Unstable PDE's

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spacedout
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Unstable PDE's

Post by spacedout »

Hello

I created a new thread since the name is more suitable than the previous thread I started on ModelPDE.

I am trying to find out where the problem arises with my plasma physics equations.

So for all solvers I replaced

Linear System Solver = Iterative

by

Linear System Solver = Direct

and for all nonlinear equations, I replaced

Nonlinear System Relaxation Factor = 1

by

Nonlinear System Relaxation Factor = .3


But to no avail: The norm in the nonlinear iterations quickly become infinite within a microsecond.

Is that almost a mathematical proof that this is not a numerical problem but that the PDE's themselves have unstable coupling terms (for example like a term with a wrong sign which grows forever) ?

Wish you all a nice weekend

P.S.

the deltaT is 1.0e-8 sec and the Mach number is .8 (272 m/sec around 1 atm and T = 300 Kelvin)
Rich_B
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Re: Unstable PDE's

Post by Rich_B »

Hello,

Can you post a minimum working example? Or at least a minimum almost working example?

Rich.
spacedout
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Re: Unstable PDE's

Post by spacedout »

Actually I cannot ! I tried Elmer's original HeatSolver and FlowSolver both with Linear System Solver = Direct on a standard NACA0012 airfoil and the norm in the nonlinear iterations of the FlowSolver still becomes infinite within microseconds.

There are no other solvers called in the .sif (They are a lot of them but with Exec Solver = Never)

(the deltaT is 1.0e-8 sec and the Mach number is .8 (272 m/sec around 1 atm and T = 300 Kelvin)
I use velocity = 0 Heat Flux = 0.0 boundary conditions on the airfoil itself. And I use
velocity = 272 m/sec, pressure = 1 atm and temperature = 300 Kelvin as boundary conditions everywhere in the farfield.
(These are also the initial conditions everywhere in the domain)

Either these conditions are unphysical or Elmer cannot handle highly compressible flows.
kevinarden
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Re: Unstable PDE's

Post by kevinarden »

FlowSolver by itself generally does not converge with high velocity flows. It can be coupled with k-epsilon for imprvement

viewtopic.php?f=3&t=6995&hilit=turbulent
spacedout
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Re: Unstable PDE's

Post by spacedout »

Turbulence models involve the energy equation but FlowSolver only handles continuity and momentum equations. Shouldn't HeatSolver also look at the turbulence keywords in the .sif ? Not according to its description under ElmerModelsManual. Moreover, I found keywords under ElmerTutorialFilesGUI that were different from those of ElmerModelsManual.

Best,
Marc
kevinarden
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Re: Unstable PDE's

Post by kevinarden »

In Elmer, I believe the heat only follows the flow (convection), except for conduction and radiation, and P=VRT. I believe since FlowSolver only handles continuity and momentum is the reason it can't converge on turbulent flow conditions. Therefore if the flow is turbulent other equations must be introduce. The software is developed and update daily, however the manuals, and tutorials are not, so the documentation lags significantly behind the development.
spacedout
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Re: Unstable PDE's

Post by spacedout »

The theory of characteristics for compressional flows in 3D shows that to avoid instabilities and other problems for subsonic flow ( M <=.8 in my case) then,

we must prescribe at most 4 characteristic variables from the freestream values at the inflow boundary (the 5th variable being extrapolated from the interior domain) whereas at the outflow boundary it is the 5th variable which is prescribed from its freestream value and and all other 4 variables are extrapolated from the interior domain.

We can of course get the primitive variables on the boundaries from the characteristic variables.

The above is standard procedure in finite-volume CFD books but Elmer utilizes a finite element approach. (although finite volume is a special case of finite element methods)

Is this theory included in Elmer ?
spacedout
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Re: Unstable PDE's

Post by spacedout »

Forget about characteristics theory since most CFD books rely on certain static and total variables on the boundaries and never make the connection between them and the theory. Anyhow, back to turbulence. I disabled it in the finite volume openFOAM software: I ran their sonicFoam solver, (the equivalent to Elmer's FlowSolver and HeatSolver) with laminar turned on in one of their dictionaries (their kind of .sif ) and now using the same geometry, initial and boundary conditions, this software is stable not only after a few microseconds (recall I said Elmer was unstable after that period of time) but up to over one millisecond ! So why would a finite element code like Elmer require turbulence parameters when openFOAM doesn't ?
raback
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Re: Unstable PDE's

Post by raback »

Hi

Finite elements excel for elliptic PDEs and finite volumes for parabolic ones. This is of course a generalization that applies to the standard formulations.

Yes, finite volumes can be seen as special case of finite elements when choosing lowest order discontinuous galerkin formulation. Yes, there are conservative formulations for fluids for FEM. But when >90% of applications that we have are not about fluids and even for the fluids we tend to excel for low-Re fluids there has not been much effort to push these avenues.

So if this case is about Navier-Stokes maybe OpenFOAM does provide a better solution for the problem. If it is about some other equation I would recommend doing some dimensional analysis. There is not just one reason for instability. You may, for example, try to solve two perfectly linear PDEs as a coupled system and this may result to divergence if the coupling is not properly addressed.

-Peter
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