Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

Please note that all times are shown in the time zone of the conference. The current conference time is: 10th Dec 2022, 09:44:56am CET

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Session Overview
MS 11c: recent developments in high-order methods for time-dependent problems
Tuesday, 13/July/2021:
4:00pm - 6:00pm

Session Chair: Reza Abedi
Session Chair: Tamas Horvath
Virtual location: Zoom 1

Session Abstract

Both hyperbolic and parabolic time-dependent problems have been of great interest in the

applied mathematics and engineering communities as they cover a wide range of applications.

To improve the accuracy both in space and time, several high order methods have been

developed in the recent years.

However, with today's exascale computing architectures we also aim to solve these prob-

lems eectively, not just accurately. Many methods have achieved great success in paral-

lelization, such as parallel-in-time and space-time methods, among many others.

In this minisymposium we aim to present the state-of-the-art theoretical and application

based results, and bring the members of this community closer to each other. Methods

of interest include, but are not limited to, space-time discontinuous Galerkin, space-time

nite element, implicit-explicit methods, asynchronous, parallel-in-time and adaptive mesh


Presentations regarding exascale or highly parallel implementations (such as application

of GPU platforms) are also welcomed.

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4:00pm - 4:30pm

A Local Approximate Mass Matrix Inversion for Continuous hp Triangular Finite Element Methods

Jay Appleton, Brian Helenbrook

Clarkson University, United States of America

Explicit time advancement for continuous finite elements requires the inversion of a global mass matrix at each time step. For spectral element simulations on quadrilaterals and hexahedra, there is an accurate approximate mass matrix which is diagonal, making it computationally efficient for explicit simulations. Such a diagonal approximate mass matrix does not exist for triangles. Instead, we develop and demonstrate an accurate and easily invertible approximation to the mass matrix, in which accuracy is defined as an exact projection of polynomial functions up to one degree less than the approximation space. The approximate mass matrix and associated local inversion process is demonstrated for p degree polynomial basis functions, 3<p<9. The resulting scheme is easily parallelized which creates the possibility of an efficient and accurate application to explicit time stepping methods in which the computational work required scales linearly with the number of elements in the mesh, the best scaling that is possible.

4:30pm - 5:00pm

A Scalable Exponential-DG approach for Nonlinear conservation laws: with application to Burger and Euler equations

Shinhoo Kang1, Tan Bui-Thanh2,3

1Argonne National Laboratory, USA; 2Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, TX 78712, USA; 3The Oden Institute for Computational Engineering And Sciences, The University of Texas at Austin, Austin, TX 78712, USA

We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number (Cr>1); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present detailed stability and convergence analyses for the exponential Euler DG scheme.

5:00pm - 5:30pm

Error analysis of a space-time hybridized discontinuous Galerkin method for the incompressible Navier–Stokes equations

Keegan Kirk1, Tamas Horvath2, Sander Rhebergen1

1University of Waterloo, Canada; 2Oakland University, USA

Much of the recent progress in the numerical solution of incompressible flow problems has concentrated on pressure-robust finite element methods, a class of mimetic methods that preserve a fundamental invariance property of the incompressible Navier–Stokes equations. In contrast to many classical finite element methods, the approximation error in the velocity for pressure-robust methods is independent of the pressure. Two essential ingredients are required for pressure-robustness: exact enforcement of the incompressibility constraint, and H(div)-conformity of the finite element solution.

In this talk, I will introduce a space-time hybridized discontinuous Galerkin finite element method for the evolutionary incompressible Navier–Stokes equations. The numerical scheme has a number of desirable properties, including point-wise mass conservation, energy stability, and higher-order accuracy in both space and time. Through the introduction of a pressure facet variable, H(div)-conformity of the discrete velocity solution is enforced, ensuring the numerical scheme is pressure-robust. The well-posedness of the resulting nonlinear algebraic system will be considered, and the uniqueness of the discrete solution is shown in two spatial dimensions under a small data assumption. A priori error estimates for smooth solutions will be presented.

5:30pm - 6:00pm

Sliding Grid Techniques for Space-time Embedded-Hybridized Discontinuous Galerkin Method

Tamas Horvath1, Sander Rhebergen2

1Oakland University, United States of America; 2University of Waterloo, Canada

The Space-time Discontinuous Galerkin (ST-DG) method is an excellent method to discretize problems on deforming domains. This method uses DG to discretize both in the spatial and temporal directions, allowing for an arbitrarily high order approximation in space and time.

We present a higher-order accurate Embedded-Hybridized Discontinuous Galerkin method for fluid-rigid body interactions. We apply a sliding grid technique or rotational movement that can handle arbitrary rotation. The numerical examples will include galloping and fluttering motion.

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