Nektar++ is an open-source software framework designed to support the development of high-performance scalable solvers for partial differential equations using the spectral/hp element method. High-order methods are gaining prominence in several engineering and biomedical applications due to their improved accuracy over low-order techniques at reduced computational cost for a given number of degrees of freedom. However, their proliferation is often limited by their complexity, which makes these methods challenging to implement and use. Nektar++ is an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretisation techniques and implementation strategies, supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/hp element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains.Program summaryProgram title: Nektar++Catalogue identifier: AEVV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: MITNo. of lines in distributed program, including test data, etc.: 1052456No. of bytes in distributed program, including test data, etc.: 42851367Distribution format: tar.gzProgramming language: C++.Computer: Any PC workstation or cluster.Operating system: Linux/UNIX, OS X, Microsoft Windows.RAM: 512 MBClassification: 12.External routines: Boost, FFTW, MPI, BLAS, LAPACK and METIS (www.cs.umn.edu)Nature of problem:The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations.Solution method:Spectral/hp element methodRunning time:The tests provided take a few minutes to run. Runtime in general depends on mesh size and total integration time

Floating wave energy converters (WECs) operating in the resonance region are strongly affected by non-linearities arising from the interaction between the waves, the WEC motion and the mooring restraints. To compute the restrained WEC motion thus requires a method which readily accounts for these effects. This paper presents a method for coupled mooring analysis using a two-phase Navier–Stokes (VOF–RANS) model and a high-order finite element model of mooring cables. The method is validated against experimental measurements of a cylindrical buoy in regular waves, slack-moored with three catenary mooring cables. There is overall a good agreement between experimental and computational results with respect to buoy motions and mooring forces. Most importantly, the coupled numerical model accurately recreates the strong wave height dependence of the response amplitude operators seen in the experiments

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate a C⁰ - continuous expansion. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use of the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed.

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third‐order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.

We present spectral/hp discontinuous Galerkin methods for modelling weakly nonlinear and dispersive water waves, described by a set of depth-integrated Boussinesq equations, on unstructured triangular meshes. When solving the equations two different formulations are considered: directly solving the coupled momentum equations and the 'scalar method', in which a wave continuity equation is solved as an intermediate step. We demonstrate that the approaches are fully equivalent and give identical results in terms of accuracy, convergence and restriction on the time step. However, the scalar method is shown to be more CPU efficient for high order expansions, in addition to requiring less storage.