Integrating Differential Equations dy ( t ) = f ( t, y ( t )) dt y ( t 0 ) = y 0 y n ≈ y ( t n ) , n = 0 , 1 , . . . h n = t n +1 − t n
� t + h y ( t + h ) = y ( t ) + f ( s, y ( s )) ds t � t n +1 y n +1 = y n + f ( s ) ds t n 1
y = dy ( t ) ˙ dt y = d 2 y ( t ) ¨ dt 2 2
Systems of Equations x ( t ) = − x ( t ) ¨ � � x ( t ) y ( t ) = x ( t ) ˙ � � x ( t ) ˙ y ( t ) ˙ = − x ( t ) � � y 2 ( t ) = − y 1 ( t ) 3
− u ( t ) /r ( t ) 3 ¨ u ( t ) = − v ( t ) /r ( t ) 3 ¨ v ( t ) = � u ( t ) 2 + v ( t ) 2 r ( t ) = u ( t ) v ( t ) y ( t ) = u ( t ) ˙ v ( t ) ˙ u ( t ) ˙ v ( t ) ˙ y ( t ) = ˙ − u ( t ) /r ( t ) 3 − v ( t ) /r ( t ) 3 4
Linearized Differential Equations f ( t, y ) = f ( t c , y c ) + α ( t − t c ) + J ( y − y c ) + . . . α = ∂f ∂t ( t c , y c ) J = ∂f ∂y ( t c , y c ) 5
y 1 ( t ) f 1 ( t, y 1 , . . . , y n ) d y 2 ( t ) f 2 ( t, y 1 , . . . , y n ) = . . . . . . dt y n ( t ) f n ( t, y 1 , . . . , y n ) ∂f 1 ∂f 1 ∂f 1 . . . ∂y 1 ∂y 2 ∂y n ∂f 2 ∂f 2 ∂f 2 . . . ∂y 1 ∂y 2 ∂y n J = . . . . . . . . . ∂f n ∂f n ∂f n . . . ∂y 1 ∂y 2 ∂y n 6
y = Jy ˙ λ k = µ k + iν k = eig( J ) Λ = diag( λ k ) J = V Λ V − 1 V x = y x k = λ k x k ˙ x k ( t ) = e λ k ( t − t c ) x ( t c ) 7
� � 0 1 y = ˙ y − 1 0 � � 0 1 J = − 1 0 Eigenvalues of J are ± i and the solutions are purely oscillatory linear combinations of e it and e − it . 8
y 3 ( t ) y 4 ( t ) y ( t ) = ˙ − y 1 ( t ) /r ( t ) 3 − y 2 ( t ) /r ( t ) 3 � y 1 ( t ) 2 + y 2 ( t ) 2 r ( t ) = r 5 0 0 0 r 5 J = 1 0 0 0 2 y 2 1 − y 2 r 5 3 y 1 y 2 0 0 2 2 y 2 2 − y 2 3 y 1 y 2 0 0 1 √ 2 1 i √ λ = r 3 / 2 2 − − i 9
Single Step Methods y n +1 = y n + hf ( t n , y n ) t n +1 = t n + h 10
s 1 = f ( t n , y n ) f ( t n + h 2 , y n + h = 2 s 1 ) s 2 y n +1 = y n + hs 2 t n +1 = t n + h = f ( t n , y n ) s 1 = f ( t n + h, y n + hs 1 ) s 2 y n + hs 1 + s 2 y n +1 = 2 t n +1 = t n + h 11
Classical Runge-Kutta s 1 = f ( t n , y n ) f ( t n + h 2 , y n + h = 2 s 1 ) s 2 f ( t n + h 2 , y n + h s 3 = 2 s 2 ) s 4 = f ( t n + h, y n + hs 3 ) y n + h y n +1 = 6( s 1 + 2 s 2 + 2 s 3 + s 4 ) t n +1 = t n + h 12
i − 1 � s i = f ( t n + α i h, y n + h β i,j s j ) j =1 i = 1 , . . . , k k � y n +1 = y n + h γ i s i i =1 k � e n +1 = h δ i s i i =1 13
The BS23 algorithm s 1 = f ( t n , y n ) f ( t n + h 2 , y n + h s 2 = 2 s 1 ) f ( t n + 3 4 h, y n + 3 = 4 hs 2 ) s 3 t n +1 = t n + h y n + h y n +1 = 9(2 s 1 + 3 s 2 + 4 s 3 ) s 4 = f ( t n +1 , y n +1 ) h e n +1 = 72( − 5 s 1 + 6 s 2 + 8 s 3 − 9 s 4 ) 14
s2 s1 s1 yn yn tn tn+h tn tn+h/2 s4 s3 ynp1 s s2 yn yn tn tn+3*h/4 tn tn+h 15
Lorenz Attractor y = Ay ˙ y 1 ( t ) y ( t ) = y 2 ( t ) y 3 ( t ) − β 0 y 2 A = 0 − σ σ − y 2 ρ − 1 16
0 − β η A = 0 − σ σ − η ρ − 1 � η = ± β ( ρ − 1) ρ − 1 y ( t 0 ) = η η 0 y ( t ) = ˙ 0 0 17
Stiffness A problem is stiff if the solution being sought is varying slowly, but there are nearby solutions that vary rapidly, so the numerical method must take small steps to obtain satisfactory results. 18
y = y 2 − y 3 ˙ y (0) = η 0 ≤ t ≤ 2 /η 19
Events y = f ( t, y ) ˙ y ( t 0 ) = y 0 g ( t ∗ , y ( t ∗ )) = 0 20
y 2 ¨ y = − 1 + ˙ y (0) = 1, ˙ y (0) = 0. 21
d ( t ) T d ( t ) g ( t, y ) = ˙ d = ( y 1 ( t ) − y 1 (0) , y 2 ( t ) − y 2 (0)) T 22
Local discretization error u n = f ( t, u n ) ˙ u n ( t n ) = y n d n = y n +1 − u n ( t n +1 ) Global discretization error e n = y n − y ( t n ) 23
� t N N − 1 � f ( τ ) dτ ≈ h n f ( t n ) t 0 0 � t n +1 d n = h n f ( t n ) − f ( τ ) dτ t n � t N N − 1 � e N = h n f ( t n ) − f ( τ ) dτ t 0 n =0 N − 1 � e N = d n n =0 24
order | d n | ≤ Ch p +1 n d n = O ( h p +1 ) n 25
y n +1 = y n + h n f ( t n , y n ) u n ( t ) = u n ( t n ) + ( t − t n ) u ′ n ( t n ) + O (( t − t n ) 2 ) u n ( t n +1 ) = y n + h n f ( t n , y n ) + O ( h 2 n ) d n = y n +1 − u n ( t n +1 ) = O ( h 2 n ) 26
N = t f − t 0 h Nǫ = Lǫ h Ch p + Lǫ h 1 � Lǫ � p +1 h ≈ C � C 1 � p +1 N ≈ L Lǫ 27
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