2.4 Other Software Implementations and Frameworks

In the last ten years a collection of software frameworks has been put forward to try to bridge the gap between the mathematics of high-order methods and their implementation. A number of software packages already exist for fluid dynamics which implement high-order finite element methods, although these packages are typically targeted at a specific domain or provide limited high-order capabilities as an extension. A major challenge many practitioners have with spectral/hp elements and high-order methods, in general, is the complexity (in terms of algorithmic design) they encounter. In this section, we give an incomplete but representative summary of several of these attempts to overcome this challenge.

The Nektar flow solver is the predecessor to Nektar++ and implements the spectral/hp element method for solving the incompressible and compressible Navier-Stokes equations in both 2D and 3D. While it is widely used and the implementation is computationally efficient on small parallel problems, achieving scaling on large HPC clusters is challenging. Semtex [11] implements the 2D spectral element method coupled with a Fourier expansion in the third direction. The implementation is highly efficient, but can only be parallelised through Fourier-mode decomposition. Nek5000 [31] is a 3D spectral element code, based on hexahedral elements, which has been used for massively parallel simulations up to 300,000 cores. The Non-hydrostatic Unified Model of the Atmosphere (NUMA) [35] is a spectral element framework that employs continuous and discontinuous Galerkin strategies for solving a particular problem of interest, but in a way on which others could adopt and build. Hermes [67] implements hp-FEM for two-dimensional problems and has been used in a number of application areas. Limited high-order finite element capabilities are also included in a number of general purpose PDE frameworks including the DUNE project [23] and deal.II [10]. FEniCS [51] is a collaborative project for the development of scientific computing tools, with a particular focus on the automated solution of differential equations by finite element methods (FEM). Through the use of concepts such as meta-programming, FEniCS tries to keep the solving of PDEs with FEM, from the application programmers’ perspective, as close to the mathematical expressions as possible without sacrificing computational efficiency. A number of codes also implement high-order finite element methods on GPGPUs including nudg++, which implments a nodal discontinuous Galerkin scheme [39], and PyFR [71], which supports a range of flux reconstruction techniques.