Mathematics for Modeling and Scientific Computing

Inbunden, Engelska, 2016

Av Thierry Goudon, France) Goudon, Thierry (Universite Cote d'Azur

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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.

Produktinformation

Utgivningsdatum2016-11-11
Mått165 x 241 x 31 mm
Vikt812 g
FormatInbunden
SpråkEngelska
Antal sidor472
FörlagISTE Ltd and John Wiley & Sons Inc
ISBN9781848219885

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Preface ixChapter 1. Ordinary Differential Equations 11.1. Introduction to the theory of ordinary differential equations 11.1.1. Existence–uniqueness of first-order ordinary differential equations 11.1.2. The concept of maximal solution 111.1.3. Linear systems with constant coefficients 161.1.4. Higher-order differential equations 201.1.5. Inverse function theorem and implicit function theorem 211.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability 271.2.1. Introduction 271.2.2. Fundamental notions for the analysis of numerical ODE methods 291.2.3. Analysis of explicit and implicit Euler schemes 331.2.4. Higher-order schemes 501.2.5. Leslie’s equation (Perron–Frobenius theorem, power method) 511.2.6. Modeling red blood cell agglomeration 781.2.7. SEI model 871.2.8. A chemotaxis problem 931.3. Hamiltonian problems 1021.3.1. The pendulum problem 1061.3.2. Symplectic matrices; symplectic schemes 1121.3.3. Kepler problem 1251.3.4. Numerical results 129Chapter 2. Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems 1412.1. Introduction 1412.1.1. The 1D model problem; elements of modeling and analysis 1442.1.2. A radiative transfer problem 1552.1.3. Analysis elements for multidimensional problems 1632.2. Finite difference approximations to elliptic equations 1662.2.1. Finite difference discretization principles 1662.2.2. Analysis of the discrete problem 1732.3. Finite volume approximation of elliptic equations 1802.3.1. Discretization principles for finite volumes 1802.3.2. Discontinuous coefficients 1872.3.3. Multidimensional problems 1892.4. Finite element approximations of elliptic equations 1912.4.1. P1 approximation in one dimension 1912.4.2. P2 approximations in one dimension 1972.4.3. Finite element methods, extension to higher dimensions 2002.5. Numerical comparison of FD, FV and FE methods 2042.6. Spectral methods 2052.7. Poisson–Boltzmann equation; minimization of a convex function, gradient descent algorithm 2172.8. Neumann conditions: the optimization perspective 2242.9. Charge distribution on a cord 2282.10. Stokes problem 235Chapter 3. Numerical Simulations of Partial Differential Equations: Time-dependent Problems 2673.1. Diffusion equations 2673.1.1. L2 stability (von Neumann analysis) and L∞ stability: convergence 2693.1.2. Implicit schemes 2763.1.3. Finite element discretization 2813.1.4. Numerical illustrations 2833.2. From transport equations towards conservation laws 2913.2.1. Introduction 2913.2.2. Transport equation: method of characteristics 2953.2.3. Upwinding principles: upwind scheme 2993.2.4. Linear transport at constant speed; analysis of FD and FV schemes 3013.2.5. Two-dimensional simulations 3263.2.6. The dynamics of prion proliferation 3293.3. Wave equation 3453.4. Nonlinear problems: conservation laws 3543.4.1. Scalar conservation laws 3543.4.2. Systems of conservation laws 3873.4.3. Kinetic schemes 393Appendices 407Appendix 1 409Appendix 2 417Appendix 3 427Appendix 4 433Appendix 5 443Bibliography 447Index 455