This book presents some of 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 a wide range of problems which are illustrated in the increasingly popular open source computer language R, which allows integration with more statistically based methods.
The book begins with standard techniques, followed by an overview of high resolution 'Flux Limiter' and 'Weighted Essentially Non-Oscillatory' (WENO) methods to solve problems with solutions exhibiting high gradient phenomena. Meshless methods using radial basis functions (RBFs) 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.
Table of Contents
- ODE Integration Methods
- Stability Analysis of ODE Integrators
- Numerical Solution of PDEs
- PDE Stability Analysis
- Dissipation and Dispersion
- High Resolution Schemes
- Meshless Methods
- Conservation Laws
- Case study: Analysis of Golf Ball Flight
- Case study: Taylor-Sedov Blast Wave
- Case study: The Carbon Cycle.
All computer code is available for download.