Mathematics for Modeling and Scientific Computing.

By: Goudon, ThierryPublisher: Newark : John Wiley & Sons, Incorporated, 2016Copyright date: ©2016Edition: 1st edDescription: 1 online resource (477 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9781119371182Subject(s): Differential equationsGenre/Form: Electronic books. Additional physical formats: Print version:: Mathematics for Modeling and Scientific ComputingDDC classification: 515/.35 2 23 LOC classification: QA371.G683 2016Online resources: Click to View
Contents:
Cover -- Title Page -- Copyright -- Contents -- Preface -- 1. Ordinary Differential Equations -- 1.1. Introduction to the theory of ordinary differential equations -- 1.1.1. Existence-uniqueness of first-order ordinary different equations -- 1.1.2. The concept of maximal solution -- 1.1.3. Linear systems with constant coefficients -- 1.1.4. Higher-order differential equations -- 1.1.5. Inverse function theorem and implicit function theorem -- 1.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability -- 1.2.1. Introduction -- 1.2.2. Fundamental notions for the analysis of numerical ODE methods -- 1.2.3. Analysis of explicit and implicit Euler schemes -- 1.2.4. Higher-order schemes -- 1.2.5. Leslie's equation (Perron-Frobenius theorem, power method) -- 1.2.5.1. Discrete time model -- 1.2.5.2. Model in continuous time -- 1.2.5.3. Proof of the Perron-Frobenius theorem -- 1.2.5.4. Computation of the leading eigenvalue -- 1.2.6. Modeling red blood cell agglomeration -- 1.2.7. SEI model -- 1.2.8. A chemotaxis problem -- 1.3. Hamiltonian problems -- 1.3.1. The pendulum problem -- 1.3.2. Symplectic matrices -- symplectic schemes -- 1.3.3. Kepler problem -- 1.3.3.1. Review of conic sections and vector product -- 1.3.4. Numerical results -- 2. Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems -- 2.1. Introduction -- 2.1.1. The 1D model problem -- elements of modeling and analysis -- 2.1.2. A radiative transfer problem -- 2.1.2.1. Uniqueness given existence -- 2.1.2.2. A priori estimates -- 2.1.2.3. Existence of solutions -- 2.1.2.4. Shooting method -- 2.1.2.5. Interpretation as a minimization problem -- 2.1.3. Analysis elements for multidimensional problems -- 2.2. Finite difference approximations to elliptic equations.
2.2.1. Finite difference discretization principles -- 2.2.2. Analysis of the discrete problem -- 2.3. Finite volume approximation of elliptic equations -- 2.3.1. Discretization principles for finite volumes -- 2.3.2. Discontinuous coefficients -- 2.3.3. Multidimensional problems -- 2.4. Finite element approximations of elliptic equations -- 2.4.1. P1 approximation in one dimension -- 2.4.2. P2 approximations in one dimension -- 2.4.3. Finite element methods, extension to higher dimensions -- 2.5. Numerical comparison of FD, FV and FE methods -- 2.6. Spectral methods -- 2.7. Poisson-Boltzmann equation -- minimization of a convex function, gradient descent algorithm -- 2.8. Neumann conditions: the optimization perspective -- 2.9. Charge distribution on a cord -- 2.10. Stokes problem -- 3. Numerical Simulations of Partial Differential Equations: Time-dependent Problems -- 3.1. Diffusion equations -- 3.1.1. L2 stability (von Neumann analysis) and L∞ stability: convergence -- 3.1.2. Implicit schemes -- 3.1.3. Finite element discretization -- 3.1.4. Numerical illustrations -- 3.2. From transport equations towards conservation laws -- 3.2.1. Introduction -- 3.2.2. Transport equation: method of characteristics -- 3.2.3. Upwinding principles: upwind scheme -- 3.2.4. Linear transport at constant speed: analysis of FD and FV schemes -- 3.2.5. Two-dimensional simulations -- 3.2.6. The dynamics of prion proliferation -- 3.3. Wave equation -- 3.4. Nonlinear problems: conservation laws -- 3.4.1. Scalar conservation laws -- 3.4.1.1. Elements of analysis -- 3.4.1.2. Lax-Friedrichs and Rusjanov schemes for scalar conservation laws -- 3.4.1.3. Numerical illustrations -- 3.4.1.4. Propagation of a forest fire -- 3.4.2. Systems of conservation laws -- 3.4.2.1. Introduction to gas dynamics equations -- 3.4.2.2. Lax-Friedrichs and Rusjanov schemes for the Euler equations.
3.4.3. Kinetic schemes -- 3.4.3.1. Scalar conservation laws -- 3.4.3.2. Kinetic scheme for Euler equations: monatomic case -- APPENDICES -- Appendix 1. Solving Linear Systems -- A1.1. Condition number of a matrix -- A1.2. Spectral radius -- A1.3. Conjugate gradient -- Appendix 2. Numerical Integration -- Appendix 3. A Peetre-Tartar Equivalence Theorem -- Appendix 4. Schauder's Theorem -- Appendix 5. Fundamental Solutions of the Laplacian in Dimension 1 and 2 -- A5.1. Dimension 1 -- A5.2. Dimension 2 -- A5.3. Higher dimensions -- Bibliography -- Index -- Other titles from iSTE in Mathematics and Statistics -- EULA.
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Cover -- Title Page -- Copyright -- Contents -- Preface -- 1. Ordinary Differential Equations -- 1.1. Introduction to the theory of ordinary differential equations -- 1.1.1. Existence-uniqueness of first-order ordinary different equations -- 1.1.2. The concept of maximal solution -- 1.1.3. Linear systems with constant coefficients -- 1.1.4. Higher-order differential equations -- 1.1.5. Inverse function theorem and implicit function theorem -- 1.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability -- 1.2.1. Introduction -- 1.2.2. Fundamental notions for the analysis of numerical ODE methods -- 1.2.3. Analysis of explicit and implicit Euler schemes -- 1.2.4. Higher-order schemes -- 1.2.5. Leslie's equation (Perron-Frobenius theorem, power method) -- 1.2.5.1. Discrete time model -- 1.2.5.2. Model in continuous time -- 1.2.5.3. Proof of the Perron-Frobenius theorem -- 1.2.5.4. Computation of the leading eigenvalue -- 1.2.6. Modeling red blood cell agglomeration -- 1.2.7. SEI model -- 1.2.8. A chemotaxis problem -- 1.3. Hamiltonian problems -- 1.3.1. The pendulum problem -- 1.3.2. Symplectic matrices -- symplectic schemes -- 1.3.3. Kepler problem -- 1.3.3.1. Review of conic sections and vector product -- 1.3.4. Numerical results -- 2. Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems -- 2.1. Introduction -- 2.1.1. The 1D model problem -- elements of modeling and analysis -- 2.1.2. A radiative transfer problem -- 2.1.2.1. Uniqueness given existence -- 2.1.2.2. A priori estimates -- 2.1.2.3. Existence of solutions -- 2.1.2.4. Shooting method -- 2.1.2.5. Interpretation as a minimization problem -- 2.1.3. Analysis elements for multidimensional problems -- 2.2. Finite difference approximations to elliptic equations.

