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Unformatted text preview: MATH 445 THE CRANK-NICOLSON SCHEME FOR THE HEAT EQUATION Consider the one-dimensional heat equation (1) u t ( x,t ) = au xx ( x,t );0 < x < L, < t ≤ T ; u (0 ,t ) = u ( L,t ) = 0; u ( x, 0) = f ( x ) , The idea is to reduce this PDE to a system of ODEs by discretizing the equation in space, and then apply a suitable numerical method to the resulting system of ODEs. Denote by Δ x = L/N the step size in space and approximate u xx ( n Δ x,t ) for every t ∈ [0 ,T ] using central differences: (2) u xx ( n Δ x,t ) ≈ 1 (Δ x ) 2 ( u (( n + 1)Δ x,t )- 2 u ( n Δ x,t ) + u (( n- 1)Δ x,t )) . Define the column vector U ( t ) = ( U ( t ) ,U 1 ( t ) ,...,U N ( t )) T as the solution of the system of equations (3) dU n ( t ) dt = a (Δ x ) 2 ( U n +1 ( t )- 2 U n ( t ) + U n- 1 ( t ) ) , n = 1 ,...,N- 1 , < t ≤ T, with initial condition U n (0) = f ( n Δ x ), and set U ( t ) = U N ( t ) = 0 for all t . By (2), it is natural to consider U n ( t ) as an approximation of u ( n Δ x,t ). Note that, from the definition of)....
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This note was uploaded on 10/06/2009 for the course MATH 445 taught by Professor Friedlander during the Fall '07 term at USC.
- Fall '07