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## Book Description

This book presents the latest numerical solutions to initial value problems and boundary value problems described by ODEs and PDEs. The author offers practical methods that can be adapted to solve wide ranges of problems and illustrates them in the increasingly popular open source computer language R, allowing integration with more statistically based methods. The book begins with standard techniques, followed by an overview of 'high resolution' flux limiters and WENO to solve problems with solutions exhibiting high gradient phenomena. Meshless methods using radial basis functions are then discussed in the context of scattered data interpolation and the solution of PDEs on irregular grids. Three detailed case studies demonstrate how numerical methods can be used to tackle very different complex problems. With its focus on practical solutions to real-world problems, this book will be useful to students and practitioners in all areas of science and engineering, especially those using R.

1. Cover
2. Half title
3. Title
5. Dedication
7. Preface
8. 1 ODE Integration Methods
1. 1.1 Introduction
2. 1.2 Euler Methods
3. 1.3 Runge–Kutta Methods
4. 1.4 Linear Multistep Methods (LMMs)
5. 1.5 Truncation Error and Order of Integration
6. 1.6 Stiffness
7. 1.7 How to Choose a Numerical Integrator
8. 1.A Installation of the R Package Ryacas
9. 1.B Installation of the R Package rSymPy
10. References
9. 2 Stability Analysis of ODE Integrators
1. 2.1 General
2. 2.2 Dahlquist Test Problem
3. 2.3 Euler Methods
4. 2.4 Runge–Kutta Methods
5. 2.5 Linear Multistep Methods (LMMs)
6. References
10. 3 Numerical Solution of PDEs
1. 3.1 Some PDE Basics
2. 3.2 Initial and Boundary Conditions
3. 3.3 Types of PDE Solutions
4. 3.4 PDE Subscript Notation
5. 3.5 A General PDE System
6. 3.6 Classification of PDEs
7. 3.7 Discretization
8. 3.8 Method of Lines (MOL)
9. 3.9 Fully Discrete Methods
10. 3.10 Finite volume method
11. 3.11 Interpretation of Results
12. 3.A Appendix: Derivative Matrix Coefficients
13. 3.B Appendix: Derivative Matrix Library
14. References
11. 4 PDE Stability Analysis
1. 4.1 Introduction
2. 4.2 The Well-Posed PDE Problem
3. 4.3 Matrix Stability Method
4. 4.4 Von Neumann Stability Method
5. 4.5 Unstructured Grids
6. 4.A Fourier Transforms
7. References
12. 5 Dissipation and Dispersion
1. 5.1 Introduction
2. 5.2 Dispersion Relation
3. 5.3 Amplification Factor
4. 5.4 Dissipation
5. 5.5 Dispersion
6. 5.6 Dissipation and Dispersion Errors
7. 5.7 Group and Phase Velocities
8. 5.8 Modified PDEs
9. References
13. 6 High-Resolution Schemes
1. 6.1 Introduction
2. 6.2 The Riemann Problem
3. 6.3 Total Variation Diminishing (TVD) Methods
4. 6.4 Godunov Method
5. 6.5 Flux Limiter Method
6. 6.6 Monotone Upstream-Centered Schemes for Conservation Laws (MUSCL)
7. 6.7 Weighted Essentially Nonoscillatory (WENO) Method
9. 6.A Eigenvalues of Euler Equations
10. 6.B R Code for Simulating 1D Scalar Equation Problems
11. 6.C R Code for Simulating 1D Euler Equations Problems
12. References
14. 7 Meshless Methods
1. 7.1 Introduction
2. 7.2 Radial Basis Functions (RBF)
3. 7.3 Interpolation
4. 7.4 Differentiation
5. 7.5 Local RBFs
6. 7.6 Application to Partial Differential Equations
7. 7.A Franke’s Function
8. 7.B Halton Sequence
9. 7.C RBF Definitions
10. References
15. 8 Conservation Laws
1. 8.1 Introduction
2. 8.2 Korteweg–de Vries (KdV) Equation
3. 8.3 Conservation Laws for Other Evolutionary Equations
4. 8.A Symbolic Algebra Computer Source Code
5. References
16. 9 Case Study: Analysis of Golf Ball Flight
1. 9.1 Introduction
2. 9.2 Drag Force
3. 9.3 Magnus Force
4. 9.4 Gravitational Force
5. 9.5 Golf Ball Construction
6. 9.6 Ambient Conditions
7. 9.7 The Shot
8. 9.8 Completing the Mathematical Description
9. 9.9 Computer Simulation
10. 9.10 Computer Code
11. References
17. 10 Case Study: Taylor–Sedov Blast Wave
1. 10.1 Brief Background to the Problem
2. 10.2 System Analysis
3. 10.3 Some Useful Gas Law Relations
4. 10.4 Shock Wave Conditions
5. 10.5 Energy
6. 10.6 Photographic Evidence
7. 10.7 Trinity Site Conditions
8. 10.8 Numerical Solution
9. 10.9 Integration of PDEs
10. 10.A Appendix: Similarity Analysis
11. 10.B Appendix: Analytical Solution
12. References
18. 11 Case Study: The Carbon Cycle
1. 11.1 Introduction
2. 11.2 The Model
3. 11.3 Simulation Results
4. 11.A Appendices
5. References
19. Appendix: A Mathematical Aide-Mémoire
20. Index