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Table of contents
1 Introduction
1.1 Motivations
1.2 Mathematical representation of living tissues
1.2.1 First steps of mathematical modelling
1.2.2 The Cahn-Hilliard model for modelling of tissues and tumours
1.2.3 The Keller-Segel model and the volume-filling approach
1.3 General assumptions and preliminaries
1.4 Numerical simulation: foundations
1.5 Summary of the thesis
1.5.1 Towards an efficient numerical scheme for the degenerate Cahn-Hilliard model for Biology
1.5.2 Modelling of specific scenarios in Biology
1.5.3 Structure-preserving numerical method for nonlinear models
1.6 Discussion and perspectives
1.6.1 Simulation of the relaxed-degenerate Cahn-Hilliard model and effect of the relaxation
1.6.2 Support a deeper understanding of key mechanisms in tumor progression
I The Cahn-Hilliard equation for Biology
2 Relaxation of the Cahn-Hilliard equation for Biology
2.1 Introduction
2.2 The regularized problem
2.2.1 Regularization procedure
2.2.2 Existence for the regularized problem
2.2.3 Energy, entropy and a priori estimates
2.2.4 Inequalities
2.3 Existence: convergence as ! 0
2.4 Convergence as ! 0
2.5 Long-time behavior
2.6 Conclusion
3 Structure-preserving numerical method for the relaxed-degenerate Cahn- Hilliard model
3.1 Introduction
3.2 Notations
3.3 Definition of the regularized problem
3.4 Nonlinear semi-implicit scheme
3.4.1 Description of the nonlinear numerical scheme
3.4.2 Well-posedness of the regularized problem and stability bounds
3.4.3 Well-posedness of the non regularized problem and stability
3.4.4 Convergence analysis
3.5 Non-linear semi-implicit multi-dimensional upwind numerical scheme
3.6 Linearized semi-implicit numerical scheme
3.7 Numerical simulations
3.7.1 Numerical results: test cases
3.7.2 Effect of the relaxation parameter
3.8 Conclusion
3.A Proof of M-matrix properties in the 1D and 2D cases
II Modification of existing nonlinear PDE models, numerical simulation, and application in Biology.
4 Treatment-induced shrinking of tumour aggregates: A nonlinear volumefilling chemotactic approach
4.1 Introduction
4.2 Description of the experiments
4.3 Mathematical model
4.3.1 Volume-filling approach for chemotaxis: first part P1
4.3.2 PDE system including the treatment: Part P2
4.4 Linear stability analysis and pattern formation
4.4.1 Dimensionless model
4.4.2 First part: Formation of the aggregates
4.4.3 Second part: Treatment
4.5 Numerical simulations
4.5.1 Biological relevance of the model parameters
4.5.2 Numerical results for a one dimensional case
4.5.3 Numerical results for a two dimensional case
4.6 Discussion of results and perspectives
4.A Derivation of the general model
4.B Stability analysis
4.C Description of the numerics
4.D One dimensional numerical results
5 Compressible Navier-Stokes-Cahn-Hilliard model for the modelling of tumor invasion in healthy tissue
5.1 Introduction
5.2 Derivation of the model
5.2.1 Notation and definitions
5.2.2 Mass balance equations
5.2.3 Balance of linear momentum
5.2.4 Energy balance
5.2.5 Entropy balance and Clausius-Duhem inequality
5.2.6 Constitutive assumptions and model equations
5.2.7 Summary of the model equations
5.3 General assumptions and biologically relevant choice of the model functions
5.3.1 General forms and assumptions
5.3.2 Biologically consistent choice of functions
5.3.3 Non-dimensionalized model
5.4 Large friction hypothesis
5.5 Finite volume numerical scheme
III Structure-preserving numerical method for nonlinear PDEs
6 The Scalar Auxiliary Variable method for the volume-filling Keller-Segel model.
6.1 Introduction
6.2 Numerical scheme
6.2.1 Finite element framework
6.2.2 Fully discrete scheme
6.2.3 Matrix formulation
6.2.4 Upwind stabilization
6.2.5 Solving Algorithm
6.3 Existence of a non-negative solution and stability bound
6.3.1 Existence of a discrete non-negative solution
6.3.2 Discrete energy a priori estimate
6.4 Numerical results
6.4.1 1D numerical results
6.5 Conclusion
7 Conservation properties and long time behavior of the Scalar Auxiliary Variable method for nonlinear dispersive equations.
7.1 Introduction
7.2 Numerical scheme
7.2.1 Time and space discretisation of the SAV model
7.2.2 The fully discrete SAV scheme
7.3 Conservation properties and inequalities
7.4 Convergence analysis
7.4.1 Notations
7.4.2 Convergence theorem
7.5 Error analysis
7.6 Numerical experiments
7.6.1 First test case: cubic nonlinearity
7.6.2 Second test case: cubic nonlinearity with non-smooth initial condition
7.6.3 Third test case: non-integer exponent
7.6.4 Computing ground states
7.A Gradient flow with discrete normalization for computing ground state
Bibliography
