International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 9th Dec 2022, 12:20:06am CET
Z8: enerygy stability, summation-by-parts techniques
4:00pm - 4:20pm
Global stability of 2D plane Couette flow beyond the energy stability limit
1Universidad Catolica de Chile, Chile; 2Cornell University, United States of America; 3University of Victoria, Canada; 4Imperial 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 high-order 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 high-order discretizations of the perturbation velocity and then numerically solving a convex optimization problem through semidefinite programming (SDP) constrained by sums-of-squares 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 summation-by-parts 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 advection-dominated systems of conservation laws, the use of the summation-by-parts (SBP) property to algebraically mimic integration by parts has been instrumental in establishing discrete conservation as well as linear and nonlinear stability properties for high-order finite-difference methods, and there have been significant advances in recent years towards extending the SBP approach to encompass a wider range of high-order and spectral methods outside of the traditional finite-difference 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 strong-form and weak-form DG methods, which was previously established for discontinuous Galerkin spectral-element methods employing collocated Legendre-Gauss and Legendre-Gauss-Lobatto quadrature rules on tensor-product elements, to more general DG formulations as well as to the Vincent-Castonguay-Jameson-Huynh (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 strong-form 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 two-dimensional linear advection and compressible Euler equations, corroborating the theoretical analysis and demonstrating the influence of certain design choices for the construction of high-order methods afforded within the proposed framework. The present contributions facilitate the extension of existing SBP-based 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 high-order methods based on such unifying principles.
4:40pm - 5:00pm
Functional Superconvergence with High-Order Tensor-Product Generalized Summation-by-Parts 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 pressure-based 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 time-consuming part of the overall process.
For discretizations constructed using tensor-product generalized summation-by-parts (SBP) operators, we previously showed in Craig Penner and Zingg (2020) that for some linear (and subsequently nonlinear) problems, Legendre-Gauss-Lobatto (LGL) operators (having boundary nodes) outperform Legendre-Gauss (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 summation-by-parts 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 high-order grids, we compare two type of mappings based on B-spline 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 two-dimensional linear convection equation and nonlinear Euler equations.
Bassi, F. and Rebay, S., “High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations,” Journal of Computational Physics, Vol. 138, Dec. 1997, pp. 251-285.
Craig Penner, D.A., and Zingg, D.W., “Superconvergent Functional Estimates from Tensor-Product Generalized Summation-by-Parts 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., “Entropy-Stable Summation-By-Parts Discretization of the Euler Equations on General Curved Elements,” Journal of Computational Physics, Vol. 356, Mar. 2018, pp. 410-438.
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 2017-3946.
5:00pm - 5:20pm
Stability and Functional Superconvergence of Narrow-Stencil Second-Derivative Generalized Summation-By-Parts Discretizations
University of Toronto Institute for Aerospace Studies
Narrow-stencil second-derivative summation-by-parts (SBP) operators provide better numerical properties compared to wide-stencil second-derivative SBP operators, which are constructed by applying first-derivative operators twice. Some of the advantages of using narrow-stencil SBP operators are smaller solution error, superior solution convergence rate, compact stencil width, and better damping of high frequency modes . Moreover, similar to wide-stencil classical SBP operators, it has been demonstrated that adjoint consistent discretizations with narrow-stencil classical SBP operators lead to functional superconvergence . In the first part of this work, we derive energy stable and adjoint consistent SATs for diagonal- and block-norm narrow-stencil second-derivative generalized SBP operators, which have one or more of the following characteristics: non-repeating interior point operators, non-uniform nodal distribution, and exclusion of one or both boundary nodes . For the energy stability analysis, we assume that the matrix approximating the first derivative at the element boundaries is invertible and the second-derivative 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 narrow-stencil or a degree p wide-stencil diagonal-norm second-derivative generalized SBP operator leads to a functional convergence rate of 2p under mesh refinement. For discretizations with degree 2p-1 block-norm wide- or narrow-stencil second-derivative 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  for implementations with narrow-stencil second-derivative SBP operators. The SATs are modified versions of the second method of Bassi and Rebay, the local discontinuous Galerkin, the Baumann-Oden, and the Carpenter-Nordstrom-Gottlieb 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 one-dimensional steady diffusion problem.
 Mattsson, K., Nordstrom, J.: Summation by parts operators for finite difference approximations of second derivatives. Journal of Computational Physics 199(2), 503-540 (2004)
 Eriksson, S.: A dual consistent finite difference method with narrow stencil second derivative operators. Journal of Scientific Computing 75(2), 906-940 (2018)
 Del Rey Fernandez, D.C., Zingg, D.W.: Generalized summation-by-parts operators for the second derivative. SIAM Journal on Scientific Computing 37(6), A2840-A2864 (2015)
 Worku, Z.A., Zingg, D.W.: Simultaneous approximation terms and functional accuracy for diffusion problems discretized with multidimensional summation-by-parts operators. Submitted to Journal of Computational Physics (2020)
5:20pm - 5:40pm
On Entropy Conservation and Kinetic Energy Preservation Methods
1Multid Analyses AB, Sweden; 2NASA Ames Research Center, Mountain View CA, USA
The Tadmor-type 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  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 Kennedy-Gruber-Pirozzoli skew-symmetric splittings.
 Ranocha. H.: Entropy Conserving and Kinetic Energy Preserving Numerical Methods for the Euler Equations Using Summation-by-Parts Operators. Proceedings of the ICOSAHOM-2018, Imperial College, London, UK, July 9-13, 2018.
5:40pm - 6:00pm
An entropy stable strong imposition of no-slip wall boundary conditions
University of Bergen, Norway
To obtain accurate approximations of the compressible Navier-Stokes equations, the numerical scheme must be stable, which in turn requires the problem to be well-posed. 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 no-slip boundary condition for the compressible Navier-Stokes equations. Together with the Simultaneous Approximation Term to impose the Neumann condition for the temperature, we demonstrate that the proposed numerical semi-discrete scheme, based on finite difference SBP operators, is stable both linearly (energy) and a non-linearly (entropy). Numerical simulations are performed and are shown to substantiate our findings.
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