Hello, Prelims: I've been using fenics for a while now and I've had great success getting quick results for simple problems. Now, I'm starting to move to more complex problems. I'm going to be analyzing a variety of coupled hyperbolic-parabolic problems where both time and space scales range over at least 4 orders of magnitude. I will certainly be using adaptive time-stepping and, possibly, grid refinement techniques. However, the problem would become computationally far more tractable if I could time-step the small-scale region independently of the large-scale region.
Problem: I'd like to use local time stepping methods to allow the small length scale and time scale to operate independently of the large length/time scale. This is clearly possible in a specific formulation of Discontinuous Galerkin methods: http://onlinelibrary.wiley.com/doi/10.1111/j.1365-246X.2007.03427.x/abstract The linked method (ADER-DG) uses a Lax-Wendroff-like process to turn time derivatives into spatial derivatives, but extends to higher orders. The result is that the derivatives are local to an element and one element can time-integrate respecting only it's local stability condition (i.e. CFL condition). I'm not an expert on the method they describe, but it doesn't seem like it impacts the spatial discretization at all -- standard DG. Since fenics leaves time-integration to the user, implementing such a time integration scheme seems entirely possible... Questions: The open question in my mind is how easy it would be for me to use local time-steps in the fenics solver. If I solve with a local deltaT variable, will I be able to force the solver to ignore all the parts of the system with 0 time step and just update the non-zero parts? If not, how difficult would it be to code this? I'm suspect I could hack something to assemble a new form each time-step that only contains the relevant components, but that seems extremely suboptimal. Finally, what would be your description of the state of support for DG methods, especially with respect to parallelization? If any of this is not currently possible but would be reasonable within the general fenics framework, I'd be interested in helping to improve the situation. I'm an experienced software developer and am quickly gaining background in the fenics codebase/DG methods. Thanks very much! Ben Thompson
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