Solve a time-independent problem

[HaukeWittich]

Is there a way to solve time-independend problem with py-pde. I would like to solve the Reynolds equation (https://en.wikipedia.org/wiki/Reynolds_equation) for the case that rho is constant in time. This yields to a system of linear equation for the unknown pressure which is easy to solve with e.g. the numpy tools. But I need to have access to the matrices. Are there any examples or ideas in this direction. Thank you.

[david-zwicker]

There is no special approach for time-independent problems (yet). If we encountered such problems, we reformulated them such that temporal relaxation yields a solutions and then did that. If you know of any good, general methods to solve time-independent methods, we could think about an implementation, but I'm afraid that solutions might not be unique, so methods are necessarily specialized.

[HaukeWittich]

No, sorry I do not have a general method. But perhaps, this is not needed. I guess and please correct if I am wrong, that you do a discretization of your grid (here for the scalar pressure p) like:

for different derivatives. This leads to a matrice for example for the gradient in x-direction like:

plus you have to incorporate the boundary conditions. My question was, does py-pde work like that and how can I get access to the matrices? Because with these matrices I only have to solve these linear equation:

[david-zwicker]

You can only express the right hand side using a matrix if it is a linear equation, which does not seem to be the case for your equation.

[HaukeWittich]

Ok, let's do it this way: 0={}*p-K. The zero means, no change in time. Now it is on the right-hand side :-)

[david-zwicker]

I don't understand what you mean here.

[HaukeWittich]

In one of the examples there is also no time deravative: Solving Poisson’s equation in 1d. I could not find how the function: solve_poisson_equation works. Any ideas?

[david-zwicker]

Yes, this is a special function that can only evaluate the Poisson equation and does not use the main machinery of py-pde. We added it mostly to for testing purposes.