Diffusion equation with pseudo time step
 

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?

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.

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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