Use a backward difference approximation for at u i j

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Use a backward difference approximation for at u i j , Δ t Δ x i j 1 + , ( ) u t ----- i j 1 + , ( ) u t ----- i j 1 + , u i j 1 + , u i j , Δ t ---------------------------- =
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CE 30125 - Lecture 16 p. 16.10 Use a central difference for at Substituting into the p.d.e.: Re-arrange such that unknown values appear on the left hand side of the equation: 2 u x 2 -------- i j 1 + , ( ) 2 u x 2 -------- i j 1 + , 1 Δ x 2 -------- u i 1 j 1 + , + 2 u i j 1 + , u i 1 j 1 + , + [ ] = 1 Δ t ----- u i j 1 + , u i j , [ ] D Δ x ( ) 2 ------------- u i 1 j 1 + , + 2 u i j 1 + , u i 1 j 1 + , + [ ] = u i 1 j 1 + , 2 Δ x ( ) 2 Δ tD ------------- + u i j 1 + , u i 1 j 1 + , + + Δ x ( ) 2 Δ tD ----------------- u i j , =
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CE 30125 - Lecture 16 p. 16.11 Notes on Implicit Solution to the transient diffusion equation , , are the unknown values. is the only known value. • FD molecule for this representation: • Equations for adjacent nodes will be dependent on adjacent values!! We can no longer “explicitly” solve for each unknown value independently but must solve for all unknowns simultaneously as a set of linear equations “Implicit solution” u i 1 j 1 + , u i j 1 + , u i 1 + j 1 + , u i j , j+1 j-1 j i-1 i i+1 known values i+2 molecule #2 molecule #1 unknown values
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CE 30125 - Lecture 16 p. 16.12 • We generate a system of equations ( nodes - 2 b.c.’s). One equation for each unknown value. In matrix form: where and n n 2 + α 1 1 α 1 1 α 1 . . . . . . . . . 1 α 1 1 α u j 1 1 , + u j 1 2 , + u j 1 3 , + . . . u j 1 n 1 , + u j 1 n , + Au j 1 , u ** 1 j 1 + Au j 2 , . . . Au j n 1 , Au j n , u ** 2 j 1 + = α 2 Δ x ( ) 2 D Δ t ------------- + = A Δ x ( ) 2 D Δ t ------------- =
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CE 30125 - Lecture 16 p. 16.13 Notes on simultaneous equation solution • The matrix is always diagonally dominant, and there are therefore no roundoff error problems in the solution of this system of simultaneous equations. Solve this tri- diagonal, symmetric set of equations by a Gauss-elimination type procedure. • If , , are constants we do not need to re-set and triangularize the matrix at every time step, otherwise we must re-set and re-solve the matrix every . • To solve this system of equations requires operations, the same order is required for an explicit formulation. This however changes for 2D and 3D prob- lems!! Implicit methods are unconditionally stable (i.e. the method is stable for all values of and ). There are still accuracy limitations on both and (which are required to limit trun- cation error!). can be many times larger for an implicit scheme than for an explicit scheme (10 to 100 times), leading to computational savings.
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  • Fall '08
  • Westerink,J
  • Numerical Analysis, Partial differential equation, transient diffusion equation, p.d.e.

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