Splitting

Splitting - 2 A 2 l 2 + ... )( I + Bl + 1 2 B 2 l 2 + ... )...

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Splitting Algorithm Sudarshan Balakrishnan August 23, 2011 Linear Ordinary Differential Equations In order to understand the splitting associated with the logistic equation, let us first consider the linear ODE given by: du dt = Au, du dt = Bu, du dt = Cu. (1) In (1), we have that C = A + B . Solving the each of them we notice that the solution for a given time l , using the separation of variables is: ϕ l ( u ) = e Al e Bl . (2) Using the solution, we can estimate the local error ρ ( u ; l ) by: ρ ( u ; l ) = ϕ l ( u ) - e Cl . (3) In order to estimate (3), we have that: ρ ( u ; l ) = ( I + Cl + 1 2 C 2 l 2 + ... ) - ( I + Al + 1
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Unformatted text preview: 2 A 2 l 2 + ... )( I + Bl + 1 2 B 2 l 2 + ... ) (4) = ( I + Cl + 1 2 C 2 l 2 )-( I + h ( A + B ) + h 2 ( AB + 1 2 B 2 + 1 2 A 2 )) + O ( h 3 ) (5) = l 2 ( 1 2 ( A + B ) 2-( AB + 1 2 A 2 + 1 2 B 2 ) + O ( h 3 ) (6) = l 2 ( 1 2 ( A ) 2 + 1 2 ( B ) 2 + AB-AB + 1 2 A 2 + 1 2 B 2 ) + O ( h 3 ) (7) l 2 2 ( AB-BA ) + O ( h 3 ) (8) l 2 2 ( AB-BA ) + O ( h 3 ) . (9) 1 In (9),we notice that if AB = BA , then we have that the error is zero and e lA e lB = e lC . (10) 2...
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Splitting - 2 A 2 l 2 + ... )( I + Bl + 1 2 B 2 l 2 + ... )...

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