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Session Overview
MS 2a: advances in high-order nonlinearly stable methods
Monday, 12/July/2021:
4:00pm - 6:00pm

Session Chair: Nail Yamaleev
Virtual location: Zoom 4

Session Abstract

High-order nonlinearly stable methods have recently attracted much attention because of their ability to

provide stronger stability estimates for numerical solutions of nonlinear partial differential equations,

such as the Euler and Navier-Stokes equations, MHD equations, shallow water equations, etc. The main

objective of this minisymposium is to bring together experts in nonlinearly stable methods to discuss

innovative approaches, unsolved problems, and future directions for constructing high-order methods

that mimic key stability properties of the governing nonlinear PDEs including entropy stability,

dissipation of kinetic energy, and others.

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4:00pm - 4:30pm

Implicit temporal methods for Entropy Stable Spectral Collocation

Mark H. Carpenter, David Del Rey Fernandez, Chris Kennedy

NASA Langley Research Center, United States of America

High-order spectral collocation operators are notoriously stiff, and inefficient when using implicit temporal methods. The linear/nonlinear solver mechanics are further complicated by the algorithmic requirements of entropy stability. The lack of efficient and robust implicit formulations, limits the use of high-order, entropy stable spectral collocation (SSSC) by the production aerodynamic community.

Resent progress towards efficient formulations for implicit high-order SSSC operators is presented.

Three complimentary strategies are investigated. First, new (E)SDIRK schemes are developed that reduce the stiffness of the linear/nonlinear algebraic systems. Next, accurate and robust stage-value predictors are designed to minimize the number of search directions required by the solver. Finally, a nonlinear controller dynamically adjusts the timestep to optimize the simulation time. Large-eddy simulations of 3D airfoils are used to assess the efficacy of the proposed algorithms.

4:30pm - 5:00pm

Addressing challenges in high-speed turbulence simulations with entropy-stable methods

Jungyeoul Bradley Maeng, Travis C. Fisher, Jerry Waktins, Michael Hansen, Jared Crean

Sandia National Laboratories, United States of America

Scale-resolved simulations of high-speed turbulent flows require robust numerical methods that are capable of efficiently resolving the spatial and temporal turbulent scales for a given turbulence model. In addition to all the issues of simulating subsonic and transonic flows, the simulation approach must be robust in the presence of shocks and efficiently resolve the effects of isothermal boundary conditions at walls. We have previously shown that entropy-stability can create a minimally-dissipative, stable baseline for numerical methods and guide how dissipation is added due to under-resolved features and shocks to maintain stability. In this work, we highlight additional issues in shock capturing and isothermal boundary layers and some potential solutions for structured and unstructured entropy stable methods. We also offer an initial assessment of the efficiency of these methods compared to standard-practice numerical methods on different computing architectures.

5:00pm - 5:30pm

Constructing a stable Cahn-Hilliard, artificial compressibility DGSEM for three dimensional two-phase incompressible flows

Juan Manzanero1, Gonzalo Rubio1, David A. Kopriva2, Esteban Ferrer1, Eusebio Valero1

1Universidad Politecnica de Madrid, Spain; 2The Florida State University, The San Diego State University

We present a three-dimensional entropy-stable high-order discontinuous Galerkin spectral element method (DGSEM) two-phase solver. The system of equations is comprised of the incompressible Navier-Stokes, the artificial compressibility model, and the Cahn-Hilliard equation. The first drives the fluid flow momentum, the second estimates the pressure, and the third computes the distribution of the phases. These equations are fully coupled since the thermodynamic properties are computed from the phases’ distribution, and the capillary pressure is introduced at the interfaces. The system is entropy-bouded as a mathematical entropy that represents the L2 norm of the solution is proven to be bounded. In particular, the entropy is constant if the physical boundaries represent no— or free—slip walls. To derive this entropy equation, the use of a non-conservative skew-symmetric version of the momentum equation has been found to be fundamental.

The continuous analysis dictates the steps used to derive a stable high-order discontinuous Galerkin approximation of the equations. Stability is guaranteed by satisfying a discrete version of the entropy equation, provided by the use of the Gauss-Lobatto points DG variant, which satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. We show that the approximation of the system with the skew-symmetric momentum equation, the Bassi-Rebay 1 scheme, and a suitable Riemann solver is stable as it follows the entropy equation discretely. These proofs hold for three—dimensional unstructured meshes with curvilinear hexahedral elements. Furthermore, the study is complemented with a stable approximation of wall physical boundary conditions.

We support these findings through two— and three—dimensional numerical experiments that assess the accuracy and robustness of the approximation. After these benchmarks and academic cases are assessed, we concentrate on a particular industrial application: the solution of two—phase flows in pipes. As an example, we present the solution of an annular flow regime.

5:30pm - 6:00pm

Entropy Stable Method for the Euler Equations Revisited: Central Differencing via Entropy Splitting and SBP

Bjorn Sjogreen1, H.C. Yee2

1Multid Analyses AB, Sweden; 2NASA Ames Research Center, Mountain View CA, USA

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure by Olsson & Oliger, Gerritsen & Olsson, and Yee et al. [1,2, 5] is revisited. The objective of this talk is to demonstrate that the entropy split methods based on physical entropies are entropy stable methods for central differencing with SBP operators for both periodic and non-periodic boundary conditions for nonlinear Euler equations. Standard high order spatial central differencing as well as high order central spatial DRP (dispersion relation preserving) spatial differencing is part of the entropy stable methodology framework. The proof is to replace the spatial derivatives by summation-by-parts (SBP) difference operators in the entropy split form of the equations using the physical entropy of the Euler equations. The numerical boundary closure follows directly from the SBP operator. No additional numerical boundary procedure is required. In contrast, Tadmor-type entropy conserving methods [4] using mathematical entropies and more recently the methods in

do not naturally come with a numerical boundary closure and a generalized SBP operator has to be developed [3]. Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also

include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy & Grubber splitting.


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1. Gerritsen, M, Olsson, P.: Designing an efficient solution strategy for fluid flows.I. A stable high order finite difference scheme and sharp shock resolution for theEuler equations. J. Comput. Phys. 129, 245–262 (1996).

2. Olsson, P., Oliger, J.: Energy and maximum norm estimates for nonlinear conservation laws. RIACS Technical Report 94.01, (1994).

3. Ranocha, H.: Generalized summation-by-Parts Operators and Variable Coefficients. Xiv:1705.10541v2 [math.NA], Feb. 2018.

4. Tadmor, E.: Entropy Stability Theory for Difference Approximations of Nonlinear Conservation Laws and Related Time-Dependent Problems. Acta Numerica

12, 451-512 (2003).

5. Yee, H. C., Vinokur, M, Djomehri, M. J.: Entropy Splitting and Numerical Dissipation. J. Comp. Phys. 162, 33–81 (2000).

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