[aspect-devel] Treatment of artificial diffusion and convection terms

[Wolfgang Bangerth]

Sylvia,

> is there a reason why the artificial diffusion term as well as the
> convection term is now treated implicitly and not, as it is written in the
> 2012-paper (High accuracy mantle convection Simulation...), explicitly?

This is something we want to clarify in a follow-up paper that we'd like to 
finish sometime this summer. The issue is that for the explicit treatment, we 
need to satisfy a CFL condition of the form

   Delta t   <=   c  h / ||u||

on every cell. The problem is how one defines 'h', the mesh size, on meshes 
that don't just consist of square or cube cells. Is it the diameter, the 
minimal or maximal edge length, the diameter of the cell in the direction of 
flow \vec u, etc? There are many options, but neither is sufficiently 
satisfactory, and to ensure that one remains stable one ends up with a 
situation where one needs to choose 'c' pretty small for non-uniform meshes. 
In the end, this makes the timestep impracticably small. We played with this 
for a while and then realized that making advection implicit does not actually 
increase the computational complexity for solving the linear system very much, 
while it allows us to choose the time step significantly larger. We thus made 
the advection implicit in ASPECT.



> Does the extrapolated temperature / velocity still occure anywhere?

Which terms specifically do you have in mind? It certainly appears in the 
definition of the artificial viscosity.

Best
  W.

-- 
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Wolfgang Bangerth               email:            bangerth at math.tamu.edu
                                 www: http://www.math.tamu.edu/~bangerth/