Conference Agenda

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Session Overview
Session
Z11: hyperbolic, kinetic equations, shock capturing
Time:
Wednesday, 14/July/2021:
2:00pm - 4:00pm

Session Chair: Václav Kučera
Virtual location: Zoom 1


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Presentations
2:00pm - 2:20pm

High Order Edge Sensors with l1 Regularization for Enhanced Discontinuous Galerkin Methods

Jan Glaubitz, Anne Gelb

Dartmouth College, United States of America

The use of l1 regularization for solving hyperbolic conservation laws based on high-order discontinuous Galerkin (DG) approximations is investigated. We first use the polynomial annihilation method to construct a high-order edge sensor which enables us to flag “troubled” elements. The DG approximation is enhanced in these troubled regions by activating l1 regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting l1 optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting l1 regularization only in troubled cells, our method remains accurate and efficient, as no additional regularization or expensive iterative procedures are needed in smooth regions. We present results for the inviscid Burgers’ equation as well as a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.



2:20pm - 2:40pm

Asymptotic preserving schemes for kinetic equations that are also stationary preserving

Christian Klingenberg

Wuerzburg University, Germany, Germany

In this work we are interested in the stationary preserving property of asymptotic preserving (AP) schemes for kinetic models. We introduce a criterion for AP schemes for kinetic equations to be uniformly stationary preserving (SP). Our key observation is that as long as the Maxwellian of the distribution function can be updated explicitly, such AP schemes are also SP. To illustrate our observation, three different AP schemes for three different kinetic models are considered. Their SP property is proved analytically and tested numerically, which confirms our observations.

This is joint work with Farah Kanbar and Min Tang.



2:40pm - 3:00pm

A low Mach well-balanced relaxation scheme for the Euler equations with gravity

Claudius Birke

Julius-Maximilians-Universität Würzburg, Germany

We present a numerical approximation of the solutions of the Euler equations with a gravitational source term. This scheme can both preserve stationary solutions and also preserves the low Mach limit to the corresponding incompressible equations. In addition our scheme satisfies a discrete entropy inequality and guarantees not to give rise to numerical checkerboard modes in the incompressible limit.

This is joint work with Christophe Chalons (Versailles, France) & Christian Klingenberg (Wuerzburg, Germany).



3:00pm - 3:20pm

A high order monotonicity preserving scheme based on Bernstein continuous finite elements for nonlinear hyperbolic systems

Sibusiso Mabuza1, Sidafa Conde2, John Shadid2,3

1Clemson University, United States of America; 2Sandia National Laboratories, United States of America; 3University of New Mexico, United States of America

An arbitrarily high order stabilized continuous finite element scheme is proposed for nonlinear hyperbolic systems. We make use of the Bernstein-Bezier finite element space which is $H^1$ conforming. The positivity and partition of unity properties of the Bernstein finite element bases makes them appealing for problems that require positivity preservation. We consider a stabilization approach in which low order minimal dissipation is added to the semi-discrete for of the problem. The result is a semi-discrete form which satisfies the system LED condition. Then an element based limiting strategy is used to make the scheme high resolution, by limiting the antidiffusive part of the scheme through nodal variation limiters. The stabilized semi-discrete scheme can then be discretized in time using various time integrators such as high order SSP Runge Kutta schemes or IMEX-RK schemes. Numerical studies are presented to illustrate the performance of the scheme on various nonlinear scalar problems and shock hydrodynamics problems.



3:20pm - 3:40pm

Fourier analysis of continuous FEM for hyperbolic PDEs: influence of approximation and stabilization terms

Sixtine Michel1, Davide Torlo1, Mario Ricchiuto1, Rémi Abgrall2

1INRIA, France; 2Institut für Mathematik & Computational Science, Universität Zürich

High order Continous Galerkin (CG) methods can be used to solve hyperbolic problems. However, it is

well known that hyperbolic problems with a standard Finite Element Method (FEM) discretization show

instabilities and there is the need of stabilization. The CG discretizations with stabilization techniques

can have dissipation levels that are comparable to the ones brought by Discontinuous Galerkin (DG)

with upwind numerical flux of the same order of accuracy. The stabilization terms play an important

role and we will compare two of them.

