Discontinuous Galerkin Spectral Element Method Research Articles

A Verification Suite of Test Cases for the Barotropic Solver of Ocean Models

AbstractThe development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.

Accuracy assessment of discontinuous Galerkin spectral element method in simulating supersonic free jets

Accuracy assessment of discontinuous Galerkin spectral element method in simulating supersonic free jets

An Entropy Stable Discontinuous Galerkin Method for the Two-Layer Shallow Water Equations on Curvilinear Meshes

We present an entropy stable nodal discontinuous Galerkin spectral element method (DGSEM) for the two-layer shallow water equations on two dimensional curvilinear meshes. We mimic the continuous entropy analysis on the semi-discrete level with the DGSEM constructed on Legendre–Gauss–Lobatto (LGL) nodes. The use of LGL nodes endows the collocated nodal DGSEM with the summation-by-parts property that is key in the discrete analysis. The approximation exploits an equivalent flux differencing formulation for the volume contributions, which generate an entropy conservative split-form of the governing equations. A specific combination of a numerical surface flux and discretization of the nonconservative terms is then applied to obtain a high-order path-conservative scheme that is entropy conservative. Furthermore, we find that this combination yields an analogous discretization for the pressure and nonconservative terms such that the numerical method is well-balanced for discontinuous bathymetry on curvilinear domains. Dissipation is added at the interfaces to create an entropy stable approximation that satisfies the second law of thermodynamics in the discrete case, while maintaining the well-balanced property. We conclude with verification of the theoretical findings through numerical tests and demonstrate results about convergence, entropy stability and well-balancedness of the scheme.

Conservation and stability in a discontinuous Galerkin method for the vector invariant spherical shallow water equations

We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, absolute vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.

Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: Global flux quadrature and cell entropy correction

We present a novel formulation of the discontinuous Galerkin spectral element method for solving balance laws, with application to the shallow water equations. The scheme proposed is constructed starting from a global flux formulation in which an additional flux term is constructed as the primitive of the source. We show that, in the context of nodal spectral finite elements, this can be translated into a simple modification of the integral of the source term, or equivalently into a modification of a mass matrix. When using Gauss-Lobatto nodal finite elements this modified integration allows to recover at steady state a well known high order Gauss collocation ODE integrator for the flux: the LobattoIIIA method. This method is superconvergent at the collocation points. We thus obtain considerable accuracy enhancements for any steady state, and a characterization via a discrete well-balanced property similar in spirit to the one proposed in [20], albeit not needing the explicit solution of the local Cauchy problem. To control the entropy production, we introduce ad-hoc artificial viscosity corrections at the cell level and incorporate them into the scheme. We provide theoretical and numerical characterizations of the accuracy and equilibrium preservation of these corrections. Through extensive numerical benchmarking we validate all the theoretical predictions, conforming great improvements in accuracy for one dimensional steady states, and robustness for more complex scenarios both in one and two dimensions.

High order entropy stable discontinuous Galerkin spectral element methods through subcell limiting

Subcell limiting strategies for discontinuous Galerkin spectral element methods do not provably satisfy a semi-discrete cell entropy inequality. In this work, we introduce an extension to the subcell and monolithic convex limiting strategies [12,11,2] that satisfies the semi-discrete cell entropy inequality by formulating the limiting factors as solutions to an optimization problem. The optimization problem is efficiently solved using a deterministic greedy algorithm. We also discuss the extension of the proposed subcell limiting strategy to preserve general convex constraints. Numerical experiments confirm that the proposed limiting strategy preserves high-order accuracy for smooth solutions and satisfies the cell entropy inequality.

