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Unformatted text preview: 1 Atms 502 CS 505 CSE 566 Numerical Fluid Dynamics Thu., 31 August 2006 9/29/06 Atms 502  Fall 2006  Jewett 2 Stable 1 full revolution .. Note timing errors Initial condition 9/29/06 Atms 502  Fall 2006  Jewett 3 vs. Unstable Unstable; First 28 steps Stable 9/29/06 Atms 502  Fall 2006  Jewett 4 vs. Nonlinear case. Nonlinear Unstable 2 9/29/06 Atms 502  Fall 2006  Jewett 5 Another view  9/29/06 Atms 502  Fall 2006  Jewett 6 Truncation error There is a discretization or truncation error for any finite difference scheme. We want to quantify this error. Substitute Taylor series into the scheme. Example: diffusive process, u t =Ku xx Approximate with the following: u j n + 1 " u j n # t = K u j + 1 n " 2 u j n + u j " 1 n ( ) # x 2 9/29/06 Atms 502  Fall 2006  Jewett 7 Review: Truncation error u t " Ku xx = "# t 2! u tt + K # x 2 12 u xxxx + higher terms ( ) $ % & ( ) The truncation error is all terms on RHS. Order of accuracy determined by lowest powers of t, x; here is O( t, x 2 ). A scheme is consistent if truncation error approaches zero as t 0 and x 0. 9/29/06 Atms 502  Fall 2006  Jewett 8 Review: Truncation error Truncation error: Tells us if the scheme reduces to the PDE as t, x (, y, z..) 0. Tells us the discretization error, and accuracy But: It does not tell us anything about what choices to make for t, x, etc It does not tell us whether we will get a reasonable solution with finite t, x 3 9/29/06 Atms 502  Fall 2006  Jewett 9 Stability Measures of stability can include:...
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This note was uploaded on 10/06/2009 for the course ATMOSPHERI 502 taught by Professor Jewett during the Spring '09 term at University of Illinois at Urbana–Champaign.
 Spring '09
 JEWETT

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