Research lectures, seminars and events
The events listed in this area are research seminars, workshops and lectures hosted by Durham University departments and research institutes. If you are not a member of the University, but wish to enquire about attending one of the events please contact the organiser or host department.
|January 2020||March 2020|
Events for 27 February 2020
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Eike Mueller: Fast semi-implicit DG solvers for fluid dynamics: hybridization and multigrid preconditioners
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.  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.
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The Poincare conjectures roughly state that any closed n-manifold that looks like the n-sphere is the n-sphere. There are various versions of the conjecture: if a manifold is homotopy equivalent to the n-sphere, is it homeomorphic to the n-sphere? If it is homeomorphic to the n-sphere, is it diffeomorphic to the n-sphere? These are referred to as the topological and smooth Poincare Conjectures, respectively. It is claimed that they have been resolved in all cases except for the 4-dimensional smooth Poincare Conjecture, which remains shrouded in mystery.
In this talk, we will explore reasons for this gap and point to the incomplete understanding of Freedman's claimed resolution of the 4-dimensional topological case. The talk will centre on a series of unanswered MathOverflow questions:
We will describe the background and motivation of these questions and explain why the claim of the Freedman Disk Theorem, being at the heart of the matter, is problematic. In addition, we will outline approaches to disproving Freedman's claim and the implications of such a disproof.