A Numerical Test Rig for Turbomachinery Flows Based on Large Eddy Simulations With a High-Order Discontinuous Galerkin Scheme—Part II: Shock Capturing and Transonic Flows

Abstract In the second paper of this three-part series, we focus on the simulation of transonic test cases for turbomachinery applications using a high-order discontinuous Galerkin spectral element method (DGSEM). High-fidelity simulations of transonic compressors and turbines are particularly challenging, as they typically occur at high Reynolds numbers and require additional treatment to reliably capture the shock waves characterizing such flows. A recently developed finite-volume subcell shock capturing scheme tailored for the DGSEM is applied and evaluated with regard to the shock sensor. To this end, we conduct implicit large eddy simulations of a high-pressure turbine cascade from the public literature and a transonic compressor cascade measured at the German Aerospace Center, both at a high Reynolds number above 106. Based on the results, we examine modal-energy and flow-feature based shock indicator functions, compare the simulation data to experimental and numerical studies, and present an analysis of the unsteady features of the flows.

A flux-differencing formula for split-form summation by parts discretizations of non-conservative systems: Applications to subcell limiting for magneto-hydrodynamics

In this paper, we show that diagonal-norm summation by parts (SBP) discretizations of general non-conservative systems of hyperbolic balance laws can be rewritten as a finite-volume-type formula, also known as flux-differencing formula, if the non-conservative terms can be written as the product of a local and a symmetric contribution. Furthermore, we show that the existence of a flux-differencing formula enables the use of recent subcell limiting strategies to improve the robustness of the high-order discretizations. The methods are valid on unstructured curvilinear grids using tensor-product basis functions.To demonstrate the utility of the novel flux-differencing formula, we construct hybrid schemes that combine high-order SBP methods (the discontinuous Galerkin spectral element method and a high-order SBP finite difference method) with a compatible low-order finite volume (FV) scheme at the subcell level. We apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD) problems featuring strong shocks.

Inverse radiative transfer with goal-oriented hp-adaptive mesh refinement: adaptive-mesh inversion

The inverse problem for radiative transfer is important in many applications, such as optical tomography and remote sensing. Major challenges include large memory requirements and computational expense, which arise from high-dimensionality and the need for iterations in solving the inverse problem. Here, to alleviate these issues, we propose adaptive-mesh inversion: a goal-oriented hp-adaptive mesh refinement method for solving inverse radiative transfer problems. One novel aspect here is that the two optimizations (one for inversion, and one for mesh adaptivity) are treated simultaneously and blended together. By exploiting the connection between duality-based mesh adaptivity and adjoint-based inversion techniques, we propose a goal-oriented error estimator, which is cheap to compute, and can efficiently guide the mesh-refinement to numerically solve the inverse problem. We use discontinuous Galerkin spectral element methods to discretize the forward and the adjoint problems. Then, based on the goal-oriented error estimator, we propose an hp-adaptive algorithm to refine the meshes. Numerical experiments are presented at the end and show convergence speed-up and reduced memory occupation by the goal-oriented mesh adaptive method.

A high-order fluid–structure interaction framework with application to shock-wave/turbulent boundary-layer interaction over an elastic panel

Within this work, a loosely-coupled high-order fluid–structure interaction (FSI) framework is developed in order to investigate the influence of an elastic panel response on shock-wave/turbulent boundary-layer interaction (SWTBLI). Since high-order methods are expected to determine the future of high-fidelity numerical simulations, they are employed in the construction of both fluid and structure solvers. Specifically, a split-form arbitrary Lagrangian-Eulerian discontinuous Galerkin spectral element method is employed in the fluid solver and a Legendre spectral finite element method in the structure solver. Shock capturing by an improved adaptive filter method, which confines the filtering effect to the vicinity of shocks, is found to be well-behaved in accuracy, efficiency and flexibility.After being validated by two benchmark FSI problems, the developed FSI framework is applied to simulate SWTBLI over an elastic panel. A comparison with a previous simulation of SWTBLI over a rigid panel reveals that: (1) A larger amplitude of the pressure variation, observed on the elastic panel surface, implies a larger threat to the structural integrity; (2) The shock-induced separation flow over the elastic panel changes both in size and shape, leading to a different skin-friction coefficient distribution; (3) A new low-frequency flow unsteadiness of the same magnitude as the elastic panel vibration is detected, which may affect the flow dynamics inside the turbulent boundary layer.

