Navier-Stokes linear stability analysis

Linear Algebra
1

Hi there !

I’m trying to implement a linear stability module in my Navier-Stokes code with no sucess so far. My idea was to take advantage of the “derivative” function to compute the jacobian then solve the eigenvalue problem.

So far I have something like that:

    wbf = Function(space)
    ubf, pbf = split(wbf)
    v, q = TestFunctions(space)

    Fs = (nu * inner(grad(ubf), grad(v))\
         + dot(dot(grad(ubf), ubf), v)\
         - pbf * div(v)\
         - q * div(ubf)\
         - dot(f, v)) * dx

I solve this steady problem for a given nu, then I calculate the jacobian applied to the baseflow previously calculated (is the derivative function doing that?):

    Js = derivative(Fs, wbf)
    Js = assemble(Js)

Then I have two options, use dolfin.SLEPcEigenSolver or export my matrix and do the calculation later.

  1. If I use dolfin.SLEPcEigenSolver how to specify that I want to solve a complex eigenvalue problem, even though the matrix is real? Looks like it’s looking for complex eigenvalues by default.

  2. For now I’m exporting the matrix and converting it to a complex one while reading it. Then I solve the eigenvalue problem.

  3. In both case, should I impose Dirichlet BC egal to zero everywhere except at the outflow?(answered myself, Yes in order to have a proper weak form obviously) If yes does Js.apply(BC) do the thing?

I don’t have any MWE but if you need one I will implement one quickly.

EDIT : Most of the code is working properly now but my eigenvalues are not what expected. Do anyone has references for Linear Stability Analisys theory in the case of PDE ?

Thank you for your help ! :smiley: