In terms of CFD, we often analyze the stability of the error using Von-Neumann analysis, especially for FDM based problems. Should we follow the same approach for a compressible fluid flow using FVM ?
No. The von Neumann analysis is constrained in terms of linear equations.
There are different questions in your message. What answer are you doing to obtain?
I would do a mesh refinement study (or change the element orders). If the oscillations are resolved and converge to a specific pattern, I would conclude they are not numerical errors. However, if they change randomly in each study, I would attribute them to numerical error.
I recommend to do what Amin Kazemi suggested above and do the same with another numerical method (concerning to question in title)
You see, in closed volumes resonance phenomena may occur, which are related to instability inside the flow. The second reason may be the following: the step in the computational grid is very small, which leads to the fact that the difference between the flow parameters at the inlet of the computational cell and at its outlet is not too different: the difference in their value is comparable to the rounding error, the accuracy of the numerical method used. The solution in this case is to use a coarser grid.
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