Thursday 20 Feb 2020: Fast semi-implicit DG solvers for fluid dynamics: hybridization and multigrid preconditioners
Eike Mueller - University of Bath
Harrison 101 14:30-16:30
For problems in Numerical Weather Prediction, time to solution is critical. Semi-implicit time-stepping methods can speed up geophysical fluid dynamics simulations by taking larger model time-steps than explicit methods. This is possible since semi-implicit integrators treat the fast (but physically less important) waves implicitly. As a consequence, the time-step size is not restricted by an overly tight CFL condition. A disadvantage of this approach is that a large system of equations has to be solved repeatedly at every time step. However, using an suitably preconditioned iterative method significantly reduces the computational cost of this solve, potentially making a semi-implicit scheme faster overall.
A good spatial discretisation is equally important. Higher-order Discontinuous Galerkin (DG) methods are known for having high arithmetic intensity and can be parallelised very efficiently, which makes them well suited for modern HPC hardware. Unfortunately, the arising linear system in semi-implicit timestepping is difficult to precondition since the numerical flux introduces off-diagonal artificial diffusion terms. Those terms prevent the traditional reduction to a Schur-complement pressure equation. This issue can be avoided by using a hybridised DG discretisation, which introduces additional flux-unknowns on the facets of the grid and results in a sparse elliptic Schur-complement problem. Recently Kang, Giraldo and Bui-Thanh  solved the resultant linear system with a direct method. However, since the cost grows with the third power of the number of unknowns, this becomes impractical for high resolution simulations.
We show how this issue can be overcome by constructing a non-nested geometric multigrid preconditioner similar to  instead. We demonstrate the effectiveness of the multigrid method for the non-linear shallow water equations, an important model system in geophysical fluid dynamics. With our solvers semi-implicit IMEX time-steppers become competitive with standard explicit Runge Kutta methods. Hybridisation and reduction to the Schur-complement system is implemented in the Slate language , which is part of the Firedrake Python framework for solving finite element problems via code generation.
(joint work with Jack Betteridge [Bath], Ivan Graham [Bath] and Thomas Gibson [Imperial, now at Monterey])
 Kang, Giraldo, Bui-Thanh (2019): "IMEX HDG-DG: a coupled implicit hybridized discontinuous Galerkin (HDG) and explicit discontinuous Galerkin (DG) approach for shallow water systems" Journal of Computational Physics, 109010, arXiv:1711.02751
 Cockburn, Dubois, Gopalakrishnan, Tan (2014): "Multigrid for an HDG method", IMA Journal of Numerical Analysis 34(4):1386-1425
 Gibson, Mitchell, Ham, Cotter, (2018): "A domain-specific language for the hybridization and static condensation of finite element methods." arXiv preprint arXiv:1802.00303.