midterm306 - CME 306 Numerical Solutions to Partial...

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Unformatted text preview: CME 306 Numerical Solutions to Partial Differential Equations http://stanford.edu/ ~ pgarapon/cme306.html Part I - Finite difference techniques 1. Prove that the scheme above is consistent with equation ( ?? ) Using Taylor expansions, one shows: u n +1 j- 2 u n j + u n- 1 j ( t ) 2 = 2 u t 2 + O ( t 2 ) 2 ( 2 ( u )) = h 4 4 u x 4 + O ( h 6 ) Which gives the consistency. 2. Rewrite the scheme without the term . One substitutes and gets: u n +1 j- 2 u n j + u n- 1 j ( t ) 2 + u n j- 2- 4 u n j- 1 + 6 u n j- 4 u n j +1 + u n j +2 ( h ) 4 = 0 3. Write the matrix D in M N ( R ) (matrix of size N , with real entries) corresponding to 2 . Write the scheme in a matrix form and explicitly define the matrices involved. Denoting ( U n ) j = u n j , one gets the vector equality: U n +1 = (2 I + cD 2 ) U n- U n- 1 where c = t 2 h 4 and D is the matrix: D = 2- 1 ... ... 1- 1 2- 1 ... ... ..- 1 2- 1 ... .. ... ... ... ... ... ... .. ... ...- 1 2- 1 1 .. ... ..- 1 2 4. Following the Von Neumann approach, write a relation between the in- tensities of the Fourier mode k at times n- 1, n and n +1: u n +1 ( k ) , u n ( k ) , u n- 1 ( k )....
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This note was uploaded on 04/05/2011 for the course CME 306 taught by Professor Ken during the Spring '10 term at University of Utah.

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midterm306 - CME 306 Numerical Solutions to Partial...

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