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Diffusion equation with pseudo time step

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Old   June 5, 2016, 00:38
Default Diffusion equation with pseudo time step
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Hi CFD people,

I am trying to solve diffusion equation by using Newtons method and pseudo time sub iterations. I am not sure how to interpolate pseudo time sub iterations. I couldn't find the detailed example for this kind of problem. Any suggestions?
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Old   June 5, 2016, 00:58
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The diffusion equation is linear, so why are you using Newton's method?

If this is a nonlinear diffusion problem, then are you looking to use explicit pseudo-timestepping as a preconditioner for (say) a Newton-Krylov method? Or are you using an implicit pseudo-timestepping with a Newton solver to solve for the next pseudo-timestep values? I assume it is the latter, because the former is pretty complicated and probably not something you'd be tackling based on your level of experience.

I assume the equation that you are solving is:

D(u) = 0

where D is some nonlinear diffusion operator.

Pseudo time stepping would replace this equation with:

du/dt = D(u)

and (pseudo) time integrate to large (pseudo) time.

The simplest form is implicit Euler:

(u_new - u_old)/dt = D(u_new)

You can build the Newton iteration of that system and iterate to solve for u_new. Then take the next pseudo timestep. You could also use other schemes other than Implicit Euler, but (pseudo) time accuracy is usually not important. The focus is on using a stable scheme to wrap around the outside of the Newton method...which is not terribly well behaved when the initial guess is far from the final answer.

Good luck.
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Old   June 5, 2016, 01:07
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Thank yo for your response. This will be really helpful. I will solve highly non linear and stiff problem. But, before solving that equation, I need to feel comfortable about pseudo time stepping which I never used previously. So, I chose non linear diffusion equation to practice. I will compute jacobian matrices for real and pseudo time stepping. I don't want to use explicit sub iterations.
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Old   June 22, 2016, 08:28
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