Contents
Part I: Ordinary Differential Equations and Their Approximations
Chapter 1: First-Order Scalar Equations
1.1 Constant coefficient linear equations
1.2 Variable coefficient linear equations
1.3 Perturbations and the concept of stability
1.4 Nonlinear equations: the possibility of blow-up
1.5 Principle of linearization
2.2 Stability of the explicit Euler method
2.3 Accuracy and truncation error
2.4 Discrete Duhamel’s principle and global error
2.6 How to test the correctness of a program
Chapter 3: Higher-Order Methods
3.1 Second-order Taylor method
3.3 Accuracy of the solution computed
3.6 Accuracy and truncation error
3.7 Difference approximations for unstable problems
Chapter 4: Implicit Euler Method
4.3 Simple variable-step-size strategy
Chapter 5: Two-Step and Multistep Methods
5.4 Stability of multistep methods
Chapter 6: Systems of Differential Equations
Part II: Partial Differential Equations and Their Approximations
Chapter 7: Fourier Series and Interpolation
7.2 L2-norm and scalar product
8.1 Examples of equations with simple wave solutions
Chapter 9: Approximations of 1-Periodic Solutions of Partial Differential Equations
9.1 Approximations of space derivatives
9.2 Differentiation of Periodic Functions
9.4 Time Discretizations and Stability Analysis
Chapter 10: Linear Initial Boundary Value Problems
10.1 Well-Posed Initial Boundary Value Problems
Chapter 11: Nonlinear Problems
11.1 Initial value problems for ordinary differential equations
11.2 Existence theorems for nonlinear partial differential equations
11.3 Nonlinear example: Burgers’ equation
Appendix A: Auxiliary Material
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