Fig 3 - available via license: Creative Commons Attribution 4.0 International
Content may be subject to copyright.
Log-log plot of the entropy conservation errors S (T ) for the Euler equations with initial data (3.10). The errors are given at time T = 13 for polynomials with degree N = 3 (solid line) and N = 4 (dotted line) on a curved moving mesh with K = 16 3 elements
Source publication
![Fig. 1 Left the reference element E = [−1, 1] 3 and on the right a...](publication/339652739/figure/fig1/AS:886082891685888@1588270032248/Left-the-reference-element-E-1-1-3-and-on-the-right-a-general-hexahedral-element-e_Q320.jpg)
![Left the reference element E=[-1,1]3\documentclass[12pt]{minimal}...](publication/339652739/figure/fig5/AS:961819485155331@1606327043974/Left-the-reference-element-E-1-13documentclass12ptminimal-usepackageamsmath_Q320.jpg)
This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre–Gauss–Lobatto points. Furthermore, the collocation of interpolation and...
Similar publications
Finite cloud method (FCM) employs the fixed kernel reproducing technique to construct the interpolation function and point collocation approach is adopted for the discretization. In this study, an improved FCM is proposed such that a node of interest is approximated with its nearest cloud. This feature enables a set of uniformly distributed clouds...
Abstract For the nonlinear predator–prey system (PPS), although a variety of numerical methods have been proposed, such as the difference method, the finite element method, and so on, but the efficient numerical method has always been the direction that scholars strive to pursue. Based on this question, a sinc function interpolation method is propo...
This paper investigates the derivation of global shape functions in T-meshed quadrilateral patches through transfinite interpolation and local elimination. The same shape functions may be alternatively derived starting from a background tensor product of Lagrange polynomials and then imposing linear constraints. Based on the nodal points of the T-m...
A new 9-DOF triangular plate finite element for linear-elastic analysis of very thin to thick plates with small deformations is presented. The element is based on Mindlin plate theory and has higher-order interpolations that are conceived to satisfy all differential equations, including the equilibrium ones based on which the shear stress resultant...
An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen’s nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid f...
Citations
... The DG method in spectral element form combined with an explicit time integration is standout due to its high efficiency for massively parallel computing. Krais et al. [91] and Schnucke et al. [130] proposed a split-form ALE DGSEM, which can deal with the nonlinear instability problem due to under-resolved turbulence. Although high-order DG methods are favorable in turbulence simulation, the appearance of strong discontinuities, e.g. ...
... Kopriva et al. [87] extended the original scheme to be an energy-stable version in order to improve its robustness. Krais et al. [91] and Schnucke et al. [130] further extended the scheme to be an entropy-stable approach. Since the nonlinear-stability problem due to under-resolved turbulence is of our interest, besides the original ALE DGSEM, the entropy-stable version is included in the following as well. ...
... More types of two-point fluxes can be found in [130,91]. As the split-form ALE DGSEM method with PI flux has proven to be stable even in under-resolved turbulent flows [89], it forms the baseline scheme for the LES framework introduced in the following chapter without further statement. ...
Within this work, an accurate and efficient fluid-structure interaction (FSI) framework is developed in order to study the influence of the elastic panel response on the shock-wave/turbulent boundary-layer interaction (SWTBLI). Of specific research interest is, in time-dependent domains, finding an accurate, efficient and robust shock-capturing method capable of obtaining sharp shock profiles as well as introducing as little dissipation into turbulent structures as possible.
Nowadays, the continuous increase in compute power, as well as the ongoing efforts to improve numerical methods and code development, allow to model more and more complex multiphysics problems. These problems introduce additional, highly non-linear and multiscale interactions between different parts of the coupled system, and therefore require the accurate description of each isolated simulation to reduce uncertainties in their modeling choices. FSI is a typical multiphysics phenomenon with crucial importance to a wide range of engineering applications including aeroelastic flutter and the deflection behavior of wind-turbine blades. The challenges associated with modeling FSI include an efficient and accurate coupling between fluid and structure solvers, that is, modeling physical phenomena at the boundary of the fluid domain through structure deformations and modeling external forces on structure surfaces through fluid flow properties. Another challenge appears along with the change of the fluid domain geometry, which necessitates either a reformulation of the governing Navier-Stokes equations or an appropriate numerical method to allow deforming and moving domains. Moreover, a mesh movement technique to handle mesh deformation inside computational domains is indispensable. In this thesis, different FSI temporal and spatial coupling techniques are explored, and subsequently these methods are implemented into a high-order discontinuous Galerkin (DG) solver in an efficient way.
