Spectral Semi-discretisations of Weakly Non-linear Wave Equations over Long Times

The long-time behaviour of spectral semi-discretisations of weakly non-linear wave equations is analysed. It is shown that the harmonic actions are approximately conserved for the semi-discretised system as well. This permits to prove that the energy of the wave equation along the interpolated semi-discrete solution remains well conserved over long times and close to the Hamiltonian of the semi-discrete equation. Although the momentum is no longer an exact invariant of the semi-discretisation, it is shown to be approximately conserved. All these results are obtained with the technique of modulated Fourier expansion

From h to p efficiently: optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method

We investigate the relative performance of a second-order Adams–Bashforth scheme and second-order and fourth-order Runge–Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/hp element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd

Hybridised multigrid preconditioners for a compatible finite element dynamical core

Compatible finite element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging since the velocity mass matrix is non-diagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate- and weather prediction model LFRic.Comment: 24 pages, 13 figures, 5 tables; accepted for publication in Quarterly Journal of the Royal Meteorological Societ

A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry

We present a fourth-order finite-volume algorithm in space and time for low Mach number reacting flow with detailed kinetics and transport. Our temporal integration scheme is based on a multi-implicit spectral deferred correction (MISDC) strategy that iteratively couples advection, diffusion, and reactions evolving subject to a constraint. Our new approach overcomes a stability limitation of our previous second-order method encountered when trying to incorporate higher-order polynomial representations of the solution in time to increase accuracy. We have developed a new iterative scheme that naturally fits within our MISDC framework that allows us to simultaneously conserve mass and energy while satisfying on the equation of state. We analyse the conditions for which the iterative schemes are guaranteed to converge to the fixed point solution. We present numerical examples illustrating the performance of the new method on premixed hydrogen, methane, and dimethyl ether flames.Comment: 27 pages, 5 figure

Self-adaptive isogeometric spatial discretisations of the first and second-order forms of the neutron transport equation with dual-weighted residual error measures and diffusion acceleration

As implemented in a new modern-Fortran code, NURBS-based isogeometric analysis (IGA) spatial discretisations and self-adaptive mesh refinement (AMR) algorithms are developed in the application to the first-order and second-order forms of the neutron transport equation (NTE). These AMR algorithms are shown to be computationally efficient and numerically accurate when compared to standard approaches. IGA methods are very competitive and offer certain unique advantages over standard finite element methods (FEM), not least of all because the numerical analysis is performed over an exact representation of the underlying geometry, which is generally available in some computer-aided design (CAD) software description. Furthermore, mesh refinement can be performed within the analysis program at run-time, without the need to revisit any ancillary mesh generator. Two error measures are described for the IGA-based AMR algorithms, both of which can be employed in conjunction with energy-dependent meshes. The first heuristically minimises any local contributions to the global discretisation error, as per some appropriate user-prescribed norm. The second employs duality arguments to minimise important local contributions to the error as measured in some quantity of interest; this is commonly known as a dual-weighted residual (DWR) error measure and it demands the solution to both the forward (primal) and the adjoint (dual) NTE. Finally, convergent and stable diffusion acceleration and generalised minimal residual (GMRes) algorithms, compatible with the aforementioned AMR algorithms, are introduced to accelerate the convergence of the within-group self-scattering sources for scattering-dominated problems for the first and second-order forms of the NTE. A variety of verification benchmark problems are analysed to demonstrate the computational performance and efficiency of these acceleration techniques.Open Acces

On the use of spectral element methods for under-resolved simulations of transitional and turbulent flows

