Numerical ODE - May 2022

Instructions

Show intermediate results at all steps!

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

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

Derive the absolute stability region for the backward Euler scheme.

Consider 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).

Ilustrate stability of the Trapezoidal method for the Harmonic oscillator:

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

Consider the Hamiltonian dynamics:

\frac{dq}{dt}=\nabla_pH(q,p),\qquad \frac{dp}{dt}=-\nabla_qH(q,p)\qquad\text{(1)}

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}).

Show that the implicit mid-point method

z^{n+1}=z^n+\Delta t J\nabla_z H\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 (1).

Formulate the Stomer-Verlet algorithm for equation (1) when $H$ is separable such that $H=T(p)+V(q)$ and show its Leap-frog formulation.