International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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MS 26a: High order multi-dimensional structure preserving methods
a rich phenomenology. Numerical methods originally developed in one-dimensional situations
require excessive grid refinement in order to capture multi-dimensional phenomena, such as
vortices, differential constraints, stationary states and other structures. These structures are
themselves mostly given as differential operators.
For being efficient, it is not sufficient to merely use high order methods. They need to be
equipped with structure-preserving properties. There is a qualitative difference in the results
obtained by methods which do not preserve any structures and by those which preserve
structures. This is true even if the structures in question are differential operators, and are
preserved only as a discretization. Such methods achieve unprecedented accuracy on poorly
resolved grids. This minisymposium shall bring together researchers dealing with structure
preservation in different contexts and foster exchange on recent and trend-setting strategies to
achieve this properties.
12:00pm - 12:30pm
Residual distribution schemes: Extensions to structure preserving schemes
1Universität Zürich, Switzerland; 2Politecnico di Milano; 3Johannes Gutenberg University
Hyperbolic conservation laws/balance laws contain several structural properties like entropy, angula momentum, preservation of kinetic energy, and more. Actually, it depends on the considered system. However, the numerical schemes should be constructed in a way to mimic these behaviors discretely and avoid to violate them. Recently, a lot of work has been done in the development and construction of such structure preserving schemes where one has to pay attention to several components.
In this talk, we consider this topic in the context of residual distribution schemes.
We present some extensions using correcton terms (cf. Ranocha's talk "Entropy Corrections and Related Methods" in MS02), focus on boundary operators in terms of entropy/energy stability, and the application of limiter strategies to guarantee the positivity of several physical quantities.
12:30pm - 1:00pm
The Active Flux method for nonlinear conservation/balance laws
Max-Planck-Institute for Plasma Physics, Munich
The talk focuses on the numerical solution of hyperbolic conservations laws (possibly endowed with a source term) using the Active Flux method. This method is an extension of the finite volume method. Instead of solving a Riemann Problem, the Active Flux method uses actively evolved point values along the cell boundary in order to compute the numerical flux. Early applications of the method were linear equations with an available exact solution operator, and Active Flux was shown to be structure preserving in such cases. For nonlinear PDEs or balance laws, exact evolution operators generally are unavailable. I will discuss strategies how sufficiently accurate approximate evolution operators can be designed which allow to make Active Flux structure preserving / well-balanced for nonlinear problems.
1:00pm - 1:30pm
On Kinetic Energy and Entropy Stable Moving Mesh DG Methods for the simulation of low Mach number turbulent flow
1Friedrich-Schiller-Universität Jena; 2Universität Stuttgart; 3Universität zu Köln
The construction of discontinuous Galerkin (DG) methods for the compressible Euler or Navier-Stokes equations (NSE) includes the approximation of non-linear flux terms in the volume integrals. The terms can lead to aliasing and stability issues in turbulence simulations with moderate Mach numbers, e.g. due to under-resolution of vortical dominated structures typical in large eddy simulations (LES). The kinetic energy or entropy are elevated in smooth, but under-resolved parts of the solution which are affected by aliasing. It is known that the kinetic energy is not a conserved quantity for compressible flows, but for small Mach numbers minor deviations from a constant progression should be expected. For this reason, in the simulation of under- resolved turbulent flow it is desirable to use kinetic energy preserving (KEP) or entropy conservative (EC) DG methods for the Euler equations and kinetic energy dissipative (KED) or entropy stable (ES) DG methods for the NSE, since these methods include by construction a de-aliasing mechanism.
In this talk, our recently developed provably high order KEP and EC DG methods [1,2] on moving curved hexahedral meshes for the three dimensional Euler equations are presented. Then these methods are used to derive KED and ES moving mesh DG methods for the NSE. Moving mesh methods are of interest to represent specific physical features of the flow or in the context of r-adaptation which involves the re-distribution of the mesh nodes in regions of rapid variation of the solution. The three dimensional Taylor-Green vortex is investigated to show and analyze numerically the theoretical findings and robustness of the developed moving mesh methods for turbulent vortex dominated flows.
1:30pm - 2:00pm
TREFFTZ METHODS FOR TRANSPORT EQUATIONS
Sorbonne University, France
Trefftz methods can be used for high order numerical discretization of linear problems with increasing complexity. We address PN angular approximation of transport equations (linear Boltzman equations with stiff coefficients), often encountered for in neutron propagation and transfer equations. Indeed Trefftz methods provide a natural way to define numerical methods with genuine balance between the differential part and the relaxation part. These methods are naturally written in the context of Discontinuous Galerkin (DG) methods, and so are called Trefftz Discontinuous Galerkin (TDG) methods.
We will report on the construction of the method and excellent behavior for boundary layers. It certain cases, the numerical accuracy of TDG just outperform more traditional polynomial based DG. New theoretical convergence estimates show the enhanced high order accuracy of TDG with respect to DG.
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