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Adaptive mesh refinement for discontinuous Galerkin method on quadrilateral non-conforming grids

Dr. Michal Kopera
Naval Postgraduate School, California, USA

In recent years there has been a significant interest in using the discontinuous Galerkin method (DG) for solving fluid dynamics problems. Studies in [1], [2], and [3] have shown that the DG method is indeed a good choice for the construction of future non-hydrostatic numerical weather prediction models. It combines high-order accuracy of the solution with geometric flexibility of unstructured grids and exhibits excellent scaling properties.

In order to increase the scale resolution capabilities of DG, as well as to take better advantage of available computing power, the use of adaptive mesh refinement (AMR) for a quadrilateral non-conforming grid is being investigated. The results of AMR implementation to a 2D version of the Non-hydrostatic Unified Model of the Atmosphere (NUMA) will be presented during the talk. Both static and dynamic h-adaptivity (element grid adaptivity) will be considered. Since increased local grid resolution has to impose severe constraints on the time-step of an explicit method, implicit-explicit (IMEX) and multi rate time integration will be discussed as well.

As the latest trend in scientific computing is to employ clusters of GPUs for high-performance computations, the implementation of CUDA kernels into AMR NUMA will be examined.

[1] Giraldo, F. & Restelli, M. (2008). A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in non-hydrostatic mesoscale atmospheric modelling: Equation sets and test cases. Journal of Computational Physics, 227, 3849-3877

[2] Restelli, M. & Giraldo, F.X. (2009). A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modelling. SIAM J. Sci. Comp., 31, 2231-2257

[3] Kelly, J.F. & Giraldo, F.X. (2012). Continuous and discontinuous Galerkin methods for scalable nonhydrostatic atmospheric models: limited-area mode. Journal of Computational Physics, in review (2012).