Energy Stable Flux Reconstruction Schemes Research Articles

An Extended Range of Stable Flux Reconstruction Schemes on Quadrilaterals for Various Polynomial Bases

An extended range of energy stable flux reconstruction schemes, developed using a summation-by-parts approach, is presented on quadrilateral elements for various sets of polynomial bases. For the maximal order bases, a new set of correction functions which result in stable schemes is found. However, for a range of orders it is shown that only a single correction function can be cast as a tensor-product. Subsequently, correction functions are identified using a generalised analytic framework that results in stable schemes for total order and approximate Euclidean order polynomial bases on quadrilaterals—which have not previously been explored in the context of flux reconstruction. It is shown that the approximate Euclidean order basis can provide similar numerical accuracy as the maximal order basis but with fewer points per element, and thus lower cost.

Provably stable flux reconstruction high-order methods on curvilinear elements

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, the curvilinear metric split-form for a linear physical flux is examined as it enables the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and non-conservative forms are inherently different–even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders of convergence. The scheme loses the orders of accuracy at the equivalent correction parameter value c as that of the one-dimensional ESFR scheme.

Nonlinearly stable flux reconstruction high-order methods in split form

The flux reconstruction (FR) method has gained popularity in the research community as it recovers promising high-order methods through modally filtered correction fields, such as the discontinuous Galerkin method, amongst others, on unstructured grids over complex geometries. Moreover, FR schemes, specifically energy stable FR (ESFR) schemes also known as Vincent-Castonguay-Jameson-Huynh schemes, have proven attractive as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms have recently seen a resurgence in research activity due to their resultant nonlinear (entropy) stability proofs. This paper derives for the first time nonlinearly stable ESFR schemes in split form that enable nonlinear stability proofs for, uncollocated, modal, ESFR split forms with different volume and surface cubature nodes. The critical enabling technology is applying the splitting to the discrete stiffness operator. This naturally leads to appropriate surface and numerical fluxes, enabling both entropy stability and conservation proofs. When these schemes are recast in strong form, they differ from schemes found in the ESFR literature as the ESFR correction functions are incorporated on the volume integral. Furthermore, numerical experiments are conducted for Burgers' equation verifying that the new class of proposed ESFR split forms is nonlinearly stable in contrast to the standard split form ESFR approach. Lastly, the new ESFR split form is shown to obtain the correct orders of accuracy.

Insights on the coercivity of the ESFR methods for elliptic problems

The Flux Reconstruction approach is a recent high-order method which has been introduced for unsteady problems. Initial energy stability has been conducted for the advection problem, leading to the well know Energy Stable Flux Reconstruction (ESFR) scheme. Using the ESFR scheme, the energy stability proof has been extended for the advection–diffusion using the Local Discontinuous Galerkin (LDG) numerical flux. Recently, stability conditions were derived for the compact Interior Penalty (IP) and Bassi–Rebay II (BR2) numerical fluxes. Here we apply ESFR schemes to elliptic problems and derive the associated bilinear form for the Poisson equation. We show that for the compact IP and BR2 numerical fluxes, the bilinear form is independent of the auxiliary correction function. Finally, we provide some insights on the coercivity of the ESFR scheme.

Stability of energy stable flux reconstruction for the diffusion problem using the interior penalty and Bassi and Rebay II numerical fluxes for linear triangular elements

The flux reconstruction (FR) method has gained popularity within the research community. The approach has been demonstrated to recover high-order methods such as the discontinuous Galerkin (DG) method. Stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction (ESFR) methods also named Vincent–Castonguay–Jameson–Huynh (VCJH) methods. ESFR schemes can be viewed as DG schemes with modally filtered correction fields. Using this class of methods, the linear advection diffusion problem has been shown to be stable using the local discontinuous Galerkin scheme (LDG) to compute the viscous numerical flux. This stability proof has been extended for linear triangular and tetrahedra elements. Although the LDG scheme is commonly used, it requires, on particular meshes, a wide stencil, which raises the computational cost.As a consequence, many prefer the compact interior penalty (IP) or the Bassi and Rebay II (BR2) numerical fluxes. This article, for the first time, derives, for both schemes, a condition on the penalty term to ensure the stability for the linear diffusion equation. The derivations are based on Gauss Legendre flux points and on periodic boundary conditions. Moreover the article proposes that for both the IP and BR2 numerical fluxes, the stability of the ESFR scheme is independent of the auxiliary correction field. A von Neumann analysis is conducted to study the maximal time step of various ESFR methods.

Spatial eigensolution analysis of energy-stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness

This study focuses on the dispersion and diffusion characteristics of high-order energy-stable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter ‘c’ dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of ‘c’, two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from “under-” to “over-upwinding”. In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving non-periodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.

