Mathematics for Modeling and Scientific Computing

This book provides the mathematical basis for investigating numerically equations from physics, life sciences or engineering. Tools for analysis and algorithms are confronted to a large set of relevant examples that show the difficulties and the limitations of the most naïve approaches. These examples not only provide the opportunity to put into practice mathematical statements, but modeling issues are also addressed in detail, through the mathematical perspective.

Thierry GOUDON, Senior research scientist INRIA, Sophia Antipolis Mediterranee Research Centre & Labo. J. A. Dieudonne, University of Nice Sophia Antipolis & CNRS, France

Preface ix

Chapter 1. Ordinary Differential Equations 1

1.1. Introduction to the theory of ordinary differential equations  1

1.1.1. Existence–uniqueness of first-order ordinary differential equations 1

1.1.2. The concept of maximal solution  11

1.1.3. Linear systems with constant coefficients  16

1.1.4. Higher-order differential equations 20

1.1.5. Inverse function theorem and implicit function theorem  21

1.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability  27

1.2.1. Introduction  27

1.2.2. Fundamental notions for the analysis of numerical ODE methods 29

1.2.3. Analysis of explicit and implicit Euler schemes  33

1.2.4. Higher-order schemes 50

1.2.5. Leslie’s equation (Perron–Frobenius theorem, power method)  51

1.2.6. Modeling red blood cell agglomeration 78

1.2.7. SEI model 87

1.2.8. A chemotaxis problem  93

1.3. Hamiltonian problems 102

1.3.1. The pendulum problem  106

1.3.2. Symplectic matrices; symplectic schemes 112

1.3.3. Kepler problem  125

1.3.4. Numerical results 129

Chapter 2. Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems  141

2.1. Introduction  141

2.1.1. The 1D model problem; elements of modeling and analysis  144

2.1.2. A radiative transfer problem 155

2.1.3. Analysis elements for multidimensional problems 163

2.2. Finite difference approximations to elliptic equations 166

2.2.1. Finite difference discretization principles  166

2.2.2. Analysis of the discrete problem 173

2.3. Finite volume approximation of elliptic equations 180

2.3.1. Discretization principles for finite volumes 180

2.3.2. Discontinuous coefficients  187

2.3.3. Multidimensional problems 189

2.4. Finite element approximations of elliptic equations  191

2.4.1. P1 approximation in one dimension 191

2.4.2. P2 approximations in one dimension  197

2.4.3. Finite element methods, extension to higher dimensions  200

2.5. Numerical comparison of FD, FV and FE methods  204

2.6. Spectral methods  205

2.7. Poisson–Boltzmann equation; minimization of a convex function, gradient descent algorithm 217

2.8. Neumann conditions: the optimization perspective  224

2.9. Charge distribution on a cord 228

2.10. Stokes problem  235

Chapter 3. Numerical Simulations of Partial Differential Equations: Time-dependent Problems  267

3.1. Diffusion equations  267

3.1.1. L2 stability (von Neumann analysis) and L∞ stability: convergence  269

3.1.2. Implicit schemes  276

3.1.3. Finite element discretization 281

3.1.4. Numerical illustrations  283

3.2. From transport equations towards conservation laws  291

3.2.1. Introduction  291

3.2.2. Transport equation: method of characteristics 295

3.2.3. Upwinding principles: upwind scheme 299

3.2.4. Linear transport at constant speed; analysis of FD and FV schemes  301

3.2.5. Two-dimensional simulations  326

3.2.6. The dynamics of prion proliferation 329

3.3. Wave equation 345

3.4. Nonlinear problems: conservation laws 354

3.4.1. Scalar conservation laws 354

3.4.2. Systems of conservation laws  387

3.4.3. Kinetic schemes  393

Appendices  407

Appendix 1  409

Appendix 2  417

Appendix 3  427

Appendix 4  433

Appendix 5  443

Bibliography 447

Index  455

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