This is the version 5bb8b4b408367b37baaa5b2f from 2008-05-28 15:03:36 comment: 'Neue Version'
Simulationsmethoden ed
Numerik ed
Differentiation ed
- forward two-point formula
- \( f'(n) = \frac{ f(n+1) - f(n) } h + O(h) \)
- backwards two-point formula
- \( f'(n) = \frac{ f(n) - f(n-1) } h + O(h) \)
- central 3-point formula
- \( f'(n) = \frac{ f(n+1) - f(n-1) }{ 2 h } + O(h^2) \)
- \( f''(n) = \frac{ f(n+1) - 2 f(n) + f(n-1) }{ h^2 } + O(h^2) \)
Integration ed
- Trapezregel
- \( S_T = h \left( \frac 12 f(0) + f(1) + \cdots + f(N-2) + \frac 12 f(N-1) \right) + O(h^2) \)
...
klassische Bewegungsgleichungen ed
Euler ed
- Euler
- \( v(n+1) = v(n) + \tau a(n) \)
- \( x(n+1) = x(n) + \tau v(n) \)
- Fehler: \( O(\tau^2) \)
- Energie nicht erhalten
Verlet ed
- Verlet
- a hängt nicht von v ab!
- \( x(n+1) = 2 x(n) - x(n-1) + \tau^2 a(n) \)
- \( v(n) = \frac{ x(n+1) - x(n-1) }{ 2 \tau } \)
- Benötigt am Anfang \( x(-1) = x(0) - \tau v(0) + \frac{ \tau^2 }2 a(0) \)
- Fehler in \( O(\tau^4) \)
- Zeit-umkehrbar
- periodische Systeme -> Fehler begrenzt
- Velocity Verlet
- \( x(n+1) = x(n) + \tau v(n) + \frac{ \tau^2 }2 a(n) \)
- \( v(n+1) = v(n) + \frac \tau 2 ( a(n) + a(n+1) ) \)
- self starting
- Leap-frog
- \( v(n+\frac 12) = v(n-\frac 12) + \tau a(n) \)
- \( x(n+1) = x(n) + \tau v(n+\frac 12) \)
- Anfang: \( v(-\frac 12) = v(0) - \frac \tau 2 a(0) \)
- mit Reibung
- lineare Abhängigkeit von v: \( a(x,v,t) = a^0(x,t) - \Gamma v \)
- \( v(n+1) = ( 1 + \frac \tau 2 \Gamma )^{-1} \left( ( 1 - \frac \tau 2 \Gamma ) v(n) + \frac \tau 2 ( a^0(n) + a^0(n+1) ) \right) \)
- \( x(n+1) = x(n) + \tau ( 1 + \frac \tau 2 \Gamma ) v(n) + \frac{ \tau^2 }2 a^0(n) \)
Stabilität ed
...exp-Funktion in Methode einsetzen... Frequenz reell->stabil, imaginär->schlecht