This is the version 5bb8b4b308367b37baaa5b2e from 2008-05-28 15:49:36 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
Runge-Kutta ed
partielle Differentialgleichungen ed
- FTCS
- \( \partial_t A(x,t) = \frac{ A(x,t+1) - A(x,t-1) } \tau \) (forward time)
- \( \partial_x A(x,t) = \frac{ A(x+1,t) - A(x-1,t) }{ 2 h } \)
- \( \bigtriangleup A(x,t) = \frac{ A(x+1,t) - 2 A(x,t) + A(x-1,t) }{ h^2 } \) (centered space)
- Dirichlet-Bedingungen -> A am Rand gegeben
- von Neumann -> Ableitungen gegeben
- periodisch...
Potential Problem ed
- Poisson: \( \bigtriangleup \phi = - \frac 1 {\epsilon_0} \rho \)
- Jakobi-Relaxation
- \( \phi^{neu}(x) = \frac 1{2d} \sum_{i=0}^d \left( \phi(x+1_i) + \phi(x-1_i) \right) + \frac{ h^2 }{ 2 d \epsilon_0 } \rho(x) \)
- \( \delta \phi = max_x | \phi(x) - \phi^{neu}(x) | \)
- benötigt \( \frac p 2 N^{2/d} \) Schritte, um Fehler um 10^-p zu verringern
- Gauss-Seidel-Relaxation
- sofort Werte setzen
- benötigt \( \frac p 4 N^{2/d} \) Schritte
- Successive Overrelaxation
- \( \phi(x) \rightarrow w \phi^{neu}(x) + (1-w) \phi(x) \)
- 1 < w < 2
- \( w_{opt} = \frac 2 { 1 + \sqrt{ 1 - r^2 } }, \ \ r = \frac{ \cos \frac{ N_x }{ \pi } + \cos \frac{ N_y }{ \pi } } 2 \)
- benötigt \( \approx \frac p3 N^{1/d} \approx \frac p3 L \) Schritte
- Chebishev Beschleunigung
- 2 Teil-Gitter (Schachbrett)
- langsam anfangen:
- \( w^0 = 1 \)
- \( w^{1/2} = \frac 1 { 1 - \frac 12 r^2 } \)
- \( w^{n+1/2} = \frac 1 { 1 - \frac 14 r^2 w^n } \)
- Matrix-Formulierung
- \( A \vec \phi = \vec b \)
- Laplace-Operator -> A, Ladung + Randwerte -> b
- A orthogonal, schwach besetzt (max 2d+1 Einträge pro Reihe), A_ii = 2d, A_ij = -1, pos. definit
- Gauss-Elimination
- Pivoting... Zeilen tauschen...
- Elimination: \( O(n^3) \) , Rücksubstitution: \( O(n^2) \)
- LU-Zerlegung
- A = LU... löse Ly = b..... dann Ux = y
- \( 2 \times O(n^2) \)
- konjugierte Gradienten-Methode
- ...