M. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures. Math. Med. Biol. 20 (2003) 341–366. | Zbl | DOI

[4] C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate. ESAIM: M2AN 33 (1999) 129–156. | MR | Zbl | Numdam | DOI

[5] S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh. Numer. Math. 66 (1993) 139–157. | MR | Zbl | DOI

[6] B. Cockburn, P. A. Gremaud and J. X. Yang, A priori error estimates for numerical methods for scalar conservation laws part III: multidimensional flux-splitting monotone schemes on non-Cartesian grids. SIAM J. Numer. Anal. 35 (1998) 1775–1803. | MR | Zbl | DOI

[7] E. Conway and J. Smoller, Global solutions of the cauchy problem for quasi-linear first-order equations in several space variables. Commun. Pure Appl. Math. 19 (1966) 95–105. | MR | Zbl | DOI

[8] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comput. 34 (1980) 1–21. | MR | Zbl | DOI

[9] B. Després, An explicit a priori estimate for a finite volume approximation of linear advection on non-Cartesian grids. SIAM J. Numer. Anal. 42 (2005) 484–504. | MR | Zbl | DOI

[10] R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–547. | MR | Zbl | DOI

[11] J. Droniou, R. Eymard, T. Gallouët, C. Guichard and R. Herbin, The Gradient Discretisation Method. In: Vol. 82 of Mathematics & Applications. Springer (2018). | MR

[12] J. Droniou, N. Nataraj and G. C. Remesan, Convergence analysis of a numerical scheme for a tumour growth model (2019).

[13] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press Inc., Florida (2015). | MR | Zbl | DOI

[14] R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, edited by P. G. Ciarlet and J.-L. Lions. In: Vol. 7 of Solution of Equation in R$$ (Part 3), Techniques of Scientific Computing (Part 3). Elsevier, Amsterdam (2000) 713–1018. | MR | Zbl | DOI

[15] S. J. Franks, H. M. Byrne, J. R. King, J. C. E. Underwood and C. E. Lewis, Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol. 47 (2003) 424–452. | MR | Zbl | DOI

[16] H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws. Springer-Verlag, Berlin Heidelberg (2015). | MR | Zbl | DOI

[17] K. H. Karlsen and J. D. Towers, Convergence of monotone schemes for conservation laws with zero-flux boundary conditions. Adv. Appl. Math. Mech. 9 (2017) 515–542. | MR | Zbl | DOI

[18] N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys. 16 (1976) 105–119. | Zbl | MR | DOI

[19] V. G. Maz’Ya and J. Rossmann, Mathematical Surveys and Monographs. American Mathematical Society (2010). | Zbl | DOI

[20] B. Merlet, $L^{\infty }$- and $L^2$ -error estimates for a finite volume approximation of linear advection. SIAM J. Numer. Anal. 46 (2007) 124–150. | MR | Zbl | DOI

[21] B. Merlet and J. Vovelle, Error estimate for finite volume scheme. Numer. Math. 106 (2007) 129–155. | MR | Zbl | DOI

[22] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comput. 40 (1983) 91–106. | MR | Zbl | DOI