[juliom]

Dear community, I read a very good paper that it is definitely worth reading. I definitely encourage the reading:" Spectral analysis of finite difference schemes for convection diffusion equation". Suman et all. Computer and Fluids 150(2017)95-114. https://www.researchgate.net/publica...usion_equation


The authors presented a great analysis in the spectral domain of different temporal-spatial discretization techniques which results are extremely interesting. However, I have a couple of questions that I'd like to know your opinion and suggestions.


1.- The authors used a lot of complex properties and operation. I think that people that work with spectral methods understand this easily. Is there any resource you could suggest. For example, the derivation of the equations shown in Part 2 was difficult to really understand.



2.- Equation (9) represents the physical amplificationf factor, taht for this specfici case (convection- diffusion) has an analytical representation.




However, the text says that the absolute of G is the amplification factor. What I understand is the following. If the value of equation (9) is greater than 1 then the system is not damped at all and it will evolve to an unstable state, therefore diverging. Similarly for negative values. Therefore, a monotonical solution will require to G be between 0<G<1. Am I right?



Each numerical implementation (scheme) has its own amplification factor and the authors exploited this fact, and therefore they obtained the ratio between the [math]G_{phys}/G_{num}[math]. However, the [math]G_{num}[math], required the derivation of its analytical counterpart. I guess that is not straight forward for all schemes, for example, FVM. Or what if I have cell center or vertex center methods, or even an averaged method. How can I get the analytical counterpart? Is it possible to get the FFT of the solution at two subsequent time steps and obtain the ration in this way? However, I think that in the initial transient part where the solution hasn't evolved to the attractor the ration could be completely wrong since it is just a numerical transient phenomenon.



3.- An striking fact, is that for Euler - central difference for the convective and diffusive term, the instability region is shown in Figure 2 is considers areas with CFL way lower than 1....Jum interesting. For the same scheme but larger Peclent number, the stable region is completely different, with the lowest CFL being exactly 1 for all wavenumbers. This lead to the next question, does the instability born and grow in a specific wavenumber? Refer to figures 2-9 from paper.


Finally, the authors referred to diffusion when my conclusion is that G is more related to the dissipation. Diffusion is driven by gradients whereas dissipation is more a numerical issue because of truncation terms. For example closing paragraph from section 3.3.