When you try to solve system of electron density and electron energy transport equations coupled with Poisson's equation, using implicit numerical finite-difference scheme, after linearization you obtain ill-conditioned Jacobian matrix ( due to the Poisson's equation). Does anybody know how to deal with this problem?

Dear Tatiana,

What do you mean by ill-conditioned? Do you mean that the Gaussian elimination becomes unstable due to lack of diagonal dominance? Or do you simply mean that the problem is stiff and fails to converge?

I've never done plasma simulations using a method involving Poisson's equation - most of the plasma work I've done involves fully coupled solution methods (i.e. simulataneously solving the chemistry and fluid equations in one implicit step) which is more robust for these types of problems.

Another thought - are your conditions close to equilibrium? If so, this can make the equations especially stiff.

If the problem is truly an ill-conditioned matrix, you may want to examine potential ways of changing your formulation - could include upwind differencing, linearlizing using the primitive (rho, u, v,w, P) variables instead of conservative (rho, rho*u, rho*v, total energy,etc.), or underrelaxing the chemistry.

Best wishes with your tackling a challenging problem.

Dear Doug!

I mean that the Gaussian elimination becomes unstable due to lack of diagonal dominance, and SVD procedure for jacobian gives values of diagonal matrix which differ 10-12 orders of magnitude due to the stiffness. I consider equilibrium conditions in chemistry part. It just gives me the source terms for transport equations.

I was thinking about changing formulation, I substitute expression of electron density from Poisson's equation (ions densities are frozen) through electric potential in electron density transport equation. It leads to five point stencil numerical scheme. But I haven't read before, does anyone do this. I hope it will be usefull.

Thank you for answer.

Dear Tatiana,

You truly have an ill-conditioned matrix!!! I don't have any further advice for you beyond what it sounds like you are already doing. Look closely at the individual terms in your Jacobian matrix to identify which terms are the offending ones (probably the chemistry source terms I'd guess). Then modify your formulation (changing the differencing scheme, adding a pseudo-time iteration, implicit smoothing, sub-iterations, etc.) and/or solution alogrithm to improve the behavior.

One thing I've seen done which might enhance stability is to use a first-order upwind LHS (especially using a flux difference or flux vector splitting) with a higher order RHS. This may slow the convergence rate, but such a formulation is generally diagonally dominant. Underrelaxation is another option.

Hope all goes well for you!

Doug