I have used this technique for the 2-D heat equation but am not sure how to apply it to a system of equations (i.e. a vector PDE). Do the errors for the dependent variables have the same Fourier decomposition? Thanks RM
You cannot apply Von Neumann stability analysis to discretizations of nonlinear problems such as the Burgers equation. You would have to linearize it, which would reduce it to an advection diffusion equation. On the other hand, you can apply the Von Neumann technique to linear systems, doing the same as in the scalar case for each component of the system.
Thank you. I realized this and linearized my system before determining some conservative stability conditions.
Dear Romit
To solve 1D viscous Burger's equation numerically and I cannot apply von Neumann analysis because the equation is non-linear. How do I predict the stability criteria for my system? I also need to predict the criteria when its inviscid(which basically makes it non-linear advection equation).
www.aei.mpg.de/~rezzolla/.../evolution_pdes_lnotes.pdf
www.che.iitm.ac.in/~sjayanti/cfd2.pdf
http://scicomp.stackexchange.com/questions/8717/stability-of-numerical-method-for-1d-burgers-equation
www.maths.lth.se/na/courses/.../Chapter6.08a__.pdf
Aly - the trick is to consider the u in the u*du/dx term as a frozen term i.e. to linearize it and proceed with the fourier decomposition of the equation into spatial and time harmonics. In simple terms, take the U in the advective term to be a constant and proceed.
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