Spatial discontinuous Galerkin spectral element method for a family of chromatography models in CADET

Packed bed liquid chromatography is widely applied in academia and industry. Model-based methods are increasingly utilized for process development and optimization, demanding multitudes of complex simulations. We derive spatial arbitrary order discontinuous Galerkin (DG) discretizations for three commonly used chromatography models, including the general rate model (GRM). The methods are integrated in the open source CADET software, making efficient implementations publicly available for the first time. The DG CADET code is validated and benchmarked against the original finite volume CADET code. We observe great performance advantages for DG, depending on the discrete problem size. For a four-component steric mass action GRM, we achieve a speed-up of an order of magnitude for an error range typical for engineering applications. We explore the performance of a collocation Legendre–Gauß–Lobatto (LGL) quadrature DG method in comparison to an exact integration DG method. Our performance benchmarks indicate a slight advantage for collocation DG.

A discontinuous Galerkin discretization of elliptic problems with improved convergence properties using summation by parts operators

Nishikawa (2007) proposed to reformulate the classical Poisson equation as a steady state problem for a linear hyperbolic system. This results in optimal error estimates for both the solution of the elliptic equation and its gradient. However, it prevents the application of well-known solvers for elliptic problems. We show connections to a discontinuous Galerkin (DG) method analyzed by Cockburn, Guzmán, and Wang (2009) that is very difficult to implement in general. Next, we demonstrate how this method can be implemented efficiently using summation by parts (SBP) operators, in particular in the context of SBP DG methods such as the DG spectral element method (DGSEM). The resulting scheme combines nice properties of both the hyperbolic and the elliptic point of view, in particular a high order of convergence of the gradients, which is one order higher than what one would usually expect from DG methods for elliptic problems.

Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method

In this work, we consider a high-order discretization of compressible viscous flows allowing parallelization both in space and time.The discontinuous Galerkin spectral element method, which is well-suited for massively parallel simulations, is used for spatial discretization. The main novelty in this work is the additional demonstration of time-parallel capabilities within an implicit two-derivative timestepping procedure to further increase the parallel speedup. Temporal parallelism is made possible by a predictor-corrector-type time discretization that allows to split the associated workload onto multiple processors.We identify a homogeneous load balance with respect to the linear (GMRES) iterations on each processor as a key for parallel efficiency. To homogenize the load and to enable practical simulations, an adaptive strategy for Newton’s method is introduced. It is shown that the time-parallel method provides a parallel efficiency of approx. 60−70% on 4−7 computational partitions. Moreover, the capabilities of the novel method for the simulation of large-scale problems are illustrated with a mixed temporal and spatial parallelization on more than 1000 processors.

High-order methods for hypersonic flows with strong shocks and real chemistry

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.

Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting.

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre-Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing.

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge–Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

Fast, low-memory numerical methods for radiative transfer via hp-adaptive mesh refinement

Numerical solutions to the radiative transfer equation are typically computationally expensive. The large expense arises because the solution has a high dimensionality with NM degrees of freedom, where the N and M arise from spatial and angular degrees of freedom, respectively. Here, a numerical method is presented that aims for fast and low-memory calculations, in the sense of computational cost and memory requirements of only O(N). The method uses a discontinuous Galerkin (DG) spectral element method and hp-adaptive mesh refinement to reduce the number of spatial degrees of freedom from N to n, thereby reducing the total cost and memory to nM, with the aim of achieving nM approximately equal to N. After this reduction in memory to O(N), in order to ensure a computational cost of O(N), a suitable preconditioner is identified and utilized. Numerical examples are presented in two spatial dimensions to allow calculation of high-resolution reference solutions for comparison, while the methodology is general and applies in either two or three spatial dimensions. The numerical examples show large memory reduction ratios N/n and fast O(N) computational cost. A variety of examples is shown, including smooth spatial variations or steep gradients, and Rayleigh (isotropic) or Mie (anisotropic) scattering. The methods could enable more tractable computations for many applications, such as medical imaging and weather and climate prediction.

