ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
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Session Overview 
Session  
Z8: enerygy stability, summationbyparts techniques
 
Presentations  
4:00pm  4:20pm
Global stability of 2D plane Couette flow beyond the energy stability limit ^{1}Universidad Catolica de Chile, Chile; ^{2}Cornell University, United States of America; ^{3}University of Victoria, Canada; ^{4}Imperial College London, United Kingdom A fundamental question in fluid stability is whether a laminar flow is nonlinearly stable to all perturbations. The typical way to verify this type of stability, called the energy method, is to show that the energy of a perturbation must decay monotonically under a certain Reynolds number called the energy stability limit. The energy method is known to be overly conservative in many systems, such as in plane Couette flow. Here, we present a methodology to computationally construct Lyapunov functions more general than the energy, which is a quadratic function of the magnitude of the perturbation velocity. These new Lyapunov functions are not restricted to being quadratic, but are instead highorder functionals that depend explicitly on the spectrum of the velocity field in the eigenbasis of the energy stability operator. The methodology involves computing and utilizing highorder discretizations of the perturbation velocity and then numerically solving a convex optimization problem through semidefinite programming (SDP) constrained by sumsofsquares polynomial ansatzes. We then apply this methodology to 2D plane Couette flow and under certain conditions we find a global stability limit higher than the energy stability limit. For this specific flow, this is the first improvement in over 110 years. 4:20pm  4:40pm
Unified analysis of discontinuous Galerkin and flux reconstruction methods based on the summationbyparts property University of Toronto Institute for Aerospace Studies The discrete preservation of vector calculus identities has emerged as a unifying thread for the construction and analysis of numerical methods for partial differential equations. Particularly in the context of hyperbolic and advectiondominated systems of conservation laws, the use of the summationbyparts (SBP) property to algebraically mimic integration by parts has been instrumental in establishing discrete conservation as well as linear and nonlinear stability properties for highorder finitedifference methods, and there have been significant advances in recent years towards extending the SBP approach to encompass a wider range of highorder and spectral methods outside of the traditional finitedifference setting. In this presentation, we describe a unifying framework for the analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods on general element types based on the algebraic properties of the matrix operators constituting such discretizations. Within this framework, a particular emphasis is placed on the role of the multidimensional SBP property in establishing conditions under which DG and FR schemes are provably conservative and energy stable, as well as those under which such methods possess discretely equivalent strong and weak formulations. Specifically, we extend the analysis of the equivalence between strongform and weakform DG methods, which was previously established for discontinuous Galerkin spectralelement methods employing collocated LegendreGauss and LegendreGaussLobatto quadrature rules on tensorproduct elements, to more general DG formulations as well as to the VincentCastonguayJamesonHuynh (VCJH) family of FR methods. Based on this equivalence, we obtain a weak formulation of the VCJH family of schemes with the potential for improved efficiency relative to the conventional strongform FR approach. We also outline algebraic proofs of conservation and stability for DG and FR methods with respect to suitable quadrature rules and discrete norms, demonstrating that the SBP property provides a unifying mechanism for establishing such results. Numerical experiments are presented for the twodimensional linear advection and compressible Euler equations, corroborating the theoretical analysis and demonstrating the influence of certain design choices for the construction of highorder methods afforded within the proposed framework. The present contributions facilitate the extension of existing SBPbased conservation, stability, and equivalence results to a broader class of methods in common use by practitioners, enable the reinterpretation and generalization of several existing results demonstrated in the literature through other techniques, and offer the potential for further development and analysis of highorder methods based on such unifying principles. 4:40pm  5:00pm
