§ 1. ODE Models in Science

1.1 Chemical reaction kinetics

• General reaction system
• Hydrolysis of Urea

1.2 Electric circuits

• Electromagnetic Oscillator with/without current source
• Colpitts Oscillator

1.3 Multibody problem

• Two-body problem
• N-body problem

1.4 Further models

• Predator-prey models: the Lotka-Volterra Equations
• Curves of Pursuit: The Fox-Rabbit Pursuit
• Epidemiology/Population Dynamics: Plants vs. Zombies
• Semidiscretization of partial differential equations: The Method of Lines
• Higher Order ODEs: Van der Pol Oscillator
• A real office problem: The Cooling Cup of Coffee

§ 2. Short Synopsis on Theory of ODEs

2.1 Linear differential equations

2.2 Existence and uniqueness

2.3 Perturbation analysis

§ 3. One-Step Methods

3.1 Preliminaries

3.2 Elementary integration schemes

3.3 Consistency and convergence

3.4 Taylor Methods for ODEs

3.5 Runge-Kutta methods

3.6 Dense output

3.7 Step-Size Control

§ 4. Multistep Methods

4.1 Techniques based on numerical quadrature

• Adams methods
• Nyström and Milne methods

4.2 Linear difference schemes

4.3 Consistency, stability and convergence

4.4 Analysis of multistep methods

• first Dahlquist barrier

4.5 Techniques based on numerical differentiation

• Backward Differentiation Formulas (BDF)
• Numerical Differentiation Formulas (NDF)

4.6 Predictor-Corrector-Methods

• Algorithm: P(EC)^mE method

4.7 Order control

§ 5. Integration of Stiff Systems

5.1 Examples

5.2 Test equations

5.3 A-stability for One-step Methods

• Definition 10: A-stability for One-step Methods
• stability function
• Definition 11: Stability domain of One-step Methods
• Example 1: Explicit/Implicit Euler Methods
• Example 2: Trapezoidal Rule
• Example 3: Explicit Midpoint Rule
• General Runge-Kutta Methods (GRK)
• Theorem 13: stability function of Runge-Kutta Method
• Definition 12: L-stability for One-step Methods
• Definition 13: Padé-approximation

5.4 Implicit Runge-Kutta Methods

• Collocation methods
• Theorem 14: Order of Collocation Methods
• Solution of Nonlinear Systems
• Diagonal Implicit Runge-Kutta Methods

5.5 Rosenbrock-Wanner Methods

• Construction of ROW Methods
• Order conditions
• A-stability
• Example 1: RWO schemes, GRK4A, GRK4T

5.6 A-stability for Multistep Methods

• Definition 14: Stability domain of Multistep Methods
• Definition 15: A-stability for Multistep Methods
• Theorem 15: convergent explicit linear multistep methods are not A-stable
• Theorem 16: second Dahlquist barrier
• Definition 16: A(α)-stability for Multistep Methods

5.7 B-stability

• Definition 17: B-stability
• Theorem 17: Gauss-Runge-Kutta methods are B-stable

5.8 Comparison of methods

• Integrators in MATLAB

§ 6. Methods for Differential Algebraic Equations

6.1 Implicit ODEs

6.2 Linear DAEs

• source terms with perturbations
• consistent choice of initial values

6.3 Index Concepts

• Differential Index
• Perturbation Index

6.4 Methods for General Systems

6.5 Methods for Semi-Explicit Systems

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• Runge Kutta Methods

6.6 Illustrative Example: Mathematical Pendulum

§ 7. Boundary Value Problems

7.1 Problem Definition

• Example 1:
• Example 2:
• Separated Boundary Conditions
• Periodic Problems of non-autonomous Systems
• Periodic Problems of autonomous Systems

7.2 Single Shooting Method

• Example 1:
• Computation of Jacobian Matrix
• Sensitivity Matrix
• Sensitivity Equations

7.3 Multiple Shooting Method

• Example 1:

7.4 Finite Difference Methods

• Periodic Problems
• ODE of second Order
• Example 1:
• Linear ODE of second Order
• Theorem 18:
• Theorem 19: Convergence of the finite difference method

7.5 Techniques with Trial Functions

§ 8. Geometric Integration

8.1 Introduction

• ODEs with conservative nature

8.2 Linear Invariants

• reversible isometry
• Example 1: chemical kinetics system
• Definition: linear invariant
• Example 2: explicit Euler method
• Example 3: linear multistep method
• Example 4: Runge-Kutta method
• Examples of linear invariants:
  • chemical kinetic systems
  • mechanical models: overall mass
  • stochastic models: "master equation", total probability
  • epidemic models: overall population

8.3 Quadratic Invariants

• Example 1: angular momentum of a free rigid body
• Definition: quadratic invariant
• Example 2: implicit mid-point rule
• Example 3: explicit Euler method
• Example 4: implicit Euler method
• Example 5: Runge-Kutta method
  • condition of Cooper
• Example 6: Kepler Problem
  • nonlinear invariant, Hamiltonoan function

8.4 Symplectic Maps

• Definition: oriented area
• oriented area preserving mappings
  • linear case
  • nonlinear case
  • condition for symplecticness

8.5 Hamiltonian ODEs

• Example 1: motion of a pendulum
• Hamiltonian ODE
• Hamiltonian function
• symplectic character of Hamiltonian ODE
• time t flow map
• Hessian of Hamiltonian function

8.6 Approximating Hamiltonian ODEs

• Example 1: forward Euler
• Example 2: backward Euler
• Example 3: symplectic Euler
• separable Hamiltonian