Toward a robust detection of viscous and turbulent flow regions using unsupervised machine learning

We propose an invariant feature space for the detection of viscous-dominated and turbulent regions (i.e., boundary layers and wakes). The developed methodology uses the principal invariants of the strain and rotational rate tensors as input to an unsupervised Machine Learning Gaussian mixture model. The selected feature space is independent of the coordinate frame used to generate the processed data, as it relies on the principal invariants of the strain and rotational rate, which are Galilean invariants. This methodology allows us to identify two distinct flow regions: a viscous-dominated, rotational region (a boundary layer and a wake region) and an inviscid, irrotational region (an outer flow region). We have tested the methodology on a laminar and a turbulent (using Large Eddy Simulation) case for flows past a circular cylinder at Re = 40 and Re = 3900 and a laminar flow around an airfoil at Re=1×105. The simulations have been conducted using a high-order nodal Discontinuous Galerkin Spectral Element Method. The results obtained are analyzed to show that Gaussian mixture clustering provides an effective identification method of viscous-dominated and rotational regions in the flow. We also include comparisons with traditional sensors to show that the proposed clustering does not depend on the selection of an arbitrary threshold, as required when using traditional sensors.

Discontinuous Galerkin spectral element method for shock capturing with summation by parts properties

This paper presents a computational methodology developed for a high-order approximation of compressible fluid dynamics equations with discontinuities. The methodology is based on a discontinuous Galerkin spectral-element method (DGSEM) built upon a split discretization framework with summation-by-parts (SBP) property, which mimics the integration-by-parts operation in a discrete sense. To extend the split DGSEM framework to discontinuous cases, we implement a shock capturing method based on the entropy viscosity formulation. The developed high-order split-form DGSEM with shock-capturing methodology is subject to a series of evaluation on both one-dimensional and two-dimensional, continuous and discontinuous cases. Convergence of the method is demonstrated both for smooth and shocked cases that have analytical solutions. The 2D Riemann problem tests illustrate an accurate representation of all the relevant flow phenomena, such as shocks, contact discontinuities, and rarefaction waves. All test cases are able to run with a polynomial order of 7 or higher. The values of the tunable parameters related to the entropy viscosity are robust for both 1D and 2D test problems. We also show that higher-order approximations yield smaller errors than lower-order approximations, for the same number of total degrees of freedom.

Entropy-stable Gauss collocation methods for ideal magneto-hydrodynamics

In this paper, we present an entropy-stable Gauss collocation discontinuous Galerkin (DG) method on 3D curvilinear meshes for the GLM-MHD equations: the single-fluid magneto-hydrodynamics (MHD) equations with a generalized Lagrange multiplier (GLM) divergence cleaning mechanism. For the continuous entropy analysis to hold and to ensure Galilean invariance in the divergence cleaning technique, the GLM-MHD system requires the use of non-conservative terms.Traditionally, entropy-stable DG discretizations have used a collocated nodal variant of the DG method, also known as the discontinuous Galerkin spectral element method (DGSEM) on Legendre-Gauss-Lobatto (LGL) points. Recently, Chan et al. [1, “Efficient Entropy Stable Gauss Collocation Methods”. SIAM (2019)] presented an entropy-stable DGSEM scheme that uses Legendre-Gauss points (instead of LGL points) for conservation laws. Our main contribution is to extend the discretization technique of Chan et al. to the non-conservative GLM-MHD system.We provide a numerical verification of the entropy behavior and convergence properties of our novel scheme on 3D curvilinear meshes. Moreover, we test the robustness and accuracy of our scheme with a magneto-hydrodynamic Kelvin-Helmholtz instability problem. The numerical experiments suggest that the entropy-stable DGSEM on Gauss points for the GLM-MHD system is more accurate than the LGL counterpart.