8.2–8.3 ODE review and dynamic form

Source: Ideas from Computational Physics (Landau, Páez, Bordeianu), §8.2–8.3. Original wording. Continues 8.1 Nonlinear oscillators. Next: 8.4 ODE algorithms.

Vocabulary

• Order of an ODE: highest derivative present. Newton’s law for $x(t)$ is second order.
• ODE vs PDE: one independent variable ($t$) vs several ($x,t$, …).
• Linear vs nonlinear: linear ODEs obey superposition ($c_1 y_1+c_2 y_2$ is a solution if each $y_i$ is). Terms like $x^{3}$, $x\dot x$, or $\sin x$ destroy that — you cannot paste harmonic solutions together.
• Initial-value (IVP) vs boundary-value (BVP): mechanics usually gives $x(t_0)$ and $v(t_0)$. Two-point boundary data in time are a different problem class.

Dynamic form

A first-order system is what steppers eat. Introduce $y^{(0)}=x$, $y^{(1)}=v$, write $\dot{\mathbf y}=\mathbf f$, and check a numeric RHS — pick each equation among lookalikes.

Harmonic benchmark used throughout

Choose $m=1$ and $k=(2\pi)^{2}$ so $\omega=2\pi$ and period $T=1$. With $x(0)=0$, $v(0)=1$,

$$x(t)=A\sin(\omega t),\qquad A=\frac{1}{\omega}=\frac{1}{2\pi}\approx 0.1591549431.$$

Every algorithm in §8.4 will be scored against that analytic orbit and against energy $E=\tfrac12 v^{2}+\tfrac12 k x^{2}$.

Quiz

ODE review and dynamic form

1 / 4

Newton’s law m x″ = F is…

Pick an answer to see if you are right.

Next: 8.4 ODE algorithms.