Fully-discrete spatial eigenanalysis of discontinuous spectral element methods: Insights into well-resolved and under-resolved vortical flows

A collocated-grid spectral difference method for compressible flows

A high-order accurate flow solution method for quadrilateral elements is developed and tested. The method employs the flux reconstruction (FR) approach proposed by Huynh. Ideal orders of accuracy are achieved for several problems on structured-based grids while some deterioration is observed on purely unstructured grids. Cares for curved wall boundary are also shown. The localized artificial diffusivity (LAD) method is incorporated to capture discontinuities that is crucial for compressible flow simulations. The FR+LAD works very well for 1D test cases but multi-dimensional shock problems requires further modifications to confine the artificial diffusivity to the shock location. Hybrid scheme of the FR and finite difference method taking advantage of 1D-feature alleviates the instability. I. Introduction Expanding demands for the computational fluid dynamics (CFD) requires more reliable numerical schemes. Specifically, turbulent flow simulations, acoustic simulation, combustion simulations and their combined problems. The key words in such simulations may be high accuracy and high resolution. In structured grid CFD, a compact finite difference scheme satisfies both requirements at higher level than existing other methods. If we put more emphasis on easier handling of complex geometries and flow-data dependent grid generation (grid adaptation), then unstructured grid method should also be explored and it should depart from 2nd-order scheme with limiter. Recent efforts in this area are development of the discontinuous Galerkin (DG) method, 1,2 the spectral finite volume (SV) method 3 and the staggered-grid multi-domain method spectral finite difference (SG) method 4,5 and its evolutional method called a spectral difference method. 6 The common and maybe the only way for those unstructured methods is to add degrees of freedom in each cell to achieve high order since the wide stencil of the original grid is not possible due to unstructured nature. The ENO-type method that uses original degrees of freedom in the grids and extends adaptive stencils to the flow solutions is also explored but the difficulty of extending stencils due to undirectional nature of unstructured grids and the unsuitability for parallel computing prohibits wide applications. A flux reconstruction (FR) scheme 7,8 studied in the paper is the new one among the former group. It uses differential form of conservation laws like the SD method but it is more general and includes equivalent counter part of the schemes in the group. In this paper, the performance of the FR is tested by a newly developed Navier-Stokes solver. Realistic flow applications of the FR in the Navier-Stokes equations are seldom seen since it is relatively new scheme. The Localized artificial diffusivity (LAD) 9–13 is incorporated for the shock capturing in the FR. The last item is motivated by the challenge and success of combining the SD and LAD in the recent study. 11,12 Wang 14 developed the lifting collocation penalty (LCP) method to extend the FR to triangular or tetrahedral grids and applied the LCP-FR combined method for hybrid unstructured grids. In the present study, only quadrilateral cells are treated to directly apply the LAD method that was developed on structured grids. The shock capturing by the FR+LAD high order scheme is tried.

The flux reconstruction approach to high-order methods is robust, efficient, simple to implement, and allows various high-order schemes, such as the nodal discontinuous Galerkin method and the spectral difference method, to be cast within a single unifying framework. Utilizing a flux reconstruction formulation, it has been proved (for one-dimensional linear advection) that the spectral difference method is stable for all orders of accuracy in a norm of Sobolev type, provided that the interior flux collocation points are located at zeros of the corresponding Legendre polynomials. In this article the aforementioned result is extended in order to develop a new class of one-dimensional energy stable flux reconstruction schemes. The energy stable schemes are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (including a particular nodal discontinuous Galerkin method and a particular spectral difference method). The analysis offers significant insight into why certain flux reconstruction schemes are stable, whereas others are not. Also, from a practical standpoint, the analysis provides a simple prescription for implementing an infinite range of energy stable high-order methods via the intuitive flux reconstruction approach.

