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<p>Preface ix</p> <p><b>Chapter 1. Ordinary Differential Equations 1</b></p> <p>1.1. Introduction to the theory of ordinary differential equations  1</p> <p>1.1.1. Existence–uniqueness of first-order ordinary differential equations 1</p> <p>1.1.2. The concept of maximal solution  11</p> <p>1.1.3. Linear systems with constant coefficients  16</p> <p>1.1.4. Higher-order differential equations 20</p> <p>1.1.5. Inverse function theorem and implicit function theorem  21</p> <p>1.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability  27</p> <p>1.2.1. Introduction  27</p> <p>1.2.2. Fundamental notions for the analysis of numerical ODE methods 29</p> <p>1.2.3. Analysis of explicit and implicit Euler schemes  33</p> <p>1.2.4. Higher-order schemes 50</p> <p>1.2.5. Leslie’s equation (Perron–Frobenius theorem, power method)  51</p> <p>1.2.6. Modeling red blood cell agglomeration 78</p> <p>1.2.7. SEI model 87</p> <p>1.2.8. A chemotaxis problem  93</p> <p>1.3. Hamiltonian problems 102</p> <p>1.3.1. The pendulum problem  106</p> <p>1.3.2. Symplectic matrices; symplectic schemes 112</p> <p>1.3.3. Kepler problem  125</p> <p>1.3.4. Numerical results 129</p> <p><b>Chapter 2. Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems  141</b></p> <p>2.1. Introduction  141</p> <p>2.1.1. The 1D model problem; elements of modeling and analysis  144</p> <p>2.1.2. A radiative transfer problem 155</p> <p>2.1.3. Analysis elements for multidimensional problems 163</p> <p>2.2. Finite difference approximations to elliptic equations 166</p> <p>2.2.1. Finite difference discretization principles  166</p> <p>2.2.2. Analysis of the discrete problem 173</p> <p>2.3. Finite volume approximation of elliptic equations 180</p> <p>2.3.1. Discretization principles for finite volumes 180</p> <p>2.3.2. Discontinuous coefficients  187</p> <p>2.3.3. Multidimensional problems 189</p> <p>2.4. Finite element approximations of elliptic equations  191</p> <p>2.4.1. P1 approximation in one dimension 191</p> <p>2.4.2. P2 approximations in one dimension  197</p> <p>2.4.3. Finite element methods, extension to higher dimensions  200</p> <p>2.5. Numerical comparison of FD, FV and FE methods  204</p> <p>2.6. Spectral methods  205</p> <p>2.7. Poisson–Boltzmann equation; minimization of a convex function, gradient descent algorithm 217</p> <p>2.8. Neumann conditions: the optimization perspective  224</p> <p>2.9. Charge distribution on a cord 228</p> <p>2.10. Stokes problem  235</p> <p><b>Chapter 3. Numerical Simulations of Partial Differential Equations: Time-dependent Problems  267</b></p> <p>3.1. Diffusion equations  267</p> <p>3.1.1. L2 stability (von Neumann analysis) and L∞ stability: convergence  269</p> <p>3.1.2. Implicit schemes  276</p> <p>3.1.3. Finite element discretization 281</p> <p>3.1.4. Numerical illustrations  283</p> <p>3.2. From transport equations towards conservation laws  291</p> <p>3.2.1. Introduction  291</p> <p>3.2.2. Transport equation: method of characteristics 295</p> <p>3.2.3. Upwinding principles: upwind scheme 299</p> <p>3.2.4. Linear transport at constant speed; analysis of FD and FV schemes  301</p> <p>3.2.5. Two-dimensional simulations  326</p> <p>3.2.6. The dynamics of prion proliferation 329</p> <p>3.3. Wave equation 345</p> <p>3.4. Nonlinear problems: conservation laws 354</p> <p>3.4.1. Scalar conservation laws 354</p> <p>3.4.2. Systems of conservation laws  387</p> <p>3.4.3. Kinetic schemes  393</p> <p>Appendices  407</p> <p>Appendix 1  409</p> <p>Appendix 2  417</p> <p>Appendix 3  427</p> <p>Appendix 4  433</p> <p>Appendix 5  443</p> <p>Bibliography 447</p> <p>Index  455</p>

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

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