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Index
Cover
Table of Contents
Title
Copyright
Preface
1 Ordinary Differential Equations
1.1. Introduction to the theory of ordinary differential equations
1.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability
1.3. Hamiltonian problems
2 Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems
2.1. Introduction
2.2. Finite difference approximations to elliptic equations
2.3. Finite volume approximation of elliptic equations
2.4. Finite element approximations of elliptic equations
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
3.2. From transport equations towards conservation laws
3.3. Wave equation
3.4. Nonlinear problems: conservation laws
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
End User License Agreement
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