Numerical ODE - May 2021

Instructions

Show intermediate results at all steps!

Problem 1

Consider the following homogeneous ODE with smooth coefficient $f(\cdot)$ :

\frac{du(t)}{dt}=f(u).\qquad\text{(1)}

Show that the Local Truncation Error of the following midpoint method is of order $O(k^2)$ :

U^{n+1}=U^n+kf\left(U^n+\frac{1}{2}kf(U^n)\right).

Problem 2

The Asymptotic stability of a numerical method can studied through the Harmonic oscillator

\frac{dz}{dt}=Az,\quad\text{where }A=\begin{bmatrix}0&1\\-\omega^2&0\end{bmatrix},\quad z=(q,v)^T.

Derive the stability condition for the Verlet algorithm:

q^{n+1}=q^n+kv^{n+1/2},

v^{n+1/2}=v^n-\frac{k}{2}\omega^2q^n,

v^{n+1}=v^{n+1/2}-\frac{k}{2}\omega^2q^{n+1}.

Problem 3

Prove the global convergence of the Euler scheme for equation (1):

U^{n+1}=U^n+kf(U^n).

Problem 4

Consider the Hamiltonian dynamics:

\frac{dq}{dt}=\nabla_p H(q,p),\qquad \frac{dp}{dt}=-\nabla_q H(q,p).

Show that the implicit Euler-B scheme is symplectic

q^{n+1}=q^n+\Delta t\nabla_p H(q^n,p^{n+1}),\qquad p^{n+1}=p^n-\Delta t\nabla_q H(q^n,p^{n+1}).

Problem 5

Consider linear equation

\frac{dz}{dt}=Az,\quad\text{where }A=\begin{bmatrix}0&\omega\\-\omega&0\end{bmatrix},\quad z=(q,p)^T.

Derive the phase shift of the explicit Euler scheme.

Problem 6

For rigid body problem $q_i=q_{cm}+Qr_i^0$ , $i=1,2,\ldots,k$ , with $Q^TQ=I_3$ . Give the Hamiltonian dynamics for $(q_{cm},Q)$ and its momentum under external potential $V_{ext}(q_{cm},Q)$ , and formulate the Rattle Scheme for simulation.