This study is directed at a rigorous characterization of the consistency and convergence of discontinuous finite element schemes formulated using Flux Reconstruction (FR). We show that the FR formulation is consistent for linear advection and converges to the exact solution for any scheme that is stable in the von Neumann sense. The numerical solution for a scheme of polynomial order P is composed of P + 1 eignemodes, of which, one and exactly one is 'physical' such that it exhibits the analytical dispersion behavior in the limit of asymptotic grid resolution. The remaining P modes are 'spurious' such that the fraction of energy received by them from the initial condition vanishes in the asymptotic limit. On grid refinement, the rate of convergence of the numerical solution is a function of time, starting from a short-time rate at t = 0 + , associated with interpolation, and asymptotically approaching a long-time rate as t → ∞, associated with numerical differentiation. Both these rates can be inferred directly from the eigensystem of the numerical derivative operator. We verify these analytical expectations using simple experiments in 1-D and 2-D.
Performance portability on heterogeneous high-performance computing (HPC) systems is a major challenge faced today by code developers: parallel code needs to be executed correctly as well as with high performance on machines with different architectures, operating systems, and software libraries. The finite element method (FEM) is a popular and flexible method for discretizing partial differential equations arising in a wide variety of scientific, engineering, and industrial applications that require HPC. This article presents some preliminary results pertaining to our development of a performance portable implementation of the FEM-based Albany code. Performance portability is achieved using the Kokkos library. We present performance results for the Aeras global atmosphere dynamical core module in Albany. Numerical experiments show that our single code implementation gives reasonable performance across three multicore/ many-core architectures: NVIDIA General Processing Units (GPU's), Intel Xeon Phis, and multicore CPUs.
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