International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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MS 12b: cut-DG methods for hyperbolic problems
The development and analysis of high-order numerical methods, which take into account non-
trivial geometric features of the domain, is an important and challenging current research area.
Especially challenging are problems in the class of nonlinear hyperbolic conservation laws,
which can model many physical problems such as fluid dynamics, aerodynamics, the
dynamics of a single star, or the plasma in a fusion reactor, and multiphase problems.
In many realistic problems complex geometric features appear, due to a complex domain or
material interface. A straightforward way to handle complex geometry within the DG
methodology is to use a computational mesh that is fitted to the boundary or interface.
However, it is very time-consuming and complex to create these body-fitted meshes with a
sufficient quality. In case of a changing geometry computational efficiency is lost due to
Therefore, in recent years methods that can be applied without fitting have gained much
interest. However, only a small portion of this work has focused on solving hyperbolic
problems using DG methods or other kinds of high-order approaches. DG methods have
demonstrated very strong performance for hyperbolic problems, but to utilize these methods
in realistic settings with complex geometric features there is a need for further development in
the direction of un-fitted methods. In this mini-symposium we will discuss ways to handle
problems caused by non-fitted computational meshes. All speakers in this minisymposium
will address high-order methods for solving hyperbolic problems on un-fitted meshes, with
the large majority dealing with DG schemes. We believe that the mini-symposium will also be
interesting for researchers working on other types of high-order cut cell methods for
hyperbolic problems, or cut-DG methods for other problem classes.
12:00pm - 12:30pm
An Active Flux Method for Cut Cell Meshes
Heinrich-Heine-Universität Düsseldorf, Germany
We present a third order accurate Cartesian grid cut cell method for advective transport in complex geometries. The method is based on the Active Flux method of Eymann and Roe, a new finite volume method, which evolves both cell average values as well as certain point values of the conserved quantities.
The evolution used in the Active Flux method leads to an automatic stabilisation of the cut cell update, i.e. the method is stable for time steps that are appropriate for the regular cells without any further stabilisation. While previous cut cell methods have been at most second order accurate, we can maintain the third order accuracy of the original scheme. However, extra care is needed in order to achieve this property.
12:30pm - 1:00pm
Domain of dependence stabilization for systems of conservation laws
1TU Dortmund University, Germany; 2University of Münster, Germany
The usage of cut cell meshes has become increasingly popular in recent years to guarantee an efficient and fast generation of accurate grids for complex geometries. One major drawback when solving hyperbolic conservation laws on cut cell meshes is the small cell problem: small cut cells lead to stability problems, if the time step for explicit time stepping schemes is chosen according to the larger background cells.
In [SIAM J. Sci. Comput., 42:A3677-A3703, 2020], Engwer, May, Nüßing and Streitbürger introduced the domain of dependence (DoD) stabilization to overcome the small cell problem for the linear advection equation in one and two space dimensions for discontinuous Galerkin schemes using piecewise linear polynomials. The DoD stabilization overcomes the small cell problem in an algebraic way by adding penalty terms to the standard discretization. These penalty terms are constructed to restore the correct domain of dependence in the neighborhood of the small cut cells by transporting information from the inflow neighbors to the outflow neighbors.
In this talk, we propose an extension of the DoD stabilization to systems of hyperbolic conservation laws for higher order polynomial degrees in one space dimension. For the scalar case, we show that the resulting stabilized scheme is monotone for piecewise constant polynomials. Moreover, we prove L2 stability for the stabilized scheme for the semi-discrete case for arbitrary polynomial degrees. We conclude the talk with numerical examples for Burgers equation and the Euler equations. These results confirm that the proposed method produces stable solutions while converging with the expected (high) order of convergence.
1:00pm - 1:30pm
High order cut discontinuous Galerkin methods for hyperbolic conservation laws in one space dimension
Uppsala University, Sweden
In this work, we develop a family of high order cut discontinuous Galerkin (DG) methods for hyperbolic conservation laws in one space dimension. The ghost penalty stabilization is used to stabilize the scheme for small cut elements. The analysis shows that our proposed methods have similar stability and accuracy properties as the standard DG methods on a regular mesh. We also prove that the cut DG method with piecewise constants in space is total variation diminishing (TVD). We use the strong stability preserving Runge-Kutta method for time discretization and the time step is independent of the size of the cut element. Numerical examples demonstrate that the cut DG methods are high order accurate for smooth problems and perform well for discontinuous problems.
1:30pm - 2:00pm
DG methods with cut elements for Wave equations with material interfaces.
1Uppsala University, Sweden; 2KTH Stockholm, Sweden
We present a high order cut discontinuous Galerkin methodology for wave-propagation in discontinuous media. Small cut elements are stabilised by ghost penalty stabilisation. The model equation is a linear hyperbolic conservation law in one space dimension with discontinuous coefficients. In focus is conservation, accuracy, and stability. We show that the properties of the method are independent of how the material interface cuts the elements. In particular there is no severe time step restriction due to the "small cell problem", and we show how to choose fluxes to achieve conservation at the interface. The methodology is extended to systems, and to problems with a moving material interface. In the latter case we use a space-time discontinuous Galerkin method.
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