High-order methods are expected to determine the future of high-fidelity numerical simulations, and hence in this thesis high-order numerical methods are employed in the construction of both fluid and structure solvers. Specifically, a discontinuous Galerkin spectral element method (DGSEM) is adopted in the fluid solver and a Legendre spectral finite element method (LSFEM) is adopted in the structure solver.
Within this thesis, complicated flow problems and simple structure problems are involved in the FSI simulations, and thus special care is taken of the computational fluid dynamics (CFD) part, especially the numerics of modeling compressible turbulence and shock waves. For the modeling of simple structures like beams and plates, the classical structure models, i.e. the geometrically- and materially-linear dynamic Timoshenko beam and the dynamic Mindlin-Reissner plate, are employed.
The adopted high-order fluid solver is based on an arbitrary Lagrangian-Eulerian (ALE) DGSEM such that mesh movement is allowed. An implicit large eddy simulation (iLES) technique is used to model turbulence, which relies on a split-form ALE DGSEM to deal with an emerging non-linear stability problem in under-resolved turbulence. These methods were originally derived and proposed by predecessors in the DG community, and they are included in this thesis for integrity. Moreover, different shock-capturing approaches based on the (split-form) ALE DGSEM are investigated and compared in detail. A novel ALE FV sub-cell method is proposed to keep sharp shock profiles in unsteady flows. An improved adaptive filter by confining its filtering effect to be near shocks is found to be better behaved in the accuracy, efficiency and flexibility of the simulation of SWTBLI over elastic panels. In this method, an accurate shock indicator capable of distinguishing solution discontinuities due to shocks and under-resolved turbulence plays the key role while determining the filter effective regions. For this purpose, two new shock indicators based on the original Jameson shock indicator and the Ducros shock indicator are proposed. Apart from the shock capturing, an efficient zonal LES framework relying on a turbulent inflow method and a non-reflecting outflow boundary condition is applied to simulate the SWTBLI over elastic panels within a computational domain of small streamwise length.
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) Larger amplitudes of pressure and temperature variations, observed on the elastic panel surface, imply a larger threat to the structural integrity; 2) The shock-induced separation flow over the elastic panel changes both in size and location, leading to a different skin-friction coefficient distribution; 3) The shock system, including the incident and reflected shocks, changes along with the continuous deformation of panel; 4) A new low-frequency flow unsteadiness of the same magnitude as the elastic panel vibration is detected, which may affect the flow dynamics inside turbulent boundary layer; 5) The panel response has not obviously changed the magnitude of the separation-induced low-frequency flow unsteadiness.
... With arbitrary Lagrangian-Eulerian (ALE) approaches, it is easy to ensure a high-quality mesh so that they are suited for problems where turbulent boundary-layers are present, such as the SWTBLI over elastic panels. Specifically, the split-form ALE DGSEM by Krais et al. (2020b) and Schnücke et al. (2020) is used since it can handle the non-linear instability due to under-resolved turbulence. Although high-order DG methods are favorable in turbulence simulation, the appearance of strong discontinuities, e.g. ...
... To allow arbitrary movements of the underlying mesh, we apply an ALE ansatz relying on the DGSEM, which was described by Minoli and Kopriva (2011). To solve the nonlinear-stability problem in under-resolved turbulence simulations, Krais et al. (2020b) and Schnücke et al. (2020) proposed a split-form ALE DGSEM based on the method by Minoli and Kopriva (2011). Since the under-resolved turbulence simulation is of our interest, the split-form ALE DGSEM is employed as the baseline. ...
... In this section, the split-form ALE DGSEM (Krais et al., 2020b;Schnücke et al., 2020), relying on the so-called strong form of the governing equations discretized with Legendre-Gauss-Lobatto (LGL) nodes, is employed for the spatial discretization. First, the physical domain is subdivided into non-overlapping unstructured hexahedral elements with optionally curved surfaces to account for complex geometries. ...