The objective of the present work is to couple computational fluid dynamics analysis with optimal control theory, which has the potential to advance the strategies for active control of complex flows including aeroacoustics. Such an objective requires a highly accurate flow solver that accommodates unstructured grids to handle complex geometries and an efficient numerical algorithm to compute cost function gradients. To this end, we develop an optimal control framework for unsteady flows, based on the adjoint formulation. We pose the trailing-edge noise control problem within this framework. Specifically, we consider the two-dimensional unsteady flow past the Blake airfoil with unsteady blowing/suction on the trailing edge surface as the control, and find an optimal temporal/spatial control to reduce the trailing edge noise. The disparate range between the convective scales and the acoustic scales presents a computational challenge. It is usually handled by using a hybrid approach, which employs the so-called acoustic analogy or integral equation methodology to deal with the fast and large scales of noise propagation while solving the partial differential equations governing the flow by a sufficiently accurate numerical method. The present approach is based on the direct computation of noise as part of the flow solution. The compressible Navier-Stokes equations and their adjoint are solved numerically using a discontinuous spectral element method for spatial discretization and fourth-order RungeKutta for temporal discretization. The control is updated using a nonlinear conjugate gradient method. It is known that the discontinuous spectral element method is particularly suited for acoustics problems, as it requires, for the resolution of waves with a given accuracy, the least amount of grid points compared to the traditional finite-element or finite-difference methods. In addition to high order accuracy (spectral accuracy), the discontinuous spectral element method has the advantage of the local hp refinement. Furthermore, it is trivially parallelizable. The current work focuses on the tonal or narrowband noise associated with laminarflow airfoils. It is found that aeroacoustic control by trailing-edge blowing/suction directly affects the transient drag and lift histories, which is not surprising, as the trailing-edge noise is of dipole type (associated with the unsteady lift and drag). Such a noise-control strategy is found to obtain significant noise reduction with a direct impact on the thickness (drag) dipole with no drastic effect on the shedding region. Physical mechanisms for noise reduction are discussed in the end.

In this work, the filtered mass density function (FMDF) methodology is implemented in the discontinuous spectral element method (DSEM) for simulation of reacting flows in complex geometries. DSEM solves the compressible Navier-Stokes equations on an unstructured grid with high orders of accuracy. The FMDF is a probability density function method that has been developed for large eddy simulation (LES) of variable-density chemically reacting turbulent flows. The solution to the modeled FMDF transport equation is obtained by solving an equivalent set of stochastic differential equations (SDEs) which yield statistically equivalent results. The set of SDEs is solved by means of a Lagrangian Monte Carlo scheme. The validity of the scalar FMDF is appraised by determining the consistency against DSEM in a temporally developing shear layer. Additional results are shown for the DSEM-LES/FMDF simulation of a hydrogen-air reaction in a temporally-developing shear layer.

In this paper we solve electromagnetic scattering problems by approximating Maxwell's equations in the time‐domain with a high‐order quadrilateral discontinuous spectral element method (DSEM). The method is a collocation form of the discontinuous Galerkin method for hyperbolic systems where the solution is approximated by a tensor product Legendre expansion and inner products are replaced with Gauss–Legendre quadratures. To increase flexibility of the method, we use a mortar element method to couple element faces. Mortars provide a means for coupling element faces along which the polynomial orders differ, which allows the flexibility to choose the approximation order within an element by considering only local resolution requirements. Mortars also permit local subdivision of a mesh by connecting element faces that do not share a full side. We present evidence showing that the convergence of the non‐conforming approximations is spectral along with examples of their use. Copyright © 2001 John Wiley & Sons, Ltd.

This dissertation, framed within an Industrial Doctorate programme, has the twofold objective of improving the understanding of numerical properties and convergence acceleration techniques for High-Order methods, and to formulate an efficient incompressible flow solver for thermal applications. The first part of this manuscript is devoted to the study of the linear multi-grid method applied to Flux Reconstruction, a relatively recent High-Order numerical method with close relations to the well-known Discontinuous Galerkin and Spectral Difference schemes. We do so by investigating the non-modal dissipation behaviour of the multi-grid scheme as a function of its parameters, in an effort to find the most efficient configurations. The analysis is extended to 2D high aspect-ratio cells, in the presence of dominating convection. The second part of this manuscript introduces a High-Order incompressible flow solver devised by the simultaneous use of the Flux Reconstruction, the Discontinuous Galerkin and the Continuous Galerkin methods, each of which is known to offer increased computational efficiency for certain types of problems. On the one end of the spectrum, the Flux Reconstruction is an efficient algorithm for the explicit evaluation of fluxes in conservative form; on the other end, the Continuous Galerkin method is comparatively very efficient for the resolution of diffusion-dominated steady problems. The different formulations are put together in a fractional-step approach and validated against known test cases – unsteady Stokes and Taylor problems, lid-driven cavity, Taylor-Green vortex and thermally-driven cavity – with and without analytical solution, demonstrating convergence with mesh, time-step and polynomial refinement. RESUMEN El presente trabajo, enmarcado en un programa de Doctorado Industrial, tiene el doble objetivo de mejorar la comprensión de las propiedades numéricas y técnicas de aceleración de convergencia para métodos de alto orden, así como de formular un solver eficiente para la resolución de flujos incompresibles con aplicaciones térmicas. La primera parte del manuscrito está dedicada al estudio del método multi-grid lineal aplicado a Flux Reconstruction, un método numérico de alto orden relativamente reciente y estrechamente relacionado con los conocidos Discontinuous Galerkin y Spectral Difference. Ello se consigue investigando la disipación numérica no-modal del esquema multi-grid en función de sus parámetros, con el objetivo de determinar aquellas configuraciones más eficientes. El análisis se extiende a 2D con elementos con alta relación de aspecto en presencia de convección dominante. La segunda parte del manuscrito presenta un solver de alto orden incompresible concebido con el uso simultáneo de los métodos Flux Reconstruction, Discontinuous Galerkin y Continuous Galerkin, cada uno de los cuales ofrece una eficiencia computacional superior para ciertos tipos de problemas. En un extremo, Flux Reconstruction es un algoritmo eficiente para la evaluación explícita de flujos en forma conservativa; en el otro extremo, Continuous Galerkin es comparativamente muy eficiente para la resolución de problemas elípticos en estado estacionario. Las diferentes formulaciones se conjugan en un método de paso fraccionado validado con conocidos casos test con y sin solución analítica, demostrando convergencia con el refinamiento en el mallado, el orden polinomial y el paso de tiempo.

