ImplSchemeWaveEq

# ImplSchemeWaveEq - 2 2 u tt x i 0 from the equation u tt x...

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MATH 445 Implicit scheme for the wave equation Consider the one-dimensional wave equation u tt ( x, t ) = c 2 u xx ( x, t ); 0 < x < L, 0 < t T ; u (0 , t ) = u ( L, t ) = 0; u ( x, 0) = f ( x ) , u t ( x, 0) = g ( x ) . Denote by Δ t = τ the step size in time, and by Δ x = h , in space; set m 2 = τ 2 c 2 /h 2 , x i = ( i - 1) h, i = 1 , . . . , M + 1; t j = ( j - 1) τ, j = 1 , . . . , N + 1 . Note that L = Mh = x M +1 , T = = t N +1 . Write u i,j for the approximation of u ( x i , t j ), We approximate u tt ( x i , t j ) by the central difference at x i : u tt ( x i , t j ) 1 τ 2 ( u i,j +1 - 2 u i,j + u i,j - 1 ) and u xx ( i Δ x, j Δ t ), by the average of the corresponding central differences at t j +1 and t j - 1 : u xx ( i Δ x, j Δ t ) 1 2 1 h 2 ( u i +1 ,j +1 - 2 u i,j +1 + u i - 1 ,j +1 ) + 1 h 2 ( u i +1 ,j - 1 - 2 u i,j - 1 + u i - 1 ,j - 1 ) The result is u i,j +1 - 2 u i,j + u i,j - 1 = 1 2 m 2 ( u i +1 ,j +1 - 2 u i,j +1 + u i - 1 ,j +1 ) + ( u i +1 ,j - 1 - 2 u i,j - 1 + u i - 1 ,j - 1 ) · or - m 2 u i +1 ,j +1 + 2(1 + m 2 ) u i,j +1 - m 2 u i - 1 ,j +1 = 4 u i,j + m 2 u i +1 ,j - 1 - 2(1 + m 2 ) u i,j - 1 + m 2 u i - 1 ,j - 1 . (1) With zero boundary conditions, you get u 1 ,j = u M +1 ,j = 0 . For i = 2 , . . . , M, use the initial conditions to get u i, 1 = u ( x i , 0) = f (( i - 1) h ),
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Unformatted text preview: 2 2 u tt ( x i , 0); from the equation, u tt ( x i , 0) = c 2 u xx ( x i , 0) = c 2 f 00 ( x i ) ≈ c 2 ( f ( ih )-2 f (( i-1) h ) + f (( i-2) h )) /h 2 . Then, for each j +1 = 3 ,...,N , (1) is a linear system for the unknown vector ( u 2 ,j +1 ,...,u M,j +1 ), and all you need is to solve this system. The matrix A of this system is of size ( M-1) × ( M-1), with 2(1+ m 2 ) on the main diagonal,-m 2 just above and below the main diagonal, and 0 elsewhere. If you write U ( j ) for the vector-column ( u 2 ,j ,...,u M-1 ,j ), then the system to solve is AU ( j + 1) = 4 U ( j )-AU ( j-1) ....
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