1. ODE Integration Methods
1.1 Introduction
1.2 Euler Methods
1.2.1 Forward Euler
1.2.2 Backward Euler
1.3 Runge–Kutta Methods
1.3.1 RK Coefficients
1.3.2 Variable Step Size Methods
1.3.3 SHK: Sommeijer, Van Der Houwen, and Kok Method
1.4 Linear Multistep Methods (LMMs)
1.4.1 General
1.4.2 Backward Differentiation Formulas (BDFs)
1.4.3 Numerical Differentiation Formulas (NDFs)
1.4.4 Convergence
1.4.5 Adams Methods
1.5 Truncation Error and Order of Integration
1.5.1 LMM Truncation Error
1.5.2 Verification of Integration Order
1.6 Stiffness
1.7 How to Choose a Numerical Integrator
1.A Installation of the R Package Ryacas
1.B Installation of the R Package rSymPy
References
2. Stability Analysis of ODE Integrators
2.1 General
2.1.1 Dahlquist Barrier Theorems
2.2 Dahlquist Test Problem
2.3 Euler Methods
2.3.1 Forward Euler
2.3.2 Backward Euler
2.4 Runge–Kutta Methods
2.4.1 RK-1: First-Order Runge–Kutta
2.4.2 RK-2: Second-Order Runge–Kutta
2.4.3 RK-4: Fourth-Order Runge–Kutta
2.4.4 RKF-54: Fehlberg Runge–Kutta
2.4.5 SHK: Sommeijer, van der Houwen, and Kok
2.5 Linear Multistep Methods (LMMs)
2.5.1 General
2.5.2 Backward Differentiation Formulas (BDFs)
2.5.3 Numerical Differentiation Formulas (NDFs)
2.5.4 Adams Methods
References
3. Numerical Solution of PDEs
3.1 Some PDE Basics
3.2 Initial and Boundary Conditions
3.3 Types of PDE Solutions
3.4 PDE Subscript Notation
3.5 A General PDE System
3.6 Classification of PDEs
3.7 Discretization
3.7.1 General Finite Difference Terminology
3.7.2 The Mesh
3.7.3 Nonuniform Grid Spacing
3.7.4 The Courant–Friedrichs–Lewy Number
3.7.5 The Stencil
3.7.6 Upwinding
3.8 Method of Lines (MOL)
3.8.1 Introduction
3.8.2 Finite Difference Matrices
3.8.3 MOL 1D: Cartesian Coordinates
3.8.4 MOL 2D: Cartesian Coordinates
3.8.5 MOL 2D: Polar Coordinates
3.9 Fully Discrete Methods
3.9.1 Introduction
3.9.2 Overview of Some Common Schemes
3.9.3 Results from Simulating a Hyperbolic Equation
3.10 Finite Volume Method
3.10.1 General
3.10.2 Application to a 1D Conservative System
3.10.3 Application to a General Conservation Law
3.11 Interpretation of Results
3.11.1 Verification
3.11.2 Validation
3.11.3 Truncation Error
3.A Appendix: Derivative Matrix Coefficients
3.A.1 First Derivative Schemes
3.A.2 Second Derivative Schemes
3.A.3 Third Derivative Schemes
3.A.4 Fourth Derivative Schemes
3.B Appendix: Derivative Matrix Library
3.B.1 Example
References
4. PDE Stability Analysis
4.1 Introduction
4.2 The Well-Posed PDE Problem
4.3 Matrix Stability Method
4.3.1 Semi-Discrete Systems
4.4 Von Neumann Stability Method
4.4.1 General
4.4.2 Fully Discrete Systems
4.4.3 Semi-Discrete Systems
4.5 Unstructured Grids
4.A Fourier Transforms
References
5. Dissipation and Dispersion
5.1 Introduction
5.2 Dispersion Relation
5.3 Amplification Factor
5.4 Dissipation
5.5 Dispersion
5.6 Dissipation and Dispersion Errors
5.6.1 The 1D Advection Equation, Semi-Discrete Upwind
5.6.2 The 1D Advection Equation, Semi-Discrete Second-Order Upwind
5.6.3 The 1D Advection Equation, Fully Discrete Upwind
5.6.4 The 1D Advection Equation, Fully Discrete Lax–Friedrichs (LxF)
5.7 Group and Phase Velocities
5.7.1 Exact Relationships for the Basic PDE
5.7.2 Semi-Discrete, First-Order Upwind Discretization