Share Link: https://authors.elsevier.com/a/1hRla3AMrTGrfW
... The method of manufactured solutions is used to test the accuracy of the entropy-split discretization of the Navier-Stokes equations. To this end, we use the manufactured solution [91] The source terms are obtained by substituting the manufactured solution into the Navier-Stokes equations, (4), and solving for the difference of the left and right hand sides. A constant viscosity, µ = 0.01, is used, the gas constant and Prandtl number are set to R = 1 and Pr = 0.71, respectively. ...
High-order Hadamard-form entropy stable multidimensional summation-by-parts discretizations of the Euler and Navier-Stokes equations are considerably more expensive than the standard divergence-form discretization. In search of a more efficient entropy stable scheme, we extend the entropy-split method for implementation on unstructured grids and investigate its properties. The main ingredients of the scheme are Harten's entropy functions, diagonal- summation-by-parts operators with diagonal norm matrix, and entropy conservative simultaneous approximation terms (SATs). We show that the scheme is high-order accurate and entropy conservative on periodic curvilinear unstructured grids for the Euler equations. An entropy stable matrix-type artificial dissipation operator is constructed, which can be added to the SATs to obtain an entropy stable semi-discretization. Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta method. Entropy stable viscous SATs, applicable to both the Hadamard-form and entropy-split schemes, are developed for the Navier-Stokes equations. In the absence of heat fluxes, the entropy-split scheme is entropy stable for the Navier-Stokes equations. Local conservation in the vicinity of discontinuities is enforced using an entropy stable hybrid scheme. Several numerical problems involving both smooth and discontinuous solutions are investigated to support the theoretical results. Computational cost comparison studies suggest that the entropy-split scheme offers substantial efficiency benefits relative to Hadamard-form multidimensional SBP-SAT discretizations.
... With arbitrary Lagrangian-Eulerian (ALE) approaches, it is easy to ensure a high-quality mesh so that they are suited for problems where turbulent boundary-layers are present, such as the SWTBLI over elastic panels. Specifically, the split-form ALE DGSEM by Krais et al. [23] and Schnucke et al. [24] is used since it can handle the non-linear instability due to under-resolved turbulence. Although high-order DG methods are favorable in turbulence simulation, the appearance of strong discontinuities, e.g. ...
... To allow aribtrary movements of the underlying mesh, we apply an ALE ansatz relying on the DGSEM, which was described by Minoli and Kopriva [45]. To solve the nonlinear-stability problem in under-resolved turbulence simulations, Krais and Schnucke et al. [23,24] proposed a splitform ALE DGSEM based on the method by Minoli and Kopriva [45]. Since the under-resolved turbulence simulation is of our interest, the split-form ALE DGSEM is employed as the baseline. ...
... In this section, the split-form ALE DGSEM [23,24], relying on the so-called strong form of the governing equations discretized with Legendre-Gauss-Lobatto (LGL) nodes, is employed for the spatial discretization. First, the physical domain is subdivided into non-overlapping unstructured hexahedral elements with optionally curved surfaces to account for complex geometries. ...
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. A zonal large eddy simulation technique, relying on a turbulent inflow method and a non-reflecting outflow boundary condition, is used to model under-resolved turbulence efficiently. 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; 4) The separation-induced low-frequency flow unsteadiness over the elastic panel is detected inside a larger streamwise extent, consistent with the larger streamwise extent of the separation flow region.
... Important contributions to the development of kinetic energy compatible and entropy compatible nite difference schemes for hyperbolic PDE that make use of skew symmetric forms and/or a discrete summation by parts (SBP) property can be found, for example, in the papers of Ducros et al. [21,22] , Fisher et al. [23,24] , Carpenter and Nordström et al. [25][26][27][28] , Pirozzoli [29,30] , Sjögreen and Yee [31][32][33] and in Reiss and Sesterhenn [34] . Without pretending completeness of the following overview, important recent developments on high order entropy-compatible schemes can be found, for example, in the work of Mishra and collaborators [35][36][37] , Gassner et al., [38][39][40][41][42] , Shu and collaborators [43,44] and in Chandrashekar and Klingenberg [45] , Ray et al. [46] , Ray and Chandrashekar [47] , Chan and Taylor [48] , Chan et al. [49] , Gaburro et al. [50] , while entropy-compatible schemes for nonconservative hyperbolic systems were presented in Fjordholm and Mishra [51] . ...
