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, 12:51:50am CET
Ms 2b: advances in high-order nonlinearly stable methods
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.
4:00pm - 4:30pm
High-order Positivity-Preserving Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations
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
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.
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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