An example of an implicit method is crank nicholson 2

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An example of an implicit method is Crank- Nicholson 2 2 x u t u = α
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13 Implicit Methods Requires Matrix Solvers. Known as the Fully Implicit Method. 2 2 x u t u = α t u u j i j i - + , 1 , 2 1 , 1 1 , 1 , 1 2 x u u u j i j i j i + - + - + + + Therefore parabolic equation can be written as: 2 , 1 , 1 1 , 1 , 1 1 , ) 2 ( x t r u u u u r u j i j i j i j i j i = = + - - + - + + + + α Time (j+1) Time (j) SCHEME is ORDER ( t, x 2 )
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14 Implicit Method - Crank Nicholson 2 2 x u t u = α t u u j i j i - + , 1 , 2 , 1 , , 1 2 x u u u j i j i j i + - - + For the explicit scheme we approximated the derivatives at (i,j) using: Forward difference Central difference Approximating the derivatives at mid-point in time (i,j+0.5) we have: 2 2 x u t u = α ) 5 . 0 ( 2 , 1 , t u u j i j i - + + - + + - + - + + + - + 2 1 , 1 1 , 1 , 1 2 , 1 , , 1 2 2 2 1 x u u u x u u u j i j i j i j i j i j i Central difference at (i,j+0.5) Central difference at (i,j+1) Central difference at (i,j)
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15 Implicit Method - Crank Nicholson Crank Nicholson in 1947 proposed the above method to allow greater time steps in their calculations. The resulting difference equation is: Time (j+1) Time (j) j i j i j i j i j i j i ru u r ru ru u r ru , 1 , , 1 1 , 1 1 , 1 , 1 ) 2 2 ( ) 2 2 ( + - + + + + - + - + = - + + - 2 x t r = α In this case we have three unknowns on the RHS of the equation. Therefore the new value u i,j+1 is not given directly in terms of known values as is the case of the Explicit method. SCHEME is ORDER ( t 2 , x 2 )
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16 Explicit / Implicit Comparisons i-1 i i+1 i-1 i i+1 j-1 j j+1 i-1 i i+1 Explicit Crank-Nicholson Fully Implicit O( t, x 2 ) O( t, x 2 ) O( t 2 , x 2 ) Computational molecules show differences between the schemes If there is more than one node at timestep j +1 the scheme is implicit j-1 j j+1 j-1 j j+1
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17 Explicit - Implicit Methods Comparisons
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