Patrice Castonguay

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.

On the Non-linear Stability of Flux Reconstruction Schemes

The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG) methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently a new range of linearly stable FR schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. In this short note non-linear stability properties of FR schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability). It is shown that linearly stable VCJH schemes (at least in their standard form) may be unstable if the flux function is non-linear. This instability is due to aliasing errors, which manifest since FR schemes (in their standard form) utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. Strategies for minimizing such aliasing driven instabilities are discussed within the context of the FR approach. In particular, it is shown that the location of the solution points will have a significant effect on non-linear stability. This result is important, since linear analysis of FR schemes implies stability is independent of solution point location. Finally, it is shown that if an exact L2 projection is employed to construct an approximation of the flux (as opposed to a collocation projection), then aliasing errors and hence aliasing driven instabilities will be eliminated. However, performing such a projection exactly, or at least very accurately, would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the FR approach. It can be noted that in all above regards, non-linear stability properties of FR schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of FR schemes, which have hitherto been developed and analyzed solely in the context of a linear flux function.

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.

Insights from von Neumann analysis of high-order flux reconstruction schemes

The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently, an infinite number of linearly stable FR schemes were identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. Identification of VCJH schemes offers significant insight into why certain FR schemes are stable (whereas others are not), and provides a simple prescription for implementing an infinite range of linearly stable high-order methods. However, various properties of VCJH schemes have yet to be analyzed in detail. In the present study one-dimensional (1D) von Neumann analysis is employed to elucidate how various important properties vary across the full range of VCJH schemes. In particular, dispersion and dissipation properties are studied, as are the magnitudes of explicit time-step limits (based on stability considerations). 1D linear numerical experiments are undertaken in order to verify results of the 1D von Neumann analysis. Additionally, two-dimensional non-linear numerical experiments are undertaken in order to assess whether results of the 1D von Neumann analysis (which is inherently linear) extend to real world problems of practical interest.

Simulation of Transitional Flow over Airfoils using the Spectral Difference Method

This work addresses the simulation of transitional flow over airfoils under low Reynolds number conditions (Rec � 60000). The flow solutions are obtained by means of an Implicit Large Eddy Simulation (ILES) using a newly developed unstructured, parallel solver that employs the high-order spectral difference (SD) method for spatial discretization. The calculations are performed on the SD7003 airfoil section at an angle of attack of 4 ◦ at Reynolds number of 10000 and 60000. The SD7003 airfoil was selected due to the availability of experimental and computational results. The use of the SD method without an added subgrid-scale model appears to be capable of accurately predicting the laminar separation, transition and reattachment locations. To the authors’s knowledge, the present study is the first att empt to analyze transitional flow using the SD method.

On the Development of a High-Order, Multi-GPU Enabled, Compressible Viscous Flow Solver for Mixed Unstructured Grids

This work discusses the development of a three-dimensional, high-order, compressible viscous ow solver for mixed unstructured grids that can run on multiple GPUs. The solver utilizes a range of so-called Vincent-Castonguay-Jameson-Huynh (VCJH) ux reconstruction schemes in both tensor-product and simplex elements. Such schemes are linearly stable for all orders of accuracy and encompass several well known high-order methods as special cases. Because of the high arithmetic intensity associated with VCJH schemes and their element-local nature, they are well suited for GPUs. The single-GPU solver developed in this work achieves speed-ups of up to 45x relative to a serial computation on a current generation CPU. Additionally, the multi-GPU solver scales well, and when running on 32 GPUs achieves a sustained performance of 2.8 Tera ops (double precision) for 6th-order accurate simulations with tetrahedral elements. In this paper, the techniques used to achieve this level of performance are discussed and a performance analysis is presented. To the authors’ knowledge, the aforementioned ow solver is the rst high-order, three-dimensional, compressible Navier-Stokes solver for mixed unstructured grids that can run on multiple GPUs.

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.

Application of High-Order Energy Stable Flux Reconstruction Schemes to the Euler Equations

The authors recently identified an infinite range of high-order energy stable flux reconstruction (FR) schemes in 1D and on triangular elements in 2D. The new flux reconstruction schemes are linearly stable for all orders of accuracy in a norm of Sobolev type. They are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (such as a collocation based nodal discontinuous Galerkin method and a spectral difference method). Identification of such schemes represents a significant advance in terms of understanding why certain FR schemes are stable, whereas others are not. However, to date there have been no studies into how these schemes perform when applied to real world non-linear problems. In this paper, stability and accuracy properties of these new schemes are studied for various two-dimensional inviscid flow problems. The results offer significant insight into the performance of energy stable FR schemes for non-linear problems. It is envisaged the results will aid scheme selection for a given problem, based on its stability and accuracy requirements.

3D Flapping Wing Simulation with High Order Spectral Difference Method on Deformable Mesh

In this paper we carry out computational studies of three-dimensional flow over flapping wings. The problems we have investigated include, firstly, three-dimensonal simulation of flow over an extruded SD7003 airfoil in plunging motion at transitional Reynolds number and, secondly, flow over a pair of flapping rectangle wings with constant NACA0012 cross-sectional airfoil profile at low Reynolds number. The three-dimensional flapping wing simulations are performed using high-order spectral difference method at low Mach number. The high-order method allows a very coarse starting mesh to be used. By using high order solution, fine flow features in the vortex-dominated flow field are effectively captured. For the plunging SD7003 airfoil, we examine the laminar to turbulence flow transitional behavior at Re 40,000. For the NACA0012 rectangular wing, we analyze and compare the flow fields and aerodynamic efficiencies of several flapping wing motions at Re 2000. The flapping motions considered include wing plunging, twisting and pitching. Some of the flapping wing motions are accommodated through dynamic mesh deformation.

An Extension of Energy Stable Flux Reconstruction to Unsteady, Non-linear, Viscous Problems on Mixed Grids

This paper extends the high-order Flux Reconstruction (FR) approach to the treatment of non-linear diffusive fluxes on triangles. The FR approach for solving diffusion problems is reviewed on quadrilaterals and extended for triangles, allowing the treatment of mixed grids. In particular, this paper examines a subset of FR schemes, referred to as VincentCastonguay-Jameson-Huynh (VCJH) schemes, which are provably stable across all orders of accuracy for linear fluxes in first order systems. The correction fields of the VCJH schemes are shown to represent a family of lifting operators which are used to enforce inter-element continuity of the solution and the diffusive flux. For diffusion problems, the lifting operators of nodal DG schemes are shown to be a subset of this family. Finally, numerical results are used to show the effectiveness of VCJH schemes for a range of problems, including the model diffusion equation and the compressible Navier-Stokes equations. Optimal orders of accuracy are obtained on unstructured mixed meshes of triangular and quadrilateral elements.