201-Lect 13 - MECH201 Engineering Analysis(2009 p 70 10.6...

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Unformatted text preview: MECH201 Engineering Analysis (2009) p. 70 10.6 The Crank-Nicolson Scheme -- an Implicit method In the computation using the implicit method half of the spatial differentiation in the "future time" and thus have to be solved by iteration. The FDA is expressed as: ( ) ( ) , 1 , 1, 1 , 1 1, 1 1, , 1, 2 2 2 2 i j i j i j i j i j i j i j i j T T T T T T T T t x α + + + +- + +-- =- + +- + Δ Δ We now sort the future time (j+1) to the LHS and the rest in RHS: ( ) 1, 1 , 1 1, 1 , 1, , 1, 2 2 2 2 1 2 2 2 2 2 2 i j i j i j i j i j i j i j t t t t T T T T T T T x x x x α α α α + + +- + +- Δ Δ Δ Δ - + +- = +- + Δ Δ Δ Δ The above equation can be solve by iteration process. We may simply write: ( ) 1, 1 , 1 1, 1 , 1 2 i j i j i j i j rT r T rT b + + +- +- + +- = where 2 2 t r x α Δ = Δ and ( ) , , 1, , 1, 2 2 2 i j i j i j i j i j t b T T T T x α +- Δ = +- + Δ is a value for each node which is constant during the iteration process. and the iteration scheme take the form: ( ) ( ) , 1 1, 1 1, 1 , 1 1 2 new i j i j i j i j T r T T b r + + +- + = + + + In a form more convenient for programming:...
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• Three '09
• WeihuaLi
• Trigraph, Partial differential equation, TI, finite difference, Finite difference method, Numerical differential equations

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201-Lect 13 - MECH201 Engineering Analysis(2009 p 70 10.6...

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