Claes Eskilsson

Nektar++: An open-source spectral/hp element framework

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 obtainable from: CPC Program Library, Queens University, Belfast, N. Ireland No. of lines in distributed program, including test data, etc.: 1052456 No. of bytes in distributed program, including test data, etc.: 42851367 External routines: Boost, PFTW, 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. Running time: The tests provided take a few minutes to run. Runtime in general depends on mesh size and total integration time.

Discontinuous Galerkin spectral/ hp element modelling of dispersive shallow water systems

Two-dimensional shallow water systems are frequently used in engineering practice to model environmental flows. The benefit of these systems are that, by integration over the water depth, a two-dimensional system is obtained which approximates the full three-dimensional problem. Nevertheless, for most applications the need to propagate waves over many wavelengths means that the numerical solution of these equations remains particularly challenging. The requirement for an accurate discretization in geometrically complex domains makes the use of spectral/hp elements attractive. However, to allow for the possibility of discontinuous solutions the most natural formulation of the system is within a discontinuous Galerkin (DG) framework. In this paper we consider the unstructured spectral/hp DG formulation of (i) weakly nonlinear dispersive Boussinesq equations and (ii) nonlinear shallow water equations (a subset of the Boussinesq equations). Discretization of the Boussinesq equations involves resolving third order mixed derivatives. To efficiently handle these high order terms a new scalar formulation based on the divergence of the momentum equations is presented. Numerical computations illustrate the exponential convergence with regard to expansion order and finally, we simulate solitary wave solutions.

A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems

Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework – both in terms of algorithms and implementations – that allows an almost seamless switch between various explicit, implicit and implicit–explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butchers unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems.

An hp/Spectral Element Model for Efficient Long-Time Integration of Boussinesq-Type Equations

We present an hp/spectral element method for modelling one-dimensional nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The model uses nodal basis functions of arbitrary order in space and the third-order Adams-Bashforth scheme to advance in time. Numerical computations are used to show that the hp/spectral element model exhibits exponential convergence. The model is compared to two numerical methods frequently used for solving Boussinesq-type equations; a finite element model using linear basis functions and a finite difference model using a five-point stencil for estimating the first-order derivatives. Using numerical examples, we show that the hp/spectral element model gives great savings in computational time, compared to the other models, if: (i) highly accurate results Eire requested, or, more importantly, (ii) results of “engineering accuracy” are called for in combination with long-time integration.

A Discontinuous Spectral Element Model for Boussinesq-Type Equations

We present a discontinuous spectral element model for simulating 1D nonlinear dispersive water waves, described by a set of enhanced Boussinesq-type equations. The advective fluxes are calculated using an approximate Riemann solver while the dispersive fluxes are obtained by centred numerical fluxes. Numerical computation of solitary wave propagation is used to prove the exponential convergence.

A Parallel High-Order Discontinuous Galerkin Shallow Water Model

The depth-integrated shallow water equations are frequently used for simulating geophysical flows, such as storm-surges, tsunamis and river flooding. In this paper a parallel shallow water solver using an unstructured high-order discontinuous Galerkin method is presented. The spatial discretization of the model is based on the Nektar++ spectral/hp library and the model is numerically shown to exhibit the expected exponential convergence. The parallelism of the model has been achieved within the Cactus Framework. The model has so far been executed successfully on up to 128 cores and it is shown that both weak and strong scaling are largely independent of the spatial order of the scheme. Results are also presented for the wave flume interaction with five upright cylinders.