The present thesis comprises a sequence of studies that investigate the suitability of spectral element methods for model-free under-resolved computations of transitional and turbulent flows. More specifically, the continuous and the discontinuous Galerkin (i.e. CG and DG) methods have their performance assessed for under-resolved direct numerical simulations (uDNS) / implicit large eddy simulations (iLES). In these approaches, the governing equations of fluid motion are solved in unfiltered form, as in a typical direct numerical simulation, but the degrees of freedom employed are insufficient to capture all the turbulent scales. Numerical dissipation introduced by appropriate stabilisation techniques complements molecular viscosity in providing small-scale regularisation at very large Reynolds numbers. Added spectral vanishing viscosity (SVV) is considered for CG, while upwind dissipation is relied upon for DG-based computations. In both cases, the use of polynomial dealiasing strategies is assumed. Focus is given to the so-called eigensolution analysis framework, where numerical dispersion and diffusion errors are appraised in wavenumber/frequency space for simplified model problems, such as the one-dimensional linear advection equation. In the assessment of CG and DG, both temporal and spatial eigenanalyses are considered. While the former assumes periodic boundary conditions and is better suited for temporally evolving problems, the latter considers inflow / outflow type boundaries and should be favoured for spatially developing flows. Despite the simplicity of linear eigensolution analyses, surprisingly useful insights can be obtained from them and verified in actual turbulence problems. In fact, one of the most important contributions of this thesis is to highlight how linear eigenanalysis can be helpful in explaining why and how to use spectral element methods (particularly CG and DG) in uDNS/iLES approaches. Various aspects of solution quality and numerical stability are discussed by connecting observations from eigensolution analyses and under-resolved turbulence computations. First, DG’s temporal eigenanalysis is revisited and a simple criterion named "the 1% rule" is devised to estimate DG’s effective resolution power in spectral space. This criterion is shown to pinpoint the wavenumber beyond which a numerically induced dissipation range appears in the energy spectra of Burgers turbulence simulations in one dimension. Next, the temporal eigenanalysis of CG is discussed with and without SVV. A modified SVV operator based on DG’s upwind dissipation is proposed to enhance CG’s accuracy and robustness for uDNS / iLES. In the sequence, an extensive set of DG computations of the inviscid Taylor-Green vortex model problem is considered. These are used for the validation of the 1% rule in actual three-dimensional transitional / turbulent flows. The performance of various Riemann solvers is also discussed in this infinite Reynolds number scenario, with high quality solutions being achieved. Subsequently, the capabilities of CG for uDNS/iLES are tested through a complex turbulent boundary layer (periodic) test problem. While LES results of this test case are known to require sophisticated modelling and relatively fine grids, high-order CG approaches are shown to deliver surprisingly good quality with significantly less degrees of freedom, even without SVV. Finally, spatial eigenanalyses are conducted for DG and CG. Differences caused by upwinding levels and Riemann solvers are explored in the DG case, while robust SVV design is considered for CG, again by reference to DG’s upwind dissipation. These aspects are then tested in a two-dimensional test problem that mimics spatially developing grid turbulence. In summary, a point is made that uDNS/iLES approaches based on high-order spectral element methods, when properly stabilised, are very powerful tools for the computation of practically all types of transitional and turbulent flows. This capability is argued to stem essentially from their superior resolution power per degree of freedom and the absence of (often restrictive) modelling assumptions. Conscientious usage is however necessary as solution quality and numerical robustness may depend strongly on discretisation variables such as polynomial order, appropriate mesh spacing, Riemann solver, SVV parameters, dealiasing strategy and alternative stabilisation techniques.Open Acces

Stability issues of entropy-stable and/or split-form high-order schemes -- Analysis of local linear stability

The focus of the present research is on the analysis of local linear stability of high-order (including split-form) summation-by-parts methods, with e.g. two-point entropy-conserving fluxes, approximating non-linear conservation laws. Our main finding is that local linear stability is not guaranteed even when the scheme is non-linearly stable and that this has grave implications for simulation results. We show that entropy-conserving two-point fluxes are inherently locally linearly unstable, as they can be dissipative or anti-dissipative. Unfortunately, these fluxes are at the core of many commonly used high-order entropy-stable extensions, including split-form summation-by-parts discontinuous Galerkin spectral element methods (or spectral collocation methods). For the non-linear Burgers equation, we demonstrate numerically that such schemes cause exponential growth of errors. Furthermore, we demonstrate a similar abnormal behaviour for the compressible Euler equations. Finally, we demonstrate numerically that other commonly used split-forms, such as the Kennedy and Gruber splitting, are also locally linearly unstable