Functional Superconvergence with HighOrder TensorProduct Generalized SummationbyParts Operators University of Toronto Institute for Aerospace Studies, Canada Motivation and Background: Functional superconvergence refers to the phenomenon whereby integral functionals based on a degree p discretization converge at a rate of at least 2p with uniform refinement, despite the numerical solution only converging at a nominal rate of about p+1 (for sufficiently smooth problems). In the context of aerodynamic shape optimization, obtaining superconvergent functionals (or at least maximizing the opportunity for obtaining optimally accurate functionals in practice) is particularly important since the objective functions driving the overall optimization procedure typically depend primarily on pressurebased functionals like lift and drag, and only implicitly depend on the numerical solution through these same functionals. The more accurate the functionals, the less time it takes to reach the specified error tolerances, which translates to faster flow solve times overall. This is significant because the flow solve portion of the optimization procedure is typically the most computationally expensive and timeconsuming part of the overall process. For discretizations constructed using tensorproduct generalized summationbyparts (SBP) operators, we previously showed in Craig Penner and Zingg (2020) that for some linear (and subsequently nonlinear) problems, LegendreGaussLobatto (LGL) operators (having boundary nodes) outperform LegendreGauss (LG) operators (having no boundary nodes) with respect to functional accuracy when the degree of the geometry (i.e., mapping) is greater than the degree of the underlying discretization, the volume metric terms are approximated using the same generalized summationbyparts operator used to discretize the flux terms, and the surface metric terms are constructed by extrapolating the volume metric terms (hereafter referred to as the standard approach for the metrics). Note that the motivation for considering geometries of higher degree relative to the SBP operators used arises from studies by Bassi and Rebay (1997) and Zwanenburg and Nadarajah (2017), for example, that demonstrate that using higher degree mappings can improve the accuracy of numerical solutions of hyperbolic conservation laws in the presence of flow tangency boundary conditions (due to the accuracy with which the boundary normals can be computed). Results and Conclusions: In the present work, we continue our investigation of functional accuracy by leveraging an alternative approach for computing the metrics, initially introduced by Crean et al. (2018), where the surface metric terms are defined using an analytic mapping and the volume metric terms are approximated by solving a local optimization problem on each element such that the metric invariants are satisfied (hereafter referred to as the modified approach for the metrics). Using Crean et al.’s modified approach for the metrics, we show that superconvergent functionals with LG operators can be obtained that are more accurate than those computed with LGL operators, in the presence of higher degree mappings. Indeed, for some hyperbolic cases, optimal 2p+1 superconvergence is obtained with LG operators (matching the superconvergence typically observed for discontinuous Galerkin type methods when solving hyperbolic conservation laws), while only 2p superconvergence is obtained with LGL operators, which is explained by the higher accuracy of the LG quadrature (order 2p+1) relative to the LGL quadrature (order 2p). Furthermore, for LG operators, the modified approach for the metrics gives a clear advantage with respect to both solution and functional accuracy. However, for LGL operators, the modified and standard approaches for the metrics typically give similar results. For constructing highorder grids, we compare two type of mappings based on Bspline and Lagrange polynomials, respectively, and show that they produce similar results. To generate the numerical results, we explore various problems governed by hyperbolic conservation laws having smooth solutions that are based on the twodimensional linear convection equation and nonlinear Euler equations. References: Bassi, F. and Rebay, S., “HighOrder Accurate Discontinuous Finite Element Solution of the 2D Euler Equations,” Journal of Computational Physics, Vol. 138, Dec. 1997, pp. 251285. Craig Penner, D.A., and Zingg, D.W., “Superconvergent Functional Estimates from TensorProduct Generalized SummationbyParts Discretizations in Curvilinear Coordinates,” Journal of Scientific Computing, Vol. 82, 41, Feb. 2020. Crean, J., Hicken, J.E., Del Rey Fernandez, D.C., Zingg, D.W., and Carpenter, M.H., “EntropyStable SummationByParts Discretization of the Euler Equations on General Curved Elements,” Journal of Computational Physics, Vol. 356, Mar. 2018, pp. 410438. Zwanenburg, P. and Nadarajah, S., “On the Necessity of Superparametric Geometry Representation for Discontinuous Galerkin Methods on Domains with Curved Boundaries,” 23rd AIAA Computational Fluid Dynamics Conference, June 2017, AIAA Paper 20173946. 5:00pm  5:20pm