5.7.3 Semi-Discrete Leapfrog Discretization
5.7.4 Fully Discrete Leapfrog Discretization
5.8 Modified PDEs
References
6. High-Resolution Schemes
6.1 Introduction
6.2 The Riemann Problem
6.3 Total Variation Diminishing (TVD) Methods
6.3.1 TVD Numerical Integration
6.4 Godunov Method
6.4.1 Godunov’s Theorem
6.5 Flux Limiter Method
6.5.1 How Limiters Work
6.5.2 Limiter Functions
6.6 Monotone Upstream-Centered Schemes for Conservation Laws (MUSCL)
6.6.1 Linear Reconstruction
6.6.2 Kurganov and Tadmor Central Scheme
6.6.3 Piecewise Parabolic Reconstruction
6.6.4 Solutions to the Euler Equations
6.7 Weighted Essentially Nonoscillatory (WENO) Method
6.7.1 Polynomial Reconstruction: Finite Volume Approach
6.7.2 Polynomial Coefficients
6.7.3 Polynomial Reconstruction: Finite Difference Reconstruction
6.7.4 WENO Reconstruction
6.7.5 Alternative Calculation for Substencil Coefficients
6.7.6 Weights
6.7.7 Smoothness Indicators
6.7.8 Calculation of Smoothness Indicator Coefficients
6.7.9 Flux Splitting
6.7.10 Implementation of a WENO Finite Volume Scheme
6.7.11 Scalar Problems
6.7.12 Euler Equation Problems
6.7.13 2D Examples
6.8 Further Reading
6.A Eigenvalues of Euler Equations
6.B R Code for Simulating 1D Scalar Equation Problems
6.B.1 The Main Program
6.B.2 The Derivative Function
6.B.3 The MUSCL Function
6.B.4 Initialization
6.C R Code for Simulating 1D Euler Equations Problems
6.C.1 The Main Routine
6.C.2 Initialization
6.C.3 The Derivative Function
6.C.4 The MUSCL Function
6.C.5 Postsimulation Calculations
References
7. Meshless Methods
7.1 Introduction
7.2 Radial Basis Functions (RBF)
7.2.1 Positive Definite RBFs
7.2.2 RBF with Compact Support (CSRBF)
7.3 Interpolation
7.3.1 Interpolation Example: 1D
7.3.2 Interpolation Example: 2D
7.3.3 Larger Interpolation Example: 2D
7.3.4 Interpolation Example: 3D
7.3.5 Interpolation with Polynomial Precision
7.4 Differentiation
7.4.1 Derivative Example: 1D
7.5 Local RBFs
7.5.1 Allocating Stencil Nodes
7.5.2 Choosing the Right Shape Parameter Value
7.6 Application to Partial Differential Equations
7.6.1 Explicit Euler Integration
7.6.2 Weighted Average Integration
7.6.3 Method of Lines
7.6.4 With Nonlinear Terms
7.6.5 Initial Conditions (ICs) and Boundary Conditions (BCs)
7.6.6 Stability Considerations
7.6.7 Time-Dependent PDEs
7.6.8 Time-Independent PDEs
7.A Franke’s Function
7.B Halton Sequence
7.C RBF Definitions
References
8. Conservation Laws
8.1 Introduction
8.2 Korteweg–de Vries (KdV) Equation
8.2.1 The First Conservation Law, u
8.2.2 The Second Conservation Law, u[sup(2)]8.2.3 The Third Conservation Law, u[sup(3)] + ½u[sup(2)][sub(x)]8.2.4 Another Conservation Law
8.2.5 An Infinity of Conservation Laws
8.2.6 KdV Equation: 2D
8.2.7 KdV Equation with Variable Coefficients (vcKdV)
8.3 Conservation Laws for Other Evolutionary Equations
8.3.1 Nonlinear Schrödinger Equation
8.3.2 Boussinesq Equation
8.A Symbolic Algebra Computer Source Code
References
9. Case Study: Analysis of Golf Ball Flight
9.1 Introduction
9.2 Drag Force
9.3 Magnus Force
9.4 Gravitational Force
9.5 Golf Ball Construction
9.6 Ambient Conditions
9.7 The Shot
9.7.1 Golf Ball Compression
9.7.2 Spin
9.7.3 Launch Angle
9.7.4 Bounce and Roll