Numerical ODE - August 2022

Instructions

Show intermediate results at all steps!

Problem 1

(10pt) For ODE $\dot{x}=f(x)$ with $f$ satifying the Lipshcitz condition with bounded 2nd order derivative, prove the global convergence of the Euler scheme:

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

Problem 2

(10pt) The Asymptotic stability of a numerical method can be 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:

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

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

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

Problem 3

(10pt) For the homogeneous ODE $\dot{u}(t)=f(u)$ , show that the Local Truncation Error of the following midpoint method is of order $O(k^3)$ :

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

Problem 4

(10pt) Consider the Hamiltonian dynamics:

\frac{dq}{dt}=\nabla_pH(q,p),\qquad\frac{dp}{dt}=-\nabla_qH(q,p).

Show that the implicit Euler-B scheme is symplectic

q^{n+1}=q^n+\Delta t\nabla_pH(q^n,p^{n+1}),\qquad p^{n+1}=p^n-\Delta t\nabla_qH(q^n,p^{n+1}).

Problem 5

(10pt) Consider the 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

(10pt) Show that the implicit mid-point method

z^{n+1}=z^n+\Delta tJ\nabla_zH\left(\frac{z^{n+1}+z^n}{2}\right),\quad z=(q,p)^T

where $J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}$ , is time reversible for solving Hamiltonian dynamics in Problem 4.