21 l where l is a spatial di erential operator that

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Unformatted text preview: y 0) + : : : = e @t u(x y 0) 2! t t tt or, using the partial di erential equation, 2 u(x y t) = u(x y 0) + t u(x y 0) + t 2u(x y 0) + : : : = e Lu(x y 0): 2! L L t The above manipulations are similar to those used to obtain the Lax-Wendro scheme of Section 3.3. 5.2. Operator Splitting 11 Unfortunately, once again, u(x y t) = e (L1 +L2) u(x y 0) = e L1 e L2 u(x y 0) t t 6 t unless the operators L1 and L2 commute. Let us verify this by using Taylor's series expansions of both sides of the above expression thus, 2 e (L1 +L2)u(x y 0) = I + t( 1 + 2) + t2 ( 2 + 1 t L L L 2 1 2 + L2 L1 + L2 ) + : : : ]u(x LL y 0) and 2 2 e L1 e L2 u(x y 0) = (I + t 1 + t2 2 + : : : )(I + t 2 + t2 2 + : : : )u(x y 0) 1 2 t or L L L 2 e L1 e L2 u(x y t) = (I + t( 1 + 2) + t2 ( 2 + 2 1 t Hence, t t L e (L1 +L2) ; e L1 e L2 ]u(x t t t L L t2 ( y 0) = 2 L 2 1 2 + L2 ) + : : : ]u(x LL 3 )]u(x 2 1 ; L1 L2 ) + O(t LL y 0): y 0): The di erence between the the two expressions is O(t2) unless L1L2u = L2 L1u. The factorization \almost" works when t is small hence, we can replace t by a small time increment t to obtain (L1 +L2 ) u(x t2 ( L1 e L2 + 3 2 L2L1 ; L1L2) + O( t )]u(x y 0): To obtain a numerical method, we (i) discretize the spatial operators L1 and L2, (ii) neglect the local error terms, and (iii) use the resulting method from time step-to-time step. Thus, u(x y t) = e t y 0) = e t U +1 = e n t L1 e L2 t t U (5.2.2) n where L1 and L2 are discrete approximations of L1 and L2 . This technique, often called the method of fractional steps or operator splitting, has several advantages: 1. If the operators L1 and L2 satisfy the von Neumann conditions e Lk k t k 1+c t k k=1 2 12 Multi-Dimensional Parabolic Problemss then the combined scheme is stable, since, using (5.2.2) U +1 k n e L1 k k e L2 t t kk kk U n k (1 + c t)kUn k: Similarly, if the individual operators are absolutely stable, the combined scheme will be absolutely stable. 2. With operator splitting, the local error is O( t2) unless the operators L1 and L2 commute, in which case it is O( t3). Let us examine some possibilities Example 5.2.1. In order to solve (5.2.1) by operator splitting, we solve u = 2u L t for a time step and then repeat the time step solving u = 1u: L t If we discretize the partial di erential equations with Crank-Nicolson approximations, we have ^ (I ; t L2 )U +1 = (I + t L2 )U 2 2 n n jk jk (5.2.3a) ^ (5.2.3b) (I ; 2t L1 )U +1 = (I + 2t L1 )U +1 : We have used a ^ to denote the \predicted solution" of (5.2.3a). Remark 1. If the operators L1 and L2 commute, then we may easily verify that...
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