Skip to main content
Ctrl
+
K
Runge-Kutta Methods
Introduction
Preliminaries
Teaching schedule
Python and MATLAB
Learning and teaching
Chapters
1. Ordinary Differential Equations (ODE)
1.1. The Taylor Series
1.2. The Euler method
1.3. Error analysis
1.4. Higher-order ODEs
1.5. ODEs Exercises
2. Chapter 2. Explicit Runge-Kutta Methods
2.3. Derivation of Explicit Runge-Kutta Methods
2.4. Deriving order conditions using trees
2.5. Derivation of a fourth-order explicit Runge-Kutta method
2.6. Solving initial value problems using explicit Runge-Kutta methods
2.7. Adaptive step size control
2.8. Explicit Runge-Kutta Methods Exercises
3. Implicit Runge-Kutta Methods
3.1. Determining the order of an implicit Runge-Kutta method
3.2. Deriving implicit Runge-Kutta methods
3.3. Solving initial value problems using implicit Runge-Kutta methods
3.4. Implicit Runge-Kutta methods exercises
4. Stability
4.2. Stability functions
4.3. Stability function of a Runge-Kutta method
4.4. Stability functions of implicit methods
4.5. Stability exercises
5. Boundary Value Problems
5.2. The shooting method
5.3. The finite-difference method
5.4. Boundary value problems exercises
6. Matrix Decomposition Methods
6.1. LU decomposition
6.2. Solving systems of linear equations using LU decomposition
6.3. LU decomposition with partial pivoting
6.4. Cholesky decomposition
6.5. QR decomposition
6.6. Calculating eigenvalues using QR decomposition
6.7. Matrix decomposition exercises
7. Indirect Methods
7.1. Jacobi method
7.2. Gauss-Seidel method
7.3. Convergence of indirect methods
7.4. The Successive Over Relaxation (SOR) method
7.5. Indirect methods exercises
Appendices
Exercise Solutions
Ordinary Differential Equations exercise solutions
Explicit Runge-Kutta methods exercise solutions
Implicit Runge-Kutta methods exercise solutions
Stability exercises solutions
Boundary value problems exercise solutions
Matrix decomposition solutions
Indirect methods solutions
Python Code
MATLAB Code
References
Index