... one may interpret the term 1 2 (p +1 + p ) · q + 1 2 as an approximation of the total energy density difference E + 1 2 , with an approximation of the path integral via the simple trapezoidal rule. Due to (39) and (40) , the energy ux including convective and diffusive terms is ...
We introduce a simple and general framework for the construction of thermodynamically compatible schemes for the numerical solution of overdetermined hyperbolic PDE systems that satisfy an extra conservation law. As a particular example in this paper, we consider the general Godunov-Peshkov-Romenski (GPR) model of continuum mechanics that describes the dynamics of nonlinear solids and viscous fluids in one single unified mathematical formalism.
A main peculiarity of the new algorithms presented in this manuscript is that the entropy inequality is solved as a primary evolution equation instead of the usual total energy conservation law, unlike in most traditional schemes for hyperbolic PDE. Instead, total energy conservation is obtained as a mere consequence of the proposed thermodynamically compatible discretization. The approach is based on the general framework introduced in Abgrall (2018) [1]. In order to show the universality of the concept proposed in this paper, we apply our new formalism to the construction of three different numerical methods. First, we construct a thermodynamically compatible finite volume (FV) scheme on collocated Cartesian grids, where discrete thermodynamic compatibility is achieved via an edge/face-based correction that makes the numerical flux thermodynamically compatible. Second, we design a first type of high order accurate and thermodynamically compatible discontinuous Galerkin (DG) schemes that employs the same edge/face-based numerical fluxes that were already used inside the finite volume schemes. And third, we introduce a second type of thermodynamically compatible DG schemes, in which thermodynamic compatibility is achieved via an element-wise correction, instead of the edge/face-based corrections that were used within the compatible numerical fluxes of the former two methods. All methods proposed in this paper can be proven to be nonlinearly stable in the energy norm and they all satisfy a discrete entropy inequality by construction. We present numerical results obtained with the new thermodynamically compatible schemes in one and two space dimensions for a large set of benchmark problems, including inviscid and viscous fluids as well as solids. An interesting finding made in this paper is that, in numerical experiments, one can observe that for smooth isentropic flows the particular formulation of the new schemes in terms of entropy density, instead of total energy density, as primary state variable leads to approximately twice the convergence rate of high order DG schemes for the entropy density.
... Since the seminal work of Tadmor [92] on entropy compatible schemes, many schemes have been developed in order to obtain thermodynamic compatibility on the discrete level according to the ideas of Friedrichs and Lax [45], i.e. discretizing the energy conservation law directly and obtaining the entropy inequality as a consequence of the other equations. For recent works on high order entropy-compatible finite volume and discontinuous Galerkin finite element schemes for systems of conservation laws the reader is referred to [27,28,30,38,43,46,48,60,70,81,82,87,89] and references therein. Entropy-compatible schemes for non-conservative hyperbolic equations were introduced, for example, in [2,42]. ...
... The new DG schemes proposed in this paper can therefore be seen as the missing dual algorithms to known entropy consistent DG schemes, which usually discretize the total energy conservation law directly and obtain the entropy inequality as a consequence, see e.g. [30,38,48,60,70,87,89]. ...
