A good starting point to see how this should be done is the sw_linear_williamson2_triangle.py example in dcore/examples The solver there is not robust for large dt relative to the Coriolis timescale, but we are working on a hybridised solver for this that will be. If you just want to look into the REXI parallelism aspects without combining with spatial parallelism then you could use a direct solver on small problems for this. Let's talk more offline about it to avoid clogging up the list. all the best --cjc On 30 April 2016 at 20:40, Colin Cotter <colin.cotter@imperial.ac.uk> wrote:
Hi Martin, I discussed this complex number issue with Terry Haut a while ago. He has an equivalent formulation that stays entirely in real numbers - this is because the complex values appear as conjugate pairs. So, it's probably easiest to move to this real-only formulation.
By the way, the Coriolis term does make a big difference to how things are solved, I think that it will only be possible to efficiently solve this system using hybridisation techniques: this was the foundation of the proposal that we put in to NERC with Beth in January.
To answer your other questions: we are converging towards a Firedrake dynamical core code, this is being developed on github: https://github.com/firedrakeproject/dcore but things are not really ready for integration with REXI yet, we still need to demonstrate that the basic discretisation works for 3D on the sphere.
We have a plan to implement a multigrid solver for the the REXI scheme for FEM shallow water, and then develop an APINT scheme using Firedrake for shallow water on the sphere with Jemma, we should probably discuss to make sure that we don't tread on each other's toes.
all the best --cjc
On 30 April 2016 at 07:35, Martin SCHREIBER <M.Schreiber@exeter.ac.uk> wrote:
Dear Firedrakers,
I discussed yesterday with Andrew if Firedrake can be (ab)used to solve PDEs with a non-traditional time stepping method (we call it REXI) and he suggested me to drop an Email to this mailing list. (@David/Colin: It's related to our parallelization-in-time work).
Let L be the linear operator of the SWE on the sphere in advective form including the average height (by putting the perturbation into the non-linear part). L also contains the longitude-varying Coriolis term, but this shouldn't make a big difference here.
As one building block of the new time stepping method we like to solve the SoEs given by (\alpha - L) U = U0 with \alpha a complex shifted pole creating a non-singular SoE and U0 the DoFs to solve for.
I don't see any problem to write this in weak formulation as it is required in Firedrake.
Now the following questions arise:
* Is Firedrake able to assemble and solve these system of equations with the complex value \alpha? It's only important to keep the real values of "U". The imaginary values of U are not important if that's of any help.
* If not, is Firedrake still able to solve a reformulation which would have a 6x6 linear operator instead of a 3x3 and 6 DoFs (3 pseudo DoFs) per cell? (Basically reformulating the SoE with real values instead of complex ones).
* Is there any Firedrake code publicly available with the GungHo FEM dyn-core on the sphere with a C-grid?
* A question related to possible future work: Is Firedrake supporting Semi-Lagrangian methods efficiently?
Cheers,
Martin
-- Dr. rer.-nat. Martin Schreiber Lecturer, proleptic Scientific and High-Performance Computing University of Exeter
College of Engineering, Mathematics and Physical Sciences Harrison Building, Room 319, North Park Road, Exeter, EX4 4QF
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