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I j i j i j i α 2 1 1 1 2 x t r u u u r u u j i j i

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i j i j i j i + - = - - + + α 2 , 1 , , 1 , 1 , ) 2 ( x t r u u u r u u j i j i j i j i j i = + - + = - + + α Difference equation is: Time (j+1) Time (j) SCHEME is ORDER ( t, x 2 )
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7 Example - Explicit Method 1 5 . 0 ) 1 ( 2 ) 0 , ( 5 . 0 0 2 ) 0 , ( 0 ) , 1 ( ) , 0 ( 1 0 2 2 - = = = = = x x x u x x x u t u t u x x u t u Consider the following rod (length 1m) between two pieces of ice. The partial differential equation governing heat loss from this rod is given by ICE ICE Boundary Conditions Initial Conditions
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8 Example Use x = 0.1 and t = 0.001. Therefore r = 0.1. Also we can assume symmetry around x = 0.5. Symmetry at i=5 means that u 6,j = u 4,j. The finite difference approximation for each node is given by. ICE i=0 1 2 3 4 5 6 0 = x u ) 2 ( 1 . 0 , 1 , , 1 , 1 , j i j i j i j i j i u u u u u - + + + - + =
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9 Example Use previous approximation to find changes in temperature for first ten time steps. i=0 1 2 3 4 5 6 x=0 0.1 0.2 0.3 0.4 0.5 0.6 t=0 0 0.2 0.4 0.6 0.8 1 0.8 t=0.001 0 0.2 0.4 0.6 0.8 0.96 0.8 t=0.002 0 0.2 0.4 0.6 0.796 0.928 0.796 t=0.003 0 0.2 0.4 0.5996 0.7896 0.9016 0.7896 t=0.004 0 0.2 0.4 0.5986 0.7818 0.8792 0.7818 t=0.005 0 0.2 0.3999 0.5971 0.7732 0.8597 0.7732 . . t=0.01 0 0.1996 0.3968 0.5822 0.7281 0.7867 0.7281 . . t=0.02 0 0.1938 0.3781 0.5373 0.6486 0.6891 0.6486
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10 Courant Number = 1 I=0 1 2 3 4 5 6 x=0 0.1 0.2 0.3 0.4 0.5 0.6 t=0 0 0.2 0.4 0.6 0.8 1 0.8 t=0.01 0 0.2 0.4 0.6 0.8 0.6 0.8 t=0.02 0 0.2 0.4 0.6 0.4 1.0 0.4 t=0.03 0 0.2 0.4 0.2 1.2 -0.2 1.2 t=0.04 0 0.2 0.0 1.4 -1.2 2.6 -1.2 Now let t = 0.01. Therefore r = 1. In this case the solution is unstable: .
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Explicit Method It can be shown (later lecture) that the Explicit Scheme is only stable when the Courant number (r) : Therefore for a given mesh density we have a restriction on the time step size. To overcome this limitation on the time step size we use implicit methods. 11 α 2 5 . 0 . 5 . 0 x t e i r
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12 Implicit Methods For explicit methods the space derivative is approximated at the previous time step. For implicit methods the space derivative is approximated at the latest time step.
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