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Session Overview
MS 12a: cut-DG methods for hyperbolic problems
Thursday, 15/July/2021:
4:00pm - 6:00pm

Session Chair: PEI FU
Session Chair: Gunilla Kreiss
Session Chair: Sandra May
Virtual location: Zoom 1

Session Abstract

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

constant re-meshing.

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.

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4:00pm - 4:30pm

The mixed explicit implicit scheme to overcome the small cell problem

Sandra May

TU Dortmund University, Germany

Cut cells methods have been developed in recent years for computing flow around bodies with complicated geometries. Cut cell methods cut the flow body out of a regular Cartesian grid resulting in so called cut cells. Cut cells can have irregular shape and may be very small. For solving time-dependent conservation laws on cut cell meshes, probably the biggest problem one has to face is the small cell problem: standard explicit schemes are not stable if the time step is chosen based on the size of the background cells. Therefore, special schemes must be developed.

In this talk we present a mixed explicit implicit approach for overcoming the small cell problem in the context of finite volume schemes, which we introduced with Berger (J. Sci. Comput. 71, pp. 919-943, 2017): We use implicit time stepping near the embedded boundary for stability and employ an explicit scheme otherwise to keep the cost low. We combine the explicit and implicit scheme by means of `flux bounding' -- this way of coupling preserves mass conservation and ensures stability in form of a TVD result. In this talk we present latest theoretical and numerical results. Special focus will be on the accuracy of the mixed scheme.

4:30pm - 5:00pm

A stabilized cut cell DG method for discretizing the linear transport equation

Christian Engwer1, Florian Streitb├╝rger2, Sandra May2

1WWU M├╝nster, Germany; 2TU Dortmund, Germany

We present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous Galerkin (DG). The stabilization allows for explicit time stepping schemes, despite the presence of cut cells. Using a method of lines approach, we start with a standard upwind DG discretization for the background mesh. We introduce penalty terms that stabilize the explicit time stepping on small cut cells in a conservative way such that the CFL condition is independend from the cut size.

The stabilization follows similar ideas as the ghost-penalty method in the sense that we weakly mimic the effect of cell merging. A major difference originates from the fact that the domain of dependence of the transport problem is taken into account, thus we call this a domain of dependence (DoD) stabilization.

In one dimension an eigenvalue analysis show stability of the resulting fully discrete method. Numerically we test the method in two dimensions with piecewise linear polynomials and observe the expected convergence rates for smooth problems.

5:00pm - 5:30pm

High-order cut discontinuous Galerkin methods with local time stepping for acoustics

Martin Kronbichler

Uppsala University, Sweden

My talk will present a method for the accurate simulation of the acoustic wave equation on an immersed domain. The space discretization is based on high-order discontinuous Galerkin methods, whereas time integration relies on explicit schemes such as Runge-Kutta or arbitrary derivative (ADER) methods. The immersed domain is described by a high-order level-set function. A cut finite element approach with consistent integration allows to retain a high order of accuracy all the way to the boundary, which is verified by some basic examples. As is well-known, a basic realization of the method is plagued by ill-conditioning caused by small cuts. For explicit time stepping, this ill-conditioning translates into exceedingly small time steps. A first question is therefore whether and to which extent local time stepping can balance this excess work. Similarly to preconditioning strategies initially developed in the context of cut methods for elliptic equations, this approach alone is however not satisfactory. Especially for high-order spatial discretizations, the methods quickly enter an intrinsically unstable regime, as I will demonstrate via eigenvalue spectra and critical time steps for representative cuts. For a robust method, the solution on small cut elements needs to be controlled in terms of the solution on the larger neighboring elements, such as the ghost-penalty or the cell-agglomeration methods. I will show the performance of cell agglomeration for the acoustic wave equation discretized with high-order discontinuous Galerkin schemes. While the method is stable overall, the largest possible time step size still considerably decreases near certain cuts. Thus, we propose to use local time stepping also in the stabilized case to make the method more efficient. My talk will close by considerations of software aspects and high-performance implementations of cut discontinuous Galerkin methods, as compared to state-of-the-art implementations on fitted meshes with sum-factorization techniques.

5:30pm - 6:00pm

A two-dimensional stabilized discontinuous Galerkin method on curvilinear embedded boundary grids

Andrew Giuliani1, Marsha Berger1,2

1New York University, United States of America; 2Flatiron Institute, United States of America

In this talk, we present a state redistribution method for high order discontinuous Galerkin methods on curvilinear embedded boundary grids. State redistribution relaxes the overly restrictive CFL condition that results from arbitrarily small cut cells when explicit time steppers are used. Thus, the scheme can take take time steps that are proportional to the size of cells in the background grid. The discontinuous Galerkin scheme is stabilized by postprocessing the numerical solution after each stage or step of an explicit time stepping method. The advantage of this approach is that it uses only basic mesh information that is already available in many cut cell codes and does not require complex geometric manipulations. We prove that state redistribution is conservative and p-exact. Finally, we solve a number of test problems that demonstrate the encouraging potential of this technique for applications on curvilinear embedded geometries.

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