In this work we propose a new family of high order accurate semi-discrete discontinuous Galerkin (DG) finite element schemes for the thermodynamically compatible discretization of overdetermined first order hyperbolic systems. In particular, we consider a first order hyperbolic model of turbulent shallow water flows, as well as the unified first order hyperbolic Godunov–Peshkov–Romenski (GPR) model of continuum mechanics, which is able to describe at the same time viscous fluids and nonlinear elastic solids at large deformations. Both PDE systems treated in this paper belong to the class of hyperbolic and thermodynamically compatible systems, since both satisfy an entropy inequality and the total energy conservation can be obtained as a direct consequence of all other governing equations via suitable linear combination with the corresponding thermodynamic dual variables. In this paper, we mimic this process for the first time also at the semi-discrete level at the aid of high order discontinuous Galerkin finite element schemes. For the GPR model we directly discretize the entropy inequality and obtain total energy conservation as a consequence of a suitable discretization of all other evolution equations. For turbulent shallow water flows we directly discretize the nonconservative evolution equations related to the Reynolds stress tensor and obtain total energy conservation again as a result of the thermodynamically compatible discretization. As a consequence, for continuum mechanics the new DG schemes satisfy a cell entropy inequality directly by construction and thanks to the discrete thermodynamic compatibility they are provably nonlinearly stable in the energy norm for both systems under consideration.
... So far, we discussed semi-discrete DGSEM-LGL variants with tensor product expansions on possible curvilinear unstructured hexahedral meshes. Direct extensions of this variant include nonconforming meshes [166,167], moving meshes [168][169][170], different related versions such as e.g., the line DG method [171], and a fully discrete space-time approach without the assumption on time continuity [172][173][174][175]. An exciting recent development are explicit modified Runge-Kutta methods that retain the semi-discrete entropy stability estimates [176]. ...
In this paper we will review a recent emerging paradigm shift in the construction and analysis of high order Discontinuous Galerkin methods applied to approximate solutions of hyperbolic or mixed hyperbolic-parabolic partial differential equations (PDEs) in computational physics. There is a long history using DG methods to approximate the solution of partial differential equations in computational physics with successful applications in linear wave propagation, like those governed by Maxwell’s equations, incompressible and compressible fluid and plasma dynamics governed by the Navier-Stokes and the Magnetohydrodynamics equations, or as a solver for ordinary differential equations (ODEs), e.g., in structural mechanics. The DG method amalgamates ideas from several existing methods such as the Finite Element Galerkin method (FEM) and the Finite Volume method (FVM) and is specifically applied to problems with advection dominated properties, such as fast moving fluids or wave propagation. In the numerics community, DG methods are infamous for being computationally complex and, due to their high order nature, as having issues with robustness, i.e., these methods are sometimes prone to crashing easily. In this article we will focus on efficient nodal versions of the DG scheme and present recent ideas to restore its robustness, its connections to and influence by other sectors of the numerical community, such as the finite difference community, and further discuss this young, but rapidly developing research topic by highlighting the main contributions and a closing discussion about possible next lines of research.
... These properties must be taken into account in order to obtain a stable numerical solution. Many numerical methods have been applied to hyperbolic equations [3]- [10]. In [3] [4] upwind difference schemes were used to solve one-dimensional and two-dimensional hyperbolic equations with constant coefficient and stability analysis was conducted by Fourier method. ...
... Kurganov et al. [8] [9] presented semi-discrete central upwind schemes for nonconvex hyperbolic conservation laws and shallow water equations, respectively. Discontinuous Galerkin schemes for hyperbolic laws were considered in [10]. Some high-order difference schemes were considered in [11] for first-order hyperbolic equations with variable coefficients, but the direction of flow rate was not taken into account. ...
... [36,42,46,50,55,68]. Furthermore, in [60] a provably ES moving mesh ALE DGSEM for the three dimensional Euler equations on curved hexahedral elements and in [70] a moving mesh ES spectral collocation scheme for the three dimensional NSE were constructed. The current work is a natural extension of the authors work presented in [60]. ...
... Furthermore, in [60] a provably ES moving mesh ALE DGSEM for the three dimensional Euler equations on curved hexahedral elements and in [70] a moving mesh ES spectral collocation scheme for the three dimensional NSE were constructed. The current work is a natural extension of the authors work presented in [60]. The focus herein is on (i) extending the approach to KEP/KED ALE DG methods based on general split forms of the compressible Euler equations, (ii) extending the approach to the viscous terms of the Navier-Stokes equations based on the Bassi and Rebay ansatz [2], and (iii) demonstrating the efficiency of the resulting novel KED ALE DG approach for a complex large scale application. ...
... The structure of the present work is as follows: First, the approaches [22,60] are used to construct a provably ES ALE DGSEM for the three dimensional NSE. Then a provable KEP ALE DGSEM for the Euler equations on curved moving elements is constructed. ...
