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A convergent explicit finite difference scheme for a mechanical model for tumor growth. (English) Zbl 1360.35197

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)].

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

Citations:

Zbl 0908.76004
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