Add a simple mixed steady/unsteady demo

[jwallwork23]

          @ddundo - I think it would help to have a simple mixed case test. Do you have one already implemented elsewhere in the code base I can use as a go-by?

Originally posted by @acse-ej321 in #149 (comment)

[ddundo]

Thanks @jwallwork23, I was going to ask about this on the meeting today. Do you have a suggestion for what could be a good problem here? I'm not sure how good my glacier example would be...

@stephankramer mentioned some in the original issue pyroteus/pyroteus#122 (comment):

Just to clarify: I think we are talking about solving equations without a time derivative. They are still time-dependent in the sense that they depend on other fields that are time-dependent and have been previously solved for - it's just that they don't depend on the previous value of the field that's solved for in that equation.

Some other examples of where this might happen:

• mantle convection modelling: each timestep you solve a steady-state Stokes system (so it doesn't depend on previous values of u and p) but it does depend on temperature in the buoyancy. The velocity that comes out is then used in a normal time-dependent advection-diffusion equation for temperature, which will feed into the Stokes solve in the next timestep solve
• when you do a continuous pressure projection approach: you timestep the momentum equation forward for a preliminary velocity, but then you solve a pressure equation whose right-hand side does depend on that preliminary velocity but the pressure equation does not depend on the previous value of p

Maybe one of these would be better?

[ddundo]

I just took a look at Davies, Kramer et al. 2022 for the mantle convection problem and it looks like a nice example to me. Also includes the code so it would be fast and easy to get a demo up. What do you think?

[jwallwork23]

I just took a look at Davies, Kramer et al. 2022 for the mantle convection problem and it looks like a nice example to me. Also includes the code so it would be fast and easy to get a demo up. What do you think?

That would be amazing, thanks! Goalie only currently supports $\theta$-method timesteppers but checking the paper it seems "For temporal integration, we apply a simple $\theta$ scheme to the energy Eq" (Crank-Nicolson), so I think we're all good.

[ddundo]

Blocked by #188