Day03-slides - Stable Atms 502 CS 505 CSE 566 Numerical...

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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
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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
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3 9/29/06 Atms 502 - Fall 2006 - Jewett 9 Stability Measures of stability can include: (1) The numerical solution values may increase with time, but not as a function of the number of time steps.
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