2.2.1. Finite difference discretization principles -- 2.2.2. Analysis of the discrete problem -- 2.3. Finite volume approximation of elliptic equations -- 2.3.1. Discretization principles for finite volumes -- 2.3.2. Discontinuous coefficients -- 2.3.3. Multidimensional problems -- 2.4. Finite element approximations of elliptic equations -- 2.4.1. P1 approximation in one dimension -- 2.4.2. P2 approximations in one dimension -- 2.4.3. Finite element methods, extension to higher dimensions -- 2.5. Numerical comparison of FD, FV and FE methods -- 2.6. Spectral methods -- 2.7. Poisson-Boltzmann equation -- minimization of a convex function, gradient descent algorithm -- 2.8. Neumann conditions: the optimization perspective -- 2.9. Charge distribution on a cord -- 2.10. Stokes problem -- 3. Numerical Simulations of Partial Differential Equations: Time-dependent Problems -- 3.1. Diffusion equations -- 3.1.1. L2 stability (von Neumann analysis) and L∞ stability: convergence -- 3.1.2. Implicit schemes -- 3.1.3. Finite element discretization -- 3.1.4. Numerical illustrations -- 3.2. From transport equations towards conservation laws -- 3.2.1. Introduction -- 3.2.2. Transport equation: method of characteristics -- 3.2.3. Upwinding principles: upwind scheme -- 3.2.4. Linear transport at constant speed: analysis of FD and FV schemes -- 3.2.5. Two-dimensional simulations -- 3.2.6. The dynamics of prion proliferation -- 3.3. Wave equation -- 3.4. Nonlinear problems: conservation laws -- 3.4.1. Scalar conservation laws -- 3.4.1.1. Elements of analysis -- 3.4.1.2. Lax-Friedrichs and Rusjanov schemes for scalar conservation laws -- 3.4.1.3. Numerical illustrations -- 3.4.1.4. Propagation of a forest fire -- 3.4.2. Systems of conservation laws -- 3.4.2.1. Introduction to gas dynamics equations -- 3.4.2.2. Lax-Friedrichs and Rusjanov schemes for the Euler equations.

3.4.3. Kinetic schemes -- 3.4.3.1. Scalar conservation laws -- 3.4.3.2. Kinetic scheme for Euler equations: monatomic case -- APPENDICES -- Appendix 1. Solving Linear Systems -- A1.1. Condition number of a matrix -- A1.2. Spectral radius -- A1.3. Conjugate gradient -- Appendix 2. Numerical Integration -- Appendix 3. A Peetre-Tartar Equivalence Theorem -- Appendix 4. Schauder's Theorem -- Appendix 5. Fundamental Solutions of the Laplacian in Dimension 1 and 2 -- A5.1. Dimension 1 -- A5.2. Dimension 2 -- A5.3. Higher dimensions -- Bibliography -- Index -- Other titles from iSTE in Mathematics and Statistics -- EULA.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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