I am trying to study the stability of lattice Boltzmann method. The the eigenvalues of the evolution matrix in wave number space need to be studied carefully.
I solved the eigenvalues using Matlab.
I thought that the fluctuation is an artifact resulting from using Matlab, since it shifts the order of the eigenvalues each time I am calculating them. However, this is not the case.
To prove that, I derived the characteristic polynomial and found analytical solutions for each mode. The analytical solution is correct, however some modes fluctuate between it self and other mode (see the picture). Note: there is another mode that complement this mode, so that they together form complete two modes.
what is the reason for this fluctuation? and what is its impact on stability and accuracy?
i think
- For real eigenvectors, the sign of the eigenvectors can change.
- For complex eigenvectors, the eigenvectors can be multiplied by any complex number of magnitude 1.
- For a multiple eigenvalue, its eigenvectors can be recombined through linear combinations. For example, if Ax = λx and Ay = λy, then A(x+y) = λ(x+y), so x+yalso is an eigenvector of A.
Thanks Ahmad. That answered part of the question. I will try to explain what I understood. The eigenvalues are fluctuating because eigenvector can be recombined through linear combinations. So, there is a specific set of eigenvectors that generate non- fluctuation eigenvalues. Please correct me If am wrong.
I want to know also if this fluctuation has any impact on stability or accuracy of the analyzed method?
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