This work will compare different high order in time and space methods and their stability and dis-

persive analysis for one dimensional problems. For each considered method, we find a von Neumann

stability region, which depends on the CFL number and on the stabilization coefficient, and we test such methods on some 1D and 2D numerical examples.

The high order methods that we consider are based on two continuous finite element discretizations

and two different time methods. The finite element spaces are generated by different polynomial spaces.

We study Bernstein polynomials, Lagrangian polynomials in equispaced nodes and Lagrangian polyno-

mials in cubature nodes. These latter are introduced by G. Cohen et al. in 2001 [7] to optimize the

accuracy of quadrature rule. The choice of the quadrature rule is of crucial importance as it allows to

obtain a diagonal mass matrix and so decrease considerably the time of computation. In this work, we

analyze the behaviour of the streamline-upwind Petrov-Galerkin (SUPG) stabilization technique [5, 2]

and the continuous interior penalty (CIP) stabilization method [4, 6, 3], combined with high order time

schemes: strong stability preserving RK (SSPRK) and the deferred correction (DeC) time integration

methods [1].

The aim of the fourier analysis is to obtain the best values for the CFL number and the stabilization

coefficient to use also for other equations. Then, we conclude about the best combination of methods

with numerical examples.

[1] R. Abgrall, High order schemes for hyperbolic problems using globally continos approximation and

avoiding mass matrices, Journal of Scientific Computing, 73 (2017).

[2] E. Burman, Consistent supg-method for transient transport problems: Stability and convergence,

Computer Methods in Applied Mechanics and Engineering - COMPUT METHOD APPL MECH

ENG, 199 (2010), pp. 1114–1123.

[3] E. Burman, A. Ern, and M. Fernández, Explicit Runge–Kutta Schemes and Finite Elements with

Symmetric Stabilization for First-Order Linear PDE Systems, SIAM Journal on Numerical Analysis,

48 (2010).

[4] E. Burman and P. Hansbo, The edge stabilization method for finite elements in cfd, (2004).

[5] E. Burman, A. Quarteroni, and B. Stamm, Stabilization strategies for high order methods for

transport dominated problems, Bolletino dell’Unione Matematica Italiana, 1 (2008).

[6] E. Burman, A. Quarteroni and B. Stamm, Interior penalty continuous and discontinuous finite element approximations of hyperbolic

equations, Journal of Scientific Computing, 43 (2010), pp. 293–312.

[7] G. Cohen, P. Joly, J. Roberts, and N. Tordjman, Higher order triangular finite elements with

mass lumping for the wave equation, Siam Journal on Numerical Analysis - SIAM J NUMER ANAL,

38 (2001).



3:40pm - 4:00pm

Discontinuous Galerkin method for macroscopic traffic flow models on networks using Godunov-like numerical fluxes

Lukáš Vacek, Václav Kučera

Faculty of Mathematics and Physics, Charles University, Czech Republic

Modelling of traffic flows will have an important role in the future. With a rising number of cars on the roads, we must optimize the traffic situation. That is the reason we started to study traffic flows. It is important to have working models which can help us to improve traffic flow. We can model real traffic situations and optimize e.g. the timing of traffic lights or local changes in the speed limit. The benefits of modelling and optimization of traffic flows are both ecological and economical.

We describe a numerical technique for the solution of macroscopic traffic flow models on networks of roads. On individual roads, we consider the standard Lighthill-Whitham-Richards model which is discretized using the discontinuous Galerkin method along with suitable limiters. In order to solve traffic flows on networks, we construct suitable numerical fluxes at junctions based on preferences of the drivers. We prove that our semi-discrete DG solution is L2 stable on several types of networks. We present numerical experiments, including a junction with complicated traffic light patterns with multiple phases.

The work of L. Vacek is supported by the Charles University, project GA UK No. 1114119. The work of V. Kučera is supported by the Czech Science Foundation, project No. 20-01074S.



 
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