An Analysis of Stability of the Flux Reconstruction Formulation on Quadrilateral Elements for the Linear Advection–Diffusion Equation

The Flux Reconstruction (FR) approach to high-order methods is a flexible and robust framework that has proven to be a promising alternative to the traditional Discontinuous Galerkin (DG) schemes on parallel architectures like Graphical Processing Units (GPUs) since it pairs exceptionally well with explicit time-stepping methods. The FR formulation was originally proposed by Huynh (AIAA Pap 2007-4079:1---42, 2007). Vincent et al. (J Sci Comput 47(1):50---72, 2011) later developed a single parameter family of correction functions which provide energy stable schemes under this formulation in 1D. These schemes, known as Vincent---Castonguay---Jameson---Huynh (VCJH) schemes, offer control over properties like stability, dispersion and dissipation through the variation of the VCJH parameter. Classical schemes like nodal-DG and Spectral Difference (SD) can also be recovered under this formulation. Following the development of the FR approach in 1D, Castonguay et al. (J Sci Comput 51(1):224---256, 2012) and Williams et al. (J Comput Phys 250:53---76, 2013) and Williams and Jameson (J Sci Comput 59(3):721---759, 2014) developed correction functions that give rise to energy stable FR formulations for triangles and tetrahedra. For the case of tensor product elements like quadrilaterals and hexahedra however, a simple extension of the 1D FR approach utilizing the 1D VCJH correction functions was possible and has been adopted by several authors (Castonguay in High-order energy stable flux reconstruction schemes for fluid flow simulations on unstructured grids, 2012; Witherden et al. in Comput Fluids 120:173---186, 2015; Comput Phys Commun 185(11):3028---3040, 2014). But whether such an extension of the 1D approach to tensor product elements is stable remained an open question. A direct extension of the 1D stability analysis fails due to certain key difficulties which necessitate the formulation of a norm different from the one utilized for stability analysis in 1D and on simplex elements. We have recently overcome these issues and shown that the VCJH schemes are stable for the linear advection equation on Cartesian meshes for any non-negative value of the VCJH parameter (Sheshadri and Jameson in J Sci Comput 67(2):769---790, 2016; J Sci Comput 67(2):791---794, 2016). In this paper, we have extended the stability analysis to the advection---diffusion equation, demonstrating that the tensor product FR formulation is stable on Cartesian meshes for the advection---diffusion case as well. The analysis in this paper also provides additional insights into the dependence on the VCJH parameter of the diffusion and stability characteristics of these schemes. Several numerical experiments that support the theoretical results are included.

On the properties of energy stable flux reconstruction schemes for implicit large eddy simulation

We begin by investigating the stability, order of accuracy, and dispersion and dissipation characteristics of the extended range of energy stable flux reconstruction (E-ESFR) schemes in the context of implicit large eddy simulation (ILES). We proceed to demonstrate that subsets of the E-ESFR schemes are more stable than collocation nodal discontinuous Galerkin methods recovered with the flux reconstruction approach (FRDG) for marginally-resolved ILES simulations of the Taylor–Green vortex. These schemes are shown to have reduced dissipation and dispersion errors relative to FRDG schemes of the same polynomial degree and, simultaneously, have increased Courant–Friedrichs–Lewy (CFL) limits. Finally, we simulate turbulent flow over an SD7003 aerofoil using two of the most stable E-ESFR schemes identified by the aforementioned Taylor–Green vortex experiments. Results demonstrate that subsets of E-ESFR schemes appear more stable than the commonly used FRDG method, have increased CFL limits, and are suitable for ILES of complex turbulent flows on unstructured grids.

Equivalence between the Energy Stable Flux Reconstruction and Filtered Discontinuous Galerkin Schemes

The aim of this paper is to demonstrate the equivalence between filtered Discontinuous Galerkin (DG) schemes and the Energy Stable Flux Reconstruction (ESFR) schemes, expanding on previous demonstrations in 1D 1 and for straight-sided elements in 3D 2. We first derive the DG and ESFR schemes in strong form and compare the respective flux penalization terms while highlighting the implications of the fundamental assumptions for stability in the ESFR formulations, notably that all ESFR scheme correction fields can be interpreted as modally filtered DG correction fields. We present the result in the general context of all higher dimensional curvilinear element formulations. Through a demonstration that there exists a weak form of the ESFR schemes which is both discretely and analytically equivalent to the strong form, we then extend the results obtained for the strong formulations to demonstrate that ESFR schemes can be interpreted as a DG scheme in weak form where discontinuous edge flux is substituted for numerical edge flux correction. Theoretical derivations are then verified with numerical results obtained from a 2D Euler testcase with curved boundaries. Given the current choice of high-order DG-type schemes and the question as to which might be best to use for a specific application, the main significance of this work is the bridge that it provides between them. Clearly outlining the similarities between the schemes results in the important conclusion that it is always less efficient to use ESFR schemes, as opposed to the weak DG scheme, when solving problems implicitly.

