Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 9th Dec 2022, 12:51:50am CET

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Session Overview
Ms 2b: advances in high-order nonlinearly stable methods
Tuesday, 13/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

High-order Positivity-Preserving Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations

Nail Yamaleev, Johnathon Upperman

Old Dominion University, United States of America

In this talk, we present a new class of positivity-preserving, entropy stable spectral collocation

schemes of arbitrary order of accuracy for the compressible Navier-Stokes equations. The key

distinctive feature of the proposed methodology is that it is proven to guarantee the pointwise

positivity of thermodynamic variables for compressible viscous flows. The new schemes are

constructed by combining a positivity-violating entropy stable method of arbitrary order of

accuracy and a novel first-order positivity-preserving entropy stable method discretized on the

same Legendre-Gauss-Lobatto collocation points used for the high-order counterpart. To provide

the positivity preservation and excellent discontinuity-capturing properties, the Navier-Stokes

equations are regularized by adding an artificial dissipation in the form of the Brenner-Navier-

Stokes diffusion operator. The high- and low-order schemes are combined by using a limiting

procedure, so that the resultant scheme provides conservation, guarantees pointwise positivity of

thermodynamic variables, preserves design-order of accuracy for smooth solutions, and satisfies

the discrete entropy inequality, thus facilitating a rigorous L 2 -stability proof for the symmetric

form of the discretized Navier-Stokes equations. Numerical results demonstrating accuracy and

positivity-preserving properties of the new schemes are presented for viscous flows with nearly

vacuum regions and very strong shocks and contact discontinuities.

4:30pm - 5:00pm

Entropy Corrections and Related Methods

Hendrik Ranocha1, Philipp Öffner2, Rémi Abgrall3

1University of Münster, Germany; 2Johannes Gutenberg Universität Mainz, Germany; 3Universität Zürich, Switzerland

For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J. Comp. Phys. 372: pp. 640-666, 2018). We characterize the correction terms as solutions of certain optimization problems and are adapt them to the summation-by-parts framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach can be applied to obtain a fully discrete entropy conservative/dissipative RD scheme using a deferred correction (DeC) time integration method.

5:00pm - 5:30pm

The role of linear stability for high-order entropy stable schemes.

Magnus Svärd

University of Bergen, Norway

For a number of years, there has been intensive research into high-order entropy conservative/diffusive schemes as they have proven to be robust for compressible flow simulations. The robustness of these schemes is a result of the non-linear bounds obtained from the entropy inequality. From linear theory we know that stability implies convergence of numerical scheme, something that at least for smooth solutions carries over to non-linear simulations. This allows us to assume that at least a fairly resolved simulation tracks the exact solution. It is tempting to assume that the robustness of entropy stable schemes implies convergence in the same sense as linear stability. In this talk, I will discuss the relation of entropy stability and linear stability, and convergence properties.

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