Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- 1 Introduction and Background
- 2 Mathematical Models of Fluid Flow
- 3 Numerical Methods for the Solution of Partial Differential Equations
- 4 Fundamental Stability Theory
- 5 Shock Capturing Schemes I: Scalar Conservation Laws
- 6 Shock Capturing Schemes II: Systems of Equations and Gas Dynamics
- 7 Discretization Schemes for Flows in Complex Multi-dimensional Domains
- 8 The Calculation of Viscous flow
- 9 Overview of Time Integration Methods
- 10 Steady State Problems
- 11 Time-Accurate Methods for Unsteady Flow
- 12 Energy Stability for Nonlinear Problems
- 13 High order Methods for Structured Meshes
- 14 High Order Methods for Unstructured Meshes
- 15 Aerodynamic Shape Optimization
- Appendix A Vector and Function Spaces
- Appendix B Approximation Theory
- Appendix C Polynomial Interpolation, Differentiation, and Integration
- Appendix D Potential Flow Methods
- Appendix E Fundamental Stability Theory II
- Appendix F Turbulence Models
- References
- Index
- Plates
3 - Numerical Methods for the Solution of Partial Differential Equations
Published online by Cambridge University Press: 12 August 2022
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgments
- 1 Introduction and Background
- 2 Mathematical Models of Fluid Flow
- 3 Numerical Methods for the Solution of Partial Differential Equations
- 4 Fundamental Stability Theory
- 5 Shock Capturing Schemes I: Scalar Conservation Laws
- 6 Shock Capturing Schemes II: Systems of Equations and Gas Dynamics
- 7 Discretization Schemes for Flows in Complex Multi-dimensional Domains
- 8 The Calculation of Viscous flow
- 9 Overview of Time Integration Methods
- 10 Steady State Problems
- 11 Time-Accurate Methods for Unsteady Flow
- 12 Energy Stability for Nonlinear Problems
- 13 High order Methods for Structured Meshes
- 14 High Order Methods for Unstructured Meshes
- 15 Aerodynamic Shape Optimization
- Appendix A Vector and Function Spaces
- Appendix B Approximation Theory
- Appendix C Polynomial Interpolation, Differentiation, and Integration
- Appendix D Potential Flow Methods
- Appendix E Fundamental Stability Theory II
- Appendix F Turbulence Models
- References
- Index
- Plates
Summary
The numerical methods that have been widely used for the solution of partial differential equations (PDEs), both in fluid dynamics and in other disciplines, fall into three main branches: finite difference methods, finite element methods, and finite volume methods. These methides are reviewed in this chapter together with basic theory of spectral methods.
This chapter examines the stability of difference schemes for initial value problems defined by ordinary or partial differential equations. Three simple examples are examined first: an ordinary differential equation, the linear advection equation, which is the prototype for hyperbolic equations, and the diffusion equations.
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- Information
- Computational Aerodynamics , pp. 42 - 96Publisher: Cambridge University PressPrint publication year: 2022