In this paper, the ability of high-order flux reconstruction numerical schemes to perform accurate and stable computations of compressible turbulent flows on coarse meshes is investigated. Two new flux reconstruction schemes, which are optimized for wave dissipation and dispersion properties, are compared to the nodal discontinuous Galerkin and spectral difference methods recovered via the energy-stable flux reconstruction method. The Taylor–Green vortex benchmark problem at is used as a simple a priori test of the numerics. Dissipation rates computed from kinetic energy, vorticity, and pressure dilatation are plotted against reference solutions. Results show that, at low mesh resolution, the flux reconstruction schemes are highly accurate across a range of orders of accuracy, although oscillations can appear in the solution at orders of six and above. Although the flux reconstruction method has a built-in stabilization mechanism, an additional means of damping these instabilities is required. The schemes vary in the amount of numerical dissipation and resolution of the turbulent spectrum. One of the optimized flux reconstruction schemes (the optimized flux reconstruction scheme) is shown to have greater spectral accuracy than any of the others tested, motivating its future usage for high-order high-fidelity computational fluid dynamics.

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

An explicit high-order semi-Lagrangian method is developed for the solution of Lagrangian transport equations in Eulerian-Lagrangian formulations. The method is consistent with an explicit, discontinuous spectral element method (DSEM) discretization of the Eulerian formulation. The semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values and optional constraints for mass and energy preservation. The stable explicit time step of the DSEM solver is sufficiently small to prevent particles seeded at the Gauss quadrature points from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian-Lagrangian formulations. Numerical tests in one and two dimensions for linear and non-linear advection show that the method converges exponentially. The use of mass and energy constraints can improve accuracy depending on the order of accuracy of the time integrator.

Discretely nonlinearly stable weight-adjusted flux reconstruction high-order method for compressible flows on curvilinear grids

Ensuring provable nonlinear stability provides bounds on the discrete solution and guarantees that the numerical scheme remains convergent. In the context of compressible flows, nonlinear stability is achieved by enforcing a secondary conservation principle—the second law of thermodynamics. For high-order numerical methods, such as discontinuous Galerkin (DG), discrete nonlinear and entropy stability have been effectively realized. These stability guarantees are typically derived using properties of the L2-norm. We have developed a nonlinearly stable flux reconstruction (NSFR) scheme for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to continue to benefit from the use of larger time-steps than DG due to the ESFR correction functions while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We employ a modified mass matrix in a weight-adjusted form that reduces the computational cost in curvilinear coordinates through a precomputed projection operator that approximates the dense matrix inversion and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum factorization. A novel, fully-discrete, nonlinearly stable flux reconstruction (FD-NSFR) framework is introduced, which guarantees entropy stability in both spatial and temporal domains for high-order methods via the relaxation Runge-Kutta scheme. We developed an FD-NSFR scheme that prevents a temporal numerical entropy change in the broken Sobolev norm if the governing equations admit a convex entropy function that can be expressed in inner-product form. Through the use of a bound-preserving limiter, positivity of thermodynamic quantities is preserved and enables the extension of this scheme to hyperbolic conservation laws. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy-conserving NSFR scheme and find that our proposed entropy-conserving NSFR scheme is computationally competitive with the conservative DG scheme.

