Numerical Approximation of ODEs

Contents

• Basic Ideas
• One Step Methods
• Consistency / Stability / Convergence
• Runge-Kutta Methods

Lecture 5-0

• Whiteboard Notes

Little Coding Exercise

Use your favorite programming language to

1. Write an implementation of the explicit Euler scheme like

def explicit_euler(f, x0, timegrid):
    # f -- the right hand side in `x' = f(t, x)`
    # x0 -- the initial value
    # timegrid -- somehow specify the stepsize and the endpoint

    # integrate the ODE with explicit Euler 

    return xN

2. Test it with the ODE \begin{equation} \begin{bmatrix} \dot x \\ \dot y \end{bmatrix} = \begin{bmatrix} e^t y \\ -e^t x \end{bmatrix}, \quad \begin{bmatrix} x(0) \\ y(0) \end{bmatrix} = \begin{bmatrix} \sin (1) \\ \cos(1) \end{bmatrix} \end{equation} on $[0, 3]$ with number of time steps $$N\in{2^{8}, 2^{10}, 2^{14}, 2^{18}}$$ by comparing the computed value of $x_N\approx x(3)$ to the actual value $x(3)=\sin (e^3)$.
3. Implement any Runge-Kutta method of higher-order¹ and repeat this numerical study.

Basics of Numerical Approximation of DAEs

Contents

• Notions and Notations of RKM
• How to apply an RKM to $E\dot x = Ax+f(t)$
• Explicit Euler fails, Implicit Euler works
• RKM and System Transformations
• Analysis via the Kronecker Normal Form

Lecture 5-1

• Whiteboard Notes

Consistency und Convergence for DAEs with Constant Coefficients

• Consistency error – will depend on the $\nu$-index
• Convergence – need extra stability condition
• Stiffly accurate schemes – exact for index-1 DAE parts

Lecture 5-2

• Chapter 5.2 in the script
• Whiteboard Notes

Exercise 4

• Python Script for the “$\kappa$“s: dae-butcher.py
• Implicit Euler for time-varying DAEs: python file Jupyter Notebook

RKM for (Nonlinear) Semi-explicit DAEs of Index 1

• Whiteboard Notes

RKM/BDF for (Nonlinear) Semi-explicit DAEs of Index 1/2

• Whiteboard Notes
• Link to the list of Theorems and Software

Numerical Methods for Index Reduction

1. For unstructured systems – use the derivative array but compute the projections on the fly.
2. For structured systems like multibody systems (like the pendulum – differentiate the relevant parts, add these equations to the system, and make the system square again by introducing new variables.

• Whiteboard Notes