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162 lines
4.0 KiB
Typst
162 lines
4.0 KiB
Typst
// Diff template
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#import "../preamble.typ": *
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#show: conf.with(num: 2)
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= Differentialgleichungen
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ODEs
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Mechankik: $dot.double(x) + omega_0 ^2 x=0$ ist eine ODE der Ordnung 2
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$m dot.double(x) = F(x, dot(x), t)$ Newtons'sche Mechanik
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zuruck fueren auf ein System von ODEs 1. Ordnung
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Hamilton'sche Mechanik:
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$
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(partial HH(q, p)) / (partial d) = p/m, HH(q, p) = (p^2 ) / (2m) + V(q)\
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dot(p)=-(partial HH) / (partial q) = - (partial V) / (partial q)
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$
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Anfangsbedingungen $q(t=0) = q_0 "und" p(t=0) = p_0 $.\
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Beispiele sind Ratengleichungen, radioaktiver Zerfall und Populationsdynamik.
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== Pearl-Unschulat Modell
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$n(t): "Anzahl der Individuen"$
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$
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(dif n) / (dif t) = sqrt(n)n\
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sqrt(n)="const"\
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n(t) = n(0)exp(sqrt(t)) \
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sqrt(n)=r_0 (1-k_n ), k="const">0
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$
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allgemeine Formulierung: $y^((n)) (t)=g (y, y', ..., y^((n+1)), t )$
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$
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y_(n)=y^((n)) \
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y^((n+1))+y^((n)) = g(y,y_1 , ..., y_(n-1), t)\
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y'_(n-2) =y_(n-1) \
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y'=g_1
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$
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$
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dot(arrow(y)) = arrow(g)(y_1, ..., y_(n-1) )\
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(dif y) / (dif t) = g(y, t)
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$
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Diskretisierung der Zeit $[0, T], t_0 = 0, t_1 =Delta t, t_n =n Delta t=T$
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$
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y (t_i ) =y(t_(i-1)) +integral_(t_i )^(t_(i-1)) d t g (y (t), t) approx^("Euler-Cauchy") y (t_(i-1))+g (y (t_(i-1) , t_(i-1) )) Delta t + O(Delta t^2 )\
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$
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mit globalem Fehler bei Integration ueber $[Q, T]$
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$
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E ~ (T) / (Delta t) Delta t^2 ~ T Delta t ~ O(Delta t)
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$
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Beispiel Ratengleichung $r (n)=r- (1-k_n )$
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$
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n (t_i ) = n (t_(i-1) )+r_0 (1-k_n / t_(i-1) )n (t_(i-1) Delta t)\
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n (t_i )=(1+Delta t r_0 )n (t_(i-1) ) (1- (Delta t r_0 k) / (1+ Delta t r_0 ) n (t_(i-1) ))
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$
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Variable $x (t_(i-1))=(Delta t r_0 k) / (1+Delta t r_0 ) n (t_(i-1) )$
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$
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x (t_i )=4 mu x (t_(i-1) ) (1- x (t_(i-1) )), 4 mu=1+Delta t r_0
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$
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Zuletzt: $Delta t$ klein
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$
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n (t)=1/k, x (t)=(Delta t r_0 ) / (1+ r_0 Delta t)=(4 mu-1) / (4 mu)
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$
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Pointcarre Abbildung mit Bifurkation.
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== Leapfrog Algorithmus
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// TODO: implement nl and Delta t in tyupstar as well as +
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$
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y'=g (y,t), y' (t_i ) (y (t_i )-y (t_(i-1) ) ) / (Delta t) + O (Delta t)\
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y' (t_i )=(y (t_(i+1)) -y (t_(i-1) )) / (Delta t) + O (Delta t)
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$
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#highlight[TODO: Implement this algorithmus as well as the global error]
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== Velocity-Verlet Algorithmus fuer klassische Mechanik
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$
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dot(arrow(t))=vec(dot(x) (t),dot(v) (t))=vec(v (t),a (x_(i) t))\
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v (t_(i+1) )=x (t_(i-1) )+2v (t_i )Delta t+O (Delta t^3 )\
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v (t_(i_1) )=v (t_(i-1) )+2a (x (t_i ),t_1 )Delta t+O(Delta t^3 )
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$
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Umschreiben der zweiten Gleichung:
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Es wird von $t_(i-1) "nach" t_(i+1)$ gegangen.
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$
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v (t_i )=v (t_(i-1) )+a (x (t_(i_1) ), t_(i-1) )Delta t\
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x (i_(i+1) )= x (t_(i-1) )+2v (t_i ) Delta t \
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v (t_(i+1) )=v (t_i )+a (x (t_(i+1) ),t_(i+1))Delta t
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$
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Das ergibt
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$
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v (t_(i+2) )=v (t_(i+1) )+a (x (t_(i+1) )t_(i+1) )Delta t
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$
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$==>$ $v (t_(i+2) )=v (t_i )+2a (x (t_(i+1) ), t_(i+1) )Delta t$
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$2 Delta t = delta t$
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$
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v (t+(delta t) / (alpha) )=v (t)+a (x (t),t) (delta t) / (2) \
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x (t+delta t)=x (t)+v (t+(delta t) / (2))delta t\
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v (t+delta t)=v (t+(delta t) / (s))+a (x (t+delta t),t+delta t)(delta t) / (2) \
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$
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== Forderungen an einen guten Integrator der klassischen Mechanik
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- Energieerhaltung
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- Zeitumkehrinvarianz
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+ der Bewegungsgleichungen sind zeitumkehrbar invariant. Jeder vernuenftige Algorithmus erfuellt dies zu der Ordnung $O(delta t^2 )$\
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$
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vec(x_0, v_0 )=vec(x (t),v (t))->^(T (Delta t)) vec(x (t+Delta t),v (t+Delta t))-> $
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+ des Algorithmus $x_0 -x_1 =0, v_0 -v_1 = 0$
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- Phasenraumerhaltung symplektisch
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$t_(i-1) -> t_(i+1) $
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Zeitumkehr
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$
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arrow(x)' (t_(i+1) )=arrow(x) (t_(i+1) )\
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arrow(v)' (t_(i+1) )= - arrow(v) (t_(i+1) )
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$
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$t_(i+1) -> t_(i+3) $
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$
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v' (t_(i+2))= v' (t_(i+1) )+a (x' (t_(i+1) ),t_(i+1) ) Delta t =-v (t_(i+1)) +a (x (t_(i-1) ))Delta t \
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x' (t_(i+1) )= x' (t_(i+1) )+2v' (t_(i+1) )Delta t\
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= x (t_(i+1) )+2 [-v (t_(i+1) )+a (x (t_(i+1) ))Delta t]Delta t \
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= x (t_(i-1) )+2 Delta t [v (t_i )-v (t_(i+1) )+a (x (t_(i+1) ))Delta t]\
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= x (t_(i-1) )
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$
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=== Takeways
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- Ableitung diskretisierung
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- Taylorentwicklung der Ableitung
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- Mit Euler-Cauchy kann am einfachsten nach vorne integriert werden
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