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January 28, 2018, 15:37 |
Stability Analysis of Hyperbolic Problem
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#1 |
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abcd efgh ijkl
Join Date: Oct 2015
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Hello everyone,
I hope this finds you in good health. I was reading the book "Numerical Computation of Internal and External Flows" Vol.1 by Charles Hirsch. In the Chapter of Von Neumann stability analysis, the author has stated two relations, one for exact dispersion (7.4.18) and the second for typical harmonic of solution u (7.4.19). Can anyone help me with their derivation or guide me where can I find more details about these relations. Please see the attached image. |
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January 28, 2018, 15:54 |
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#2 |
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Filippo Maria Denaro
Join Date: Jul 2010
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Considering the linear equation on a periodic domain you can see that, according to a Fourier series, a real solution u(x,t) can be written as a sum of products of two complex Uk(t)*exp(i*k*x). By substituting it in the equation, you can easily see that Uk(t)=Uk(0)*exp(-i*k*a*t). Therefore:
u(x,t) = Uk(0)*exp[i*k*(x-a*t)] The unitary amplification factor simply states that the initial function u(x,0)=Uk(0)*exp(i*k*x) is convected rigidily at a constant velocity a. Now, consider any discretization of the previous equation. Substitute the Fourier components in it and see what happens to the resolved field by means of the amplification factor |
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January 29, 2018, 10:37 |
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#3 | |
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abcd efgh ijkl
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Quote:
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stability analysis, von neumaa |
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