Stability and Functional Superconvergence of NarrowStencil SecondDerivative Generalized SummationByParts Discretizations University of Toronto Institute for Aerospace Studies Narrowstencil secondderivative summationbyparts (SBP) operators provide better numerical properties compared to widestencil secondderivative SBP operators, which are constructed by applying firstderivative operators twice. Some of the advantages of using narrowstencil SBP operators are smaller solution error, superior solution convergence rate, compact stencil width, and better damping of high frequency modes [1]. Moreover, similar to widestencil classical SBP operators, it has been demonstrated that adjoint consistent discretizations with narrowstencil classical SBP operators lead to functional superconvergence [2]. In the first part of this work, we derive energy stable and adjoint consistent SATs for diagonal and blocknorm narrowstencil secondderivative generalized SBP operators, which have one or more of the following characteristics: nonrepeating interior point operators, nonuniform nodal distribution, and exclusion of one or both boundary nodes [3]. For the energy stability analysis, we assume that the matrix approximating the first derivative at the element boundaries is invertible and the secondderivative operator is nullspace consistent. Both assumptions are satisfied by most of the SBP operators in the literature. In the second part of this work, we show that if the primal and adjoint solutions are sufficiently smooth, then an adjoint consistent discretization of the steady diffusion problem with a degree p+1 narrowstencil or a degree p widestencil diagonalnorm secondderivative generalized SBP operator leads to a functional convergence rate of 2p under mesh refinement. For discretizations with degree 2p1 blocknorm wide or narrowstencil secondderivative SBP operators, we show that functionals converge at a rate of 2p irrespective of whether or not the scheme is adjoint consistent. Finally, we specialize four types of SAT presented in [4] for implementations with narrowstencil secondderivative SBP operators. The SATs are modified versions of the second method of Bassi and Rebay, the local discontinuous Galerkin, the BaumannOden, and the CarpenterNordstromGottlieb methods. All of the SATs are analyzed and implemented without writing the second order partial differential equation (PDE) as a system of first order PDEs. The theoretical results are verified numerically using the onedimensional steady diffusion problem. References [1] Mattsson, K., Nordstrom, J.: Summation by parts operators for finite difference approximations of second derivatives. Journal of Computational Physics 199(2), 503540 (2004) [2] Eriksson, S.: A dual consistent finite difference method with narrow stencil second derivative operators. Journal of Scientific Computing 75(2), 906940 (2018) [3] Del Rey Fernandez, D.C., Zingg, D.W.: Generalized summationbyparts operators for the second derivative. SIAM Journal on Scientific Computing 37(6), A2840A2864 (2015) [4] Worku, Z.A., Zingg, D.W.: Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summationbyparts operators. Submitted to Journal of Computational Physics (2020) 5:20pm  5:40pm
On Entropy Conservation and Kinetic Energy Preservation Methods ^{1}Multid Analyses AB, Sweden; ^{2}NASA Ames Research Center, Mountain View CA, USA The Tadmortype entropy conservative method using the mathematical logarithmic entropy function and two forms of the Sjogreen & Yee entropy conservative methods using the Harten entropy function are examined for their nonlinear stability and accuracy in very long time integration of the Euler equations of compressible gas dynamics. Following the same procedure as Ranocha [1] these entropy conservative methods can be made kinetic energy preserving with minimum added computational effort. The focus of this work is to examine the nonlinear stability and accuracy of these newly introduced high order entropy conserving and kinetic energy preserving methods for very long time integration of selected test cases when compared with their original methods. Computed entropy, and kinetic energy errors for these methods are compared with the Ducros et al. and the KennedyGruberPirozzoli skewsymmetric splittings. References   [1] Ranocha. H.: Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using SummationbyParts Operators. Proceedings of the ICOSAHOM2018, Imperial College, London, UK, July 913, 2018. 5:40pm  6:00pm
An entropy stable strong imposition of noslip wall boundary conditions University of Bergen, Norway To obtain accurate approximations of the compressible NavierStokes equations, the numerical scheme must be stable, which in turn requires the problem to be wellposed. For this purpose, the boundary conditions play an important role. After they have been properly determined in the continuous setting, an implementation technique that sustains the overall stability of the numerical scheme must be chosen. In this work, we focus the attention to a technique  the injection method  for strongly imposing the noslip boundary condition for the compressible NavierStokes equations. Together with the Simultaneous Approximation Term to impose the Neumann condition for the temperature, we demonstrate that the proposed numerical semidiscrete scheme, based on finite difference SBP operators, is stable both linearly (energy) and a nonlinearly (entropy). Numerical simulations are performed and are shown to substantiate our findings. 
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