21 for a time step indeed were we to discretize the

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Unformatted text preview: .3) as n (5.2.5) n jk jk (I ; 2t L1 )U +1 2 = (I + 2t L2 )U n = (5.2.6a) n jk jk (I ; 2t L2 )U +1 = (I + 2t L1 )U +1 2 : (5.2.6b) When applied to the heat conduction equation with centered spatial di erences, this scheme is also identical to the ADI scheme (5.1.5). Let us verify that (5.2.5) and (5.2.6) are equivalent. Thus, operate on (5.2.6b) with I ; tL1 =2 to obtain n n jk jk = (I ; 2t L1 )(I ; 2t L2 )U +1 = (I ; 2t L1 )(I + 2t L1 )U +1 2 : n n jk jk = 5.2. Operator Splitting 15 The operators on the right may be interchanged to obtain (I ; 2t L1 )(I ; 2t L2 )U +1 = (I + 2t L1 )(I ; 2t L1 )U +1 2 : n n jk jk = Using (5.2.6a) yields (5.2.5). Example 5.2.3. D'Yakonov (cf. 4], Section 2.12) introduced the following scheme for solving (5.2.5) ^ (I ; t L1 )U +1 = (I + t L1 )(I + t L2 )U 2 2 2 n n jk jk ^ (I ; 2t L2 )U +1 = U +1: n (5.2.7b) n jk (5.2.7a) jk This scheme has the same order of accuracy and characteristics as the Peaceman-Rachford ADI scheme (5.2.6). Example 5.2.4. Douglas and Rachford 2] developed an alternative scheme for (5.2.1) using backward-di erence approximations. Thus, consider integrating (5.2.1) for a time step by the backward Euler method to obtain (I ; tL1 ; tL2 )u +1 = u + O( t)2: n n jk jk Let us rewrite this as (I ; tL1 ; tL2 + t2 L1L2 )u +1 = (I + t2 L1L2)u + n t2 1 2 (u LL +1 ; u n jk jk n n jk jk ) + O( t)3: As in Example 5.2.2, we may show that the next-to-last term on the right is O( t3) and, hence, may be neglected. Also neglecting the temporal discretization error term, discretizing the operators L1 and L2, and factoring the left side gives (I ; tL1 )(I ; tL2 )U +1 = (I + t2 L1 L2 )U : n n jk jk Douglas and Rachford 2] factored this as ^ (I ; tL1 )U +1 = (I + tL2 )U n n jk jk (5.2.8a) 16 Multi-Dimensional Parabolic Problemss ^ (I ; tL2 )U +1 = U +1 ; tL2 U n n jk (5.2.8b) n jk jk Assuming that the discrete spatial operators are second-order accurate, the local discretization error is O( t) + O( x2) + O( y2). This is lower order than the O( t2) local discretization error that would be obtained from the ADI factorization (5.2.6) however, backward di erencing gives greater stability than (5.2.6) which may be useful for nonlinear problems. Example 5.2.5. The previous examples suggest a simplicity that is not always present. Boundary conditions must be treated very carefully since the intermediate solutions or ^ U +1 2 or U +1 need not be consistent approximations of u(x (n + 1=2) t) or u(x (n + 1) t). Yanenko 8] presents a good example of the complications that can arise at boundaries. Strikwerda 7], Section 7.3, suggests using a combination of U and U +1 to...
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