Trivisa, Konstantina; Weber, Franziska A convergent explicit finite difference scheme for a mechanical model for tumor growth. (English) Zbl 1360.35197 ESAIM, Math. Model. Numer. Anal. 51, No. 1, 35-62 (2017). Summary: Mechanical models for tumor growth have been used extensively in recent years for the analysis of medical observations and for the prediction of cancer evolution based on image analysis. This work deals with the numerical approximation of a mechanical model for tumor growth and the analysis of its dynamics. The system under investigation is given by a multi-phase flow model: The densities of the different cells are governed by a transport equation for the evolution of tumor cells, whereas the velocity field is given by a Brinkman regularization of the classical Darcy’s law. An efficient finite difference scheme is proposed and shown to converge to a weak solution of the system. Our approach relies on convergence and compactness arguments in the spirit of P.-L. Lions [Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)]. Cited in 3 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 92C37 Cell biology 35D30 Weak solutions to PDEs 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92C50 Medical applications (general) 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs Keywords:tumor growth models; cancer progression; mixed models; multi-phase flow; finite difference scheme; existence Citations:Zbl 0908.76004 PDFBibTeX XMLCite \textit{K. Trivisa} and \textit{F. Weber}, ESAIM, Math. Model. Numer. Anal. 51, No. 1, 35--62 (2017; Zbl 1360.35197) Full Text: DOI arXiv