Efficient uncertainty quantification of a fully nonlinear and dispersive water wave model with random inputs

A major challenge in next-generation industrial applications is to improve numerical analysis by quantifying uncertainties in predictions. In this work we present a formulation of a fully nonlinear and dispersive potential flow water wave model with random inputs for the probabilistic description of the evolution of waves. The model is analyzed using random sampling techniques and nonintrusive methods based on generalized polynomial chaos (PC). These methods allow us to accurately and efficiently estimate the probability distribution of the solution and require only the computation of the solution at different points in the parameter space, allowing for the reuse of existing simulation software. The choice of the applied methods is driven by the number of uncertain input parameters and by the fact that finding the solution of the considered model is computationally intensive. We revisit experimental benchmarks often used for validation of deterministic water wave models. Based on numerical experiments and assumed uncertainties in boundary data, our analysis reveals that some of the known discrepancies from deterministic simulation in comparison with experimental measurements could be partially explained by the variability in the model input. Finally, we present a synthetic experiment studying the variance-based sensitivity of the wave load on an offshore structure to a number of input uncertainties. In the numerical examples presented the PC methods exhibit fast convergence, suggesting that the problem is amenable to analysis using such methods.

Method of moving frames to solve the shallow water equations on arbitrary rotating curved surfaces

A novel numerical scheme is proposed to solve the shallow water equations (SWEs) on arbitrary rotating curved surfaces. Based on the method of moving frames (MMF) in which the geometry is represented by orthonormal vectors, the proposed scheme not only has the fewest dimensionality both in space and time, but also does not require either of metric tensors, composite meshes, or the ambient space. The MMF–SWE formulation is numerically discretized using the discontinuous Galerkin method of arbitrary polynomial order pin space and an explicit Runge–Kutta scheme in time. The numerical model is validated against six standard tests on the sphere and the optimal order of convergence of p +1 is numerically demonstrated. The MMF–SWE scheme is also demonstrated for its efficiency and stability on the general rotating surfaces such as ellipsoid, irregular, and non-convex surfaces.

On devising Boussinesq-type models with bounded eigenspectra

The propagation of water waves in the nearshore region can be described by depth-integrated Boussinesq-type equations. The dispersive and nonlinear characteristics of the equations are governed by tuneable parameters. We examine the associated linear eigenproblem both analytically and numerically using a spectral element method of arbitrary spatial order p. It is shown that existing sets of parameters, found by optimising the linear dispersion relation, give rise to unbounded eigenspectra which govern stability. For explicit time-stepping schemes the global CFL time-step restriction typically requires Δ t ? p - 2 . We derive and present conditions on the parameters under which implicitly-implicit Boussinesq-type equations will exhibit bounded eigenspectra. Two new bounded versions having comparable nonlinear and dispersive properties as the equations of Nwogu (1993) and Schaffer and Madsen (1995) are introduced. Using spectral element simulations of stream function waves it is illustrated that (i) the bounded equations capture the physics of the wave motion as well as the standard unbounded equations, and (ii) the bounded equations are computationally more efficient when explicit time-stepping schemes are used. Thus the bounded equations were found to lead to more robust and efficient numerical schemes without compromising the accuracy.

Estimation of numerical uncertainty in computational fluid dynamics simulations of a passively controlled wave energy converter

The wave loads and the resulting motions of floating wave energy converters are traditionally computed using linear radiation–diffraction methods. Yet for certain cases such as survival conditions, phase control and wave energy converters operating in the resonance region, more complete mathematical models such as computational fluid dynamics are preferred and over the last 5 years, computational fluid dynamics has become more frequently used in the wave energy field. However, rigorous estimation of numerical errors, convergence rates and uncertainties associated with computational fluid dynamics simulations have largely been overlooked in the wave energy sector. In this article, we apply formal verification and validation techniques to computational fluid dynamics simulations of a passively controlled point absorber. The phase control causes the motion response to be highly nonlinear even for almost linear incident waves. First, we show that the computational fluid dynamics simulations have acceptable agreement to experimental data. We then present a verification and validation study focusing on the solution verification covering spatial and temporal discretization, iterative and domain modelling errors. It is shown that the dominating source of errors is, as expected, the spatial discretization, but temporal and iterative errors cannot be neglected. Using hexahedral cells with low aspect ratio and 30 cells per wave height, we obtain results with less than 5% uncertainty in motion response (except for surge) and restraining forces for the buoy without phase control. The amplified nonlinear response due to phase control caused a large increase in numerical uncertainty, illustrating the difficulty to obtain reliable solutions for highly nonlinear responses, and that much denser meshes are required for such cases.