Truncation order of implicit and explicit methods is

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Truncation order of implicit and explicit methods is the same (order in space and order in time). However the actual error will vary due to the coefficients of the trun- cation terms. Δ x Δ t D Δ t O n ( ) Δ t Δ x Δ t Δ x Δ t Δ x ( ) 2 Δ t
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CE 30125 - Lecture 16 p. 16.14 Crank-Nicolson Implicit (C-N) Method • Evaluate time derivative at point using a forward difference (or at point using a backward difference). • Evaluate the 2nd spatial derivative using the average of the central difference expres- sions at and . Applying these two steps to the transient diffusion equation leads to: Arranging knowns and unknowns: i j , ( ) i j 1 + , i j , ( ) i j 1 + , ( ) 1 Δ t ----- u i j 1 + , u i j , [ ] D Δ x ( ) 2 ------------- 1 2 -- u i 1 j 1 + , + 2 u i j 1 + , u i 1 j 1 + , + ( ) u i 1 j , + 2 u i j , u i 1 j , + ( ) + [ ] = u i 1 j 1 + , 2 2 Δ x ( ) 2 Δ tD ----------------- + u i j 1 + , u i 1 j 1 + , + + u i 1 j , 2 2 Δ x ( ) 2 Δ tD ----------------- u i j , u i 1 j , + + =
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CE 30125 - Lecture 16 p. 16.15 The FD molecule for this solution: Since the unknowns are coupled (at the new time level), the method is implicit! • This C-N solution to the transient diffusion equation is accurate in time and accurate in space. Stability of the C-N solution to the transient diffusion equation is unconditional for all . j+1 j-1 j i-1 i i+1 known values unknown values fictional node j+1/2 x x O Δ t 2 ( ) O Δ x 2 ( ) Δ t
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CE 30125 - Lecture 16 p. 16.16 An alternative interpretation of the C-N solution is to estimate the p.d.e. at : • The time derivative term can be thought of as being a central representation of at . • The 2nd spatial derivative may be thought of as being a central representation of at . • Values of , and are then estimated using values at full nodes with an interpolation procedure. • Defining full and intermediate nodes as: i j 1 2 -- + , u t ----- i j 1 2 -- + , u 2 x 2 -------- i j 1 2 -- + , u i 1 j 1 2 -- + , u i j 1 2 -- + , u i 1 + j 1 2 -- + , j+1 j i-1 i i+1 x x x j+1/2
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CE 30125 - Lecture 16 p. 16.17 • We can interpret the C-N solution as: • Now we can use linear interpolation: • Substituting leads to: u i j 1 2 + , t ----------------------- D u 2 i j 1 2 + , x 2 ------------------------- = u i j 1 + , u i j , Δ t ---------------------------- D u i 1 j 1 2 + , + 2 u i j 1 2 + , u i 1 j 1 2 + , + Δ x 2 ------------------------------------------------------------------------------------------ = u i 1 + j 1 2 + , 1 2 -- u i 1 + j 1 + , u i 1 + j , + ( ) = u i j 1 2 + , 1 2 -- u i j 1 + , u i j , + ( ) = u i 1 j 1 2 + , 1 2 -- u i 1 j 1 + , u i 1 j , + ( ) = 1 Δ t ----- u i j 1 + , u i j , [ ] D Δ x ( ) 2 ------------- 1 2 -- u i 1 j 1 + , + 2 u i j 1 + , u i 1 j 1 + , + ( ) u i 1 j , + 2 u i j , u i 1 j , + ( ) + [ ] =
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CE 30125 - Lecture 16 p. 16.18 Weighted Average Approximation
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  • Fall '08
  • Westerink,J
  • Numerical Analysis, Partial differential equation, transient diffusion equation, p.d.e.

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