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ABAtanasov / GalerkinSparseGrids.jl

Sparse Grid Discretization with the Discontinuous Galerkin Method for solving PDEs
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Sparse Space-Time Grid Discretizations #6

Open miguelraz opened 5 years ago

miguelraz commented 5 years ago

From the paper, page 14:

Since the derivative and Laplacian operators are sparse matrices, and since the maximum allowable time step scales with the resolution h and thus with the linear problem size N due to the Courant-Friedrichs-Lewy (CFL) criterion, the overall cost to calculate a solution scales as O(N^2 log^(D−1)(N)), which is significantly more efficient than the O(N^(D+1)) scaling for a full grid. Here D is the number of spatial dimensions. In principle it is possible to employ a sparse space-time discretisation as well, reducing the cost further to O(N log^D (N)). However,this will require a more complex time evolution algorithm instead of the ODE solver we employ, and we thus do not investigate this approach in this paper.

But the paper doesn't go into what the ODE solvers would need to satisfy so be able to get the optimization. Is there any chance this could be implemented with DifferentialEquations.jl internals?

eschnett commented 5 years ago

No, this cannot be done with DifferentialEquations, since this requires using a sparse structure in space-time. The way to do this would be to set up a sparse (D+1)-dimensional grid, and solve the hyperbolic equation via a global (in space-time!) solver, supplying the usual required initial and boundary conditions.