Peter E. Vincent

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.

A Randomized Optical Coherence Tomography Study of Coronary Stent Strut Coverage and Luminal Protrusion With Rapamycin-Eluting Stents

OBJECTIVES We used optical coherence tomography, which has a resolution of <20 microm, to analyze thin layers of neointima in rapamycin-eluting coronary stents. BACKGROUND Lack of neointimal coverage has been implicated in the pathogenesis of drug-eluting coronary stent thrombosis. Angiography and intracoronary ultrasound lack the resolution to examine this. METHODS We conducted a randomized trial in patients receiving polymer-coated rapamycin-eluting stents (Cypher, Cordis, Johnson & Johnson, Miami, Florida) and nonpolymer rapamycin-eluting stents (Yukon, Translumina, Hechingen, Germany) to examine neointimal thickness, stent strut coverage, and protrusion at 90 days. Twenty-four patients (n = 12 for each group) underwent stent deployment and invasive follow-up at 90 days with optical coherence tomography. The primary end point was binary stent strut coverage. Coprimary end points were neointimal thickness and stent strut luminal protrusion. RESULTS No patient had angiographic restenosis. For polymer-coated and nonpolymer rapamycin-eluting stents, respectively, mean (SD), neointimal thickness was 77.2 (25.6) microm versus 191.2 (86.7) mum (p < 0.001). Binary stent strut coverage was 88.3% (11.8) versus 97.2% (6.1) (p = 0.030). Binary stent strut protrusion was 26.5% (17.5) versus 4.8% (8.6) (p = 0.001). CONCLUSIONS Mean neointimal thickness for the polymer-coated rapamycin-eluting stent was significantly less than the nonpolymer rapamycin-eluting stent but as a result coverage was not homogenous, with >10% of struts being uncovered. High-resolution imaging allowed development of the concept of the protrusion index, and >25% of struts protruded into the vessel lumen with the polymer-coated rapamycin-eluting stent compared with <5% with the nonpolymer rapamycin-eluting stent. These findings may have important implications for the risk of stent thrombosis and, therefore, future stent design. (An optical coherence tomography study to determine stent coverage in polymer coated versus bare metal rapamycin eluting stents. ORCA 1, from the Optimal Revascularization of the Coronary Arteries group; ISRCTN42475919).

Blood flow in the rabbit aortic arch and descending thoracic aorta

The distribution of atherosclerotic lesions within the rabbit vasculature, particularly within the descending thoracic aorta, has been mapped in numerous studies. The patchy nature of such lesions has been attributed to local variation in the pattern of blood flow. However, there have been few attempts to model and characterize the flow. In this study, a high-order continuous Galerkin finite-element method was used to simulate blood flow within a realistic representation of the rabbit aortic arch and descending thoracic aorta. The geometry, which was obtained from computed tomography of a resin corrosion cast, included all vessels originating from the aortic arch (followed to at least their second generation) and five pairs of intercostal arteries originating from the proximal descending thoracic aorta. The simulations showed that small geometrical undulations associated with the ductus arteriosus scar cause significant deviations in wall shear stress (WSS). This finding highlights the importance of geometrical accuracy when analysing WSS or related metrics. It was also observed that two Dean-type vortices form in the aortic arch and propagate down the descending thoracic aorta (along with an associated skewed axial velocity profile). This leads to the occurrence of axial streaks in WSS, similar in nature to the axial streaks of lipid deposition found in the descending aorta of cholesterol-fed rabbits. Finally, it was observed that WSS patterns within the vicinity of intercostal branch ostia depend not only on local flow features caused by the branches themselves, but also on larger-scale flow features within the descending aorta, which vary between branches at different locations. This result implies that disease and WSS patterns in the vicinity of intercostal ostia are best compared on a branch-by-branch basis.

Dealiasing techniques for high-order spectral element methods on regular and irregular grids

High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.

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.

On the identification of symmetric quadrature rules for finite element methods

In this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad v1.0 we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.