Erratum to: Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

Erratum to: Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra

The flux reconstruction (FR) methodology provides a unifying description of many high-order schemes, including a particular discontinuous Galerkin (DG) scheme and several spectral difference (SD) schemes. In addition, the FR methodology has been used to generate new classes of high-order schemes, including the recently discovered `energy stable' FR schemes. These schemes, which are often referred to as VCJH (Vincent---Castonguay---Jameson---Huynh) schemes, are provably stable for linear advection---diffusion problems in 1D and on triangular elements. The VCJH schemes have been successfully applied to a wide variety of problems in 1D and 2D, ranging from linear advection---diffusion problems, to fluid mechanics problems requiring the solution of the compressible Navier---Stokes equations. Based on the results of these numerical experiments, it has been shown that certain VCJH schemes maintain the expected order of spatial accuracy and possess explicit time-step limits which rival those of the collocation-based nodal DG scheme. However, it remained to be seen whether the VCJH schemes could be extended to 3D on tetrahedral elements, enabling their convenient application to the complex geometries that arise in many real-world problems. For the first time, this article presents an extension of the VCJH schemes to tetrahedral elements. This work provides a formal proof of the stability of the new schemes and assesses their performance via numerical experiments on model problems.

Energy stable flux reconstruction schemes for advection–diffusion problems

High-order methods for unstructured grids provide a promising option for solving challenging problems in computational fluid dynamics. Flux reconstruction (FR) is a framework which unifies a number of these high-order methods, such as the spectral difference (SD) and collocation-based nodal discontinuous Galerkin (DG) methods, allowing for their more concise and flexible implementation. Additionally, the FR approach can be used to facilitate development of new numerical methods that offer arbitrary orders of accuracy on unstructured grids. In previous work, it has been shown that a particular range of FR schemes, referred to as Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, are guaranteed to be stable for linear advection problems for all orders of accuracy. There have remained questions, however, regarding the stability of FR schemes for advection–diffusion problems. In this study a new class of VCJH schemes is developed for solving one-dimensional advection–diffusion problems. For the first time, it is shown that the schemes are linearly stable for linear advection–diffusion problems for all orders of accuracy on nonuniform grids. Linear and nonlinear numerical experiments are performed in 1D and 2D to investigate the accuracy and stability properties of the new schemes. The results indicate that certain VCJH schemes for advection–diffusion problems possess significantly higher explicit time-step limits than discontinuous Galerkin schemes, while still maintaining the expected order of accuracy.

Energy stable flux reconstruction schemes for advection–diffusion problems on triangles

The Flux Reconstruction (FR) approach unifies several well-known high-order schemes for unstructured grids, including a collocation-based nodal discontinuous Galerkin (DG) method and all types of Spectral Difference (SD) methods, at least for linear problems. The FR approach also allows for the formulation of new families of schemes. Of particular interest are the energy stable FR schemes, also referred to as the Vincent–Castonguay–Jameson–Huynh (VCJH) schemes, which are an infinite family of high-order schemes parameterized by a single scalar. VCJH schemes are of practical importance because they provide a stable formulation on triangular elements which are often required for numerical simulations over complex geometries. In particular, VCJH schemes are provably stable for linear advection problems on triangles, and include the collocation-based nodal DG scheme on triangles as a special case. Furthermore, certain VCJH schemes have Courant–Friedrichs–Lewy (CFL) limits which are approximately twice those of the collocation-based nodal DG scheme. Thus far, these schemes have been analyzed primarily in the context of pure advection problems on triangles. For the first time, this paper constructs VCJH schemes for advection–diffusion problems on triangles, and proves the stability of these schemes for linear advection–diffusion problems for all orders of accuracy. In addition, this paper uses numerical experiments on triangular grids to verify the stability and accuracy of VCJH schemes for linear advection–diffusion problems and the nonlinear Navier–Stokes equations.

A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements

The flux reconstruction (FR) approach allows various well-known high-order schemes, such as collocation based nodal discontinuous Galerkin (DG) methods and spectral difference (SD) methods, to be cast within a single unifying framework. Recently, the authors identified a new class of FR schemes for 1D conservation laws, which are simple to implement, efficient and guaranteed to be linearly stable for all orders of accuracy. The new schemes can easily be extended to quadrilateral elements via the construction of tensor product bases. However, for triangular elements, such a construction is not possible. Since numerical simulations over complicated geometries often require the computational domain to be tessellated with simplex elements, the development of stable FR schemes on simplex elements is highly desirable. In this article, a new class of energy stable FR schemes for triangular elements is developed. The schemes are parameterized by a single scalar quantity, which can be adjusted to provide an infinite range of linearly stable high-order methods on triangular elements. Von Neumann stability analysis is conducted on the new class of schemes, which allows identification of schemes with increased explicit time-step limits compared to the collocation based nodal DG method. Numerical experiments are performed to confirm that the new schemes yield the optimal order of accuracy for linear advection on triangular grids.

A New Class of High-Order Energy Stable Flux Reconstruction Schemes

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.