The construction of discontinuous Galerkin (DG) methods for the compressible Euler or Navier-Stokes equations (NSE) includes the approximation of non-linear flux terms in the volume integrals. The terms can lead to aliasing and stability issues in turbulence simulations with moderate Mach numbers (Ma≲0.3), e.g. due to under-resolution of vortical dominated structures typical in large eddy simulations (LES). The kinetic energy or entropy are elevated in smooth, but under-resolved parts of the solution which are affected by aliasing. It is known that the kinetic energy is not a conserved quantity for compressible flows, but for small Mach numbers minor deviations from a conserved evolution can be expected. While it is formally possible to construct kinetic energy preserving (KEP) and entropy conserving (EC) DG methods for the Euler equations, due to the viscous terms in case of the NSE, we aim to construct kinetic energy dissipative (KED) or entropy stable (ES) DG methods on moving curved hexahedral meshes. The Arbitrary Lagrangian-Eulerian (ALE) approach is used to include the effect of mesh motion in the split form DG methods. First, we use the three dimensional Taylor-Green vortex to investigate and analyze our theoretical findings and the behavior of the novel split form ALE DG schemes for a turbulent vortical dominated flow. Second, we apply the framework to a complex aerodynamics application. An implicit LES split form ALE DG approach is used to simulate the transitional flow around a plunging SD7003 airfoil at Reynolds number Re=40,000 and Mach number Ma=0.1. We compare the standard nodal ALE DG scheme, the ALE DG variant with consistent overintegration of the non-linear terms and the novel KED and ES split form ALE DG methods in terms of robustness, accuracy and computational efficiency.
... In order to keep the discussion concise and to the point, we have focused on aspects that are (a) already available as open-source and (b) that are directly relevant to single-phase, compressible turbulent flows. There are however a number of ongoing extension to FLEXI towards a full multi-physics framework with a strong focus on applicability to complex engineering problems which have not been • Multiphase and multicomponent capabilities, in which phase boundaries are tracked with a sharp interface approach [99] • Complex equations of state based on realistic models or tabulated data [100] • A Lagrangian particle tracking method for high order geometry, used in LES and DNS of particle laden flows [101] • Asymptotic consistent low Mach number schemes based on IMEX splittings [34] • Semi-implicit and fully implicit time integration schemes [35] • Mesh deformation and mesh moving based on ALE formulations [102] • An overset/Chimera mesh module • Sliding mesh interface for stator/rotor flows • A coupled particle-in-cell and direct simulation Monte Carlo solver for reactive plasma flows [103,104] • Intrusive and non-intrusive methods for uncertainty quantification of the Navier-Stokes equation [105] • A flexible and modular framework for the creation and management of simulation stacks and automatic, optimal scheduling on HPC systems • A data-exchange interface to OpenFoam 4 and a coupling to the preCICE library [106] for coupled multiphysics simulations We plan on incorporate most of these features into the open-source version of FLEXI in order to provide a mature and feature-rich HO framework to the community. ...
High order (HO) schemes are attractive candidates for the numerical solution of multiscale problems occurring in fluid dynamics and related disciplines. Among the HO discretization variants, discontinuous Galerkin schemes offer a collection of advantageous features which have lead to a strong increase in interest in them and related formulations in the last decade. The methods have matured sufficiently to be of practical use for a range of problems, for example in direct numerical and large eddy simulation of turbulence. However, in order to take full advantage of the potential benefits of these methods, all steps in the simulation chain must be designed and executed with HO in mind. Especially in this area, many commercially available closed-source solutions fall short. In this work, we therefore present the FLEXI framework, a HO consistent, open-source simulation tool chain for solving the compressible Navier–Stokes equations on CPU clusters. We describe the numerical algorithms and implementation details and give an overview of the features and capabilities of all parts of the framework. Beyond these technical details, we also discuss the important but often overlooked issues of code stability, reproducibility and user-friendliness. The benefits gained by developing an open-source framework are discussed, with a particular focus on usability for the open-source community. We close with sample applications that demonstrate the wide range of use cases and the expandability of FLEXI and an overview of current and future developments.
