[alibaig1991]

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.

[FMDenaro]

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

[alibaig1991]

Originally Posted by FMDenaro

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

Dear Sir, thank you so much. Have a good day!