29 29 29 29 2 1 1 1 5 67 i i i i Cor Pr y y hf x y O h Oh 3 Add eqs 557 and 563

29 29 29 29 2 1 1 1 5 67 i i i i cor pr y y hf x y o

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+ = + + ( 29 ( 29 ( 29 ( 29 2 1 1 1 5 67 i i i i Cor Pr y y hf ( x , y ) O h . + + + = + + O(h 3 )
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Add eqs. (5.57) and (5.63) to get Generalize this method where The Crank-Nicolson method ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 1 1 3 1 1 1 5 57 5 63 1 5 68 2 1 1 5 70 2 2 i i i i i i i i i i i i i i i i y y y . y y y . y y y y . y y hf x , y hf x , y O h . + + + + + + + + = + ∆ = + = + + = + + + ( 29 1 1 1 2 2 5 71 i i y y w k w k . + = + + ( 29 ( 29 ( 29 ( 29 1 2 2 21 1 5 72 5 73 i i i i k hf x , y . k hf x c h, y a k . = = + +
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Graphical interpretation ( 29 1 1 1 2 2 5 71 i i y y w k w k . + = + + ( 29 ( 29 ( 29 ( 29 1 2 2 21 1 2 5 72 5 73 i i i i i k hf x , y hf . k hf x c h, y a k hf . = = = + + = 1 i i y y + 1 2 i i i x x x + + h d 1 1 2 2 w k w k + 21 1 i i y a k y + Exact  y 2 i i x x c h + i f 2 f Exact  y
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Second order Runge-Kutta (same as Crank-Nicolson) Third order Runge-Kutta The Runge-Kutta methods ( 29 ( 29 ( 29 ( 29 3 1 1 2 1 2 1 1 2 i i i i i i y y k k O h k hf x , y k hf x h, y k + = + + + = = + + ( 29 ( 29 ( 29 ( 29 4 1 1 2 3 1 1 2 3 2 1 1 4 6 2 2 2 i i i i i i i i y y k k k O h k hf x , y k h k hf x , y k hf x h, y k k + = + + + + = = + + = + + -
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The Runge-Kutta methods (cont.) ( 29 ( 29 ( 29 ( 29 5 1 1 2 3 4 1 1 2 2 3 4 3 1 2 2 6 2 2 2 2 i i i i i i i i i i y y k k k k O h k hf x , y k h k hf x , y k h k hf x , y k hf x h, y k + = + + + + + = = + + = + + = + + Fourth order Runge-Kutta
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The Runge-Kutta methods (cont.) ( 29 ( 29 ( 29 6 1 1 3 4 5 6 1 1 2 1 2 3 3 4 3 2 4 5 3 5 1 2 4 6 1 7 32 12 32 7 90 2 2 3 4 16 16 2 2 6 3 9 3 4 16 16 16 6 8 4 12 7 7 7 7 7 i i i i i i i i i i i i i i y y k k k k k O h k hf x , y k h k hf x , y k k h k hf x , y k h k hf x , y k k k h k hf x , y k k k k k k hf x h, y + = + + + + + + = = + + = + + + = + + = + - + + ° = + + + + - + ° Fifth order Runge-Kutta
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Simultaneous differential equations The methods developed above for a single ODE apply to  sets of ODEs: The fourth-order Runge-Kutta formula is ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 1 2 2 2 1 2 1 2 5 1 1 2 3 4 1 1 1 11 12 2 1 2 5 98 1 2 2 1 2 6 1 2 1 2 5 99 2 2 2 2 n n n n n i , j i,j j j j j j j i i ij in n j j i i i in dy f x, y , y , , y dx dy f x, y , y , , y . dx dy f x, y , y , , y dx y y k k k k O h j , , ,n k hf x , y , y , , y j , , ,n k k k h k hf x , y , y , y j , , ,n . + = = = = + + + + + = = = = + + + + = K K M K K K K K K ( 29 2 21 22 3 1 2 4 1 31 2 32 3 1 2 2 2 2 2 1 2 n j j i i i in j j i i i in n k k k h k hf x , y , y , y j , , ,n k hf x h, y k , y k , , y k j , , ,n = + + + + = = + + + + = K K K K
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Stability, error propagation, and convergence For differential equations, there are two types of  stability:  Inherent and numerical Linear ODEs : Inherent stability is a function of the  eigenvalues of the ODEs.  Nonlinear ODEs : Inherent stability is a function of  the eigenvalues of the Jacobian of the ODEs.  Example of inherent stability: A 1 A 2 B B 1 A 2 B 3 B 4 C C 3 B 4 C (0) 1 (0) 0 (0) 0 A B C dC k C k C C dt dC k C k C k C k C C dt dC k C k C C dt = - + = = - - + = = - = 1 2 3 5 0 1 , , λ λ λ = - = = -
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Inherent stability (cont.)
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