On 3/13/20 12:59 PM, Krishnakumar Gopalakrishnan wrote:
I have a non-linear diffusion equation of the form
du/dt = \nabla.( D(u) \grad u)
The non-linearity appears because of the dependence of the diffusion
coefficient on the solution.
When discretising by the Rothe method, applying backward Euler method in
the strictest sense:
(u^n - u^{n-1})/k^n - \nabla. ( D(u^n) ) * \grad{u^n}) = 0
This would require Newton iterations and such complications. Is it okay
to simply use the numerical value of D from the previous time-step?
(u^n - u^{n-1})/k^n - \nabla. ( D(u^{n-1}) ) * \grad{u^n}) = 0
In this case, we get a nice linear equation, and most of step-26 can be
used as. D(u) is a continuous function of u. Is this semi-explicit type
of usage of diffusion coefficient, a reasonable way to tackle this problem?
Yes, it's a good step. But it implies that you get a restriction on the
time step size.
You can also replace D(u^n) by D(u^*) where u^* is extrapolated from the
previous two or three time steps. That's a more accurate approximation
than just using u^{n-1}.
Best
W.
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