Exponentially accurate spectral and spectral element methods for fractional ODEs

En esta tesis se presentan varias técnicas de solución para métodos de Galerkin discontinuo (DG, p.s.s.e.i.1) de alto orden. Si bien el objetivo principal es el desarrollo de técnicas precisas y computacionalmente eficientes para resolver las ecuaciones de Navier-Stokes de flujo compresible en 3D por medio del método Discontinuous Galerkin Spectral Element Method (DGSEM), todos los métodos desarrollados en esta tesis se pueden aplicar a otras ecuaciones de conservación no lineales y a otros esquemas DG. Las técnicas de solución acá presentadas pueden ser clasificadas en dos grupos: (i) las que están orientadas a acelarar la integración temporal, y (ii) las que mejoran la discretización espacial con adaptación-p local. El primer grupo de técnicas disminuye el tiempo de cálculo dada una discretización espacial con un número fijo de grados de libertad (NDOF, p.s.s.e.i.) mediante la mejora de los algoritmos de solución. Por ese motivo, se presta especial atención al desarrollo de algoritmos eficientes para integración temporal implícita y de métodos multimalla. También se detalla una metodología para obtener el Jacobiano analítico DG para ecuaciónes de advección-difusión. Además, se demuestra que la discretización implícita del DGSEM en nodos de Gauss-Lobatto puede ser resuelta eficientemente mediante el método static condensation. Lo anterior permite resolver el problema con sistemas lineales más pequeños y mejor condicionados, lo cual disminuye los tiempos de cálculo para órdenes moderados. El segundo grupo de técnicas consiste en modificar localmente la discretización espacial para reducir NDOF sin deteriorar la precisión del método. Para lograr este obetivo, en esta tesis se desarrolla un nuevo estimador de error para el método DGSEM p-anisitrópico basado en el método _-estimation, el cual es más preciso y requiere menos recursos computacionales que enfoques previos. Posteriormente, este estimador se usa con éxito para realizar adaptación-p anisotrópica en casos estacionaros y no estacionarios, y se demuestra que éste se puede acoplar con métodos multimalla para acelerar más las simulaciones. Adicionalmente, se desarrollan dos técnicas de adaptación-p para casos no estacionarios: un algoritmo dinámico y un algoritmo estático, que es computacionalmente más eficiente en algunos casos estadísticamente estacionarios. ----------ABSTRACT---------- In this thesis, we develop space and time solution techniques for high-order Discontinuous Galerkin (DG) methods. Even though the main focus is the accurate and computationally efficient solution of the 3D compressible Navier-Stokes equations using the Discontinuous Galerkin Spectral Element Method (DGSEM), the methods developed here can be applied to other systems of nonlinear conservation laws and other DG methods. The solution techniques can be classified into two groups: (i) improved solution techniques for temporal discretizations, and (ii) local p-adaptation techniques that enhance the spatial discretization. The first group of techniques decreases the computational time for a given spatial discretization with a fixed number of degrees of freedom (NDOF) by improving the linear and nonlinear solver algorithms. In that regard, special attention is paid to the development of efficient implicit time-integration solvers and to multigrid methods (both explicit and implicit in time). We derive a methodology to obtain the analytical Jacobian of DG discretizations of nonlinear advectiondiffusion equations. Furthermore, we show that time-implicit DGSEM discretizations on Gauss-Lobatto points can be formulated as Schur complement problems and solved using the static-condensation method. This enables one to solve smaller, better-conditioned problems, which yields shorter computation times for moderate polynomial orders. The second group of techniques modifies the spatial discretization by reducing the NDOF while maintaining similar accuracy. To do that, we develop a novel truncation error estimator that uses the _-estimation method for the p-anisotropic DGSEM. The new method is more accurate and computationally cheaper than previous approaches. The error estimator is then successfully applied to perform local anisotropic p-adaptation for steady and unsteady flows. We show that the truncation error estimator can be readily coupled with multigrid techniques to obtain further speed-ups. Moreover, we develop two p-adaptation strategies for unsteady flows. One of them is a dynamic _-based anisotropic p-adaptation method. The other one is a static _-based anisotropic p-adaptation method, which may be computationally cheaper in statistically steady flows.

A high-order spectral difference (SD) method has been developed with graphics processing units (GPUs) using compute unified device architecture (CUDA). It solves the three-dimensional Navier-Stokes equations on unstructured hexahedral grids with RungeKutta time integration. The method is efficient since operations are completed in a onedimensional fashion and the equations are solved in differential form, removing explicit surface and volume integral calculations. Additionally, solution and flux reconstructions are completed locally per cell, increasing the parallelization of the implementation. Due to this efficiency, the application of GPU computing is appealing. This paper presents the SD method implementation with GPU CUDA computing and presents accuracy studies with isotropic vortex propagation and Couette flow, verifies the high-order accuracy of the solver with a numerical sensitive aero-acoustic problem, and compares the developed solver and a high-order finite difference solver with a case presented in the 1st International Workshop on High-Order CFD Methods. Finally, the GPU solver is compared to a similar central processing unit (CPU) solver, where speed-ups ranging from 20-40x faster are illustrated.