Numerical ODE - August 2025

Instructions

Show intermediate results at all steps!

Problem 1

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

Problem 2

(10pt) Derive the absolute stability region for the backward Euler scheme.

Problem 3

(10pt) Consider the homogeneous ODE $\dot{u}(t)=f(u)$, derive the Local Truncation Error of the following midpoint method:

$$x^{n+1}=x^n+hf\left(x^n+\frac{1}{2}hf(x^n)\right).$$

Problem 4

(10pt) For ODE $dz/dt=f(z)\in\mathbb{R}^d$, show that the implicit mid-point method

$$z^{n+1}=z^n+\Delta t f\left(\frac{z^{n+1}+z^n}{2}\right),$$

conserves linear invariant $z(t)^TC=\mathrm{const.}$ of the equation.

Problem 5

(10pt) Drive the modified equation of the backward Euler’s method for following equation

$$\frac{dx}{dt}=2x^2-1,\quad x\in\mathbb{R}.$$

Problem 6

(10pt) Consider the Hamiltonian dynamics for $(q,p)\in\mathbb{R}^2$:

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

Show that the verlet scheme is symplectic.