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Day09-slides - Computer problem 2 Plotting Atms 502...

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1 Atms 502 Numerical Fluid Dynamics Thu., Sep. 21, 2006 9/29/06 Atms 502 - Fall 2006 2 Computer problem 2 Plotting Routine doplot2 is available from the class web page; - it will overlay the true and numerical fields It is available in C and also Fortran . You will need to save the initial condition field, to use as true solution when plotting 9/29/06 Atms 502 - Fall 2006 3 Computer problem 2 C programmers : Error in p1demoV2.c - in ic routine; for loop should have read: for (i=i1,j=1; i<=i2; i++) Has been corrected - available from class page Program p1demoV3.c includes this fix and also has calls to doplot2 routine written in C. Make sure you use i1 and i2 throughout your code - watch out for old loops from 1…nx 9/29/06 Atms 502 - Fall 2006 4 Explicit vs. Implicit: advection At right: Group velocities for 1-D advection for explicit (dashed) and implicit (solid) Explicit always more accurate for waves longer than 4 x Both schemes have wrong sign for C g for short (<4 x) waves!! Grotjahn and O’Brien, MWR 104 (1976), Fig. 6 ν =10
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2 9/29/06 Atms 502 - Fall 2006 5 Explicit vs. Implicit: advection Relative phase velocity shown in 3D for implicit and explicit schemes Log 10 ( μ ) on abscissa; β =k x on ordinate. Implicit schemes maintain stability by slowing down the waves . Grotjahn and O’Brien, MWR 104 (1976), Fig. 3 Explicit Implicit β β log 10 μ log 10 μ 9/29/06 Atms 502 - Fall 2006 6 Fully implicit Computational molecule Amplification factor u j n + 1 " u j n # t = " c u j + 1 n + 1 " u j " 1 n + 1 2 # t Implicit Schemes: FTCS Explicit Computational molecule Amplification factor u j n + 1 " u j n # t = " c u j + 1 n " u j " 1 n 2 # t j n j n " = 1 # i μ sin $ 1 + μ 2 sin 2 $ (unstable) (stable) 9/29/06 Atms 502 - Fall 2006 7 Fully implicit Computational molecule Amplification factor u j n + 1 " u j n # t = " c u j + 1 n + 1 " u j " 1 n + 1 2 # t Implicit Schemes: FTCS j n " = 1 # i μ sin $ 1 + μ 2 sin 2 $ (stable) This scheme is known as the Euler Implicit Method Accuracy : O( t, x 2 ) Unconditionally stable but large t produces large truncation errors System of algebraic equations solved on each time step
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