Conference Agenda

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Session Overview
Session
MS 11a: recent developments in high-order methods for time-dependent problems
Time:
Monday, 12/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

renement.

Presentations regarding exascale or highly parallel implementations (such as application

of GPU platforms) are also welcomed.


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Presentations
4:00pm - 4:30pm

The solution of elliptic PDEs by using a hyperbolic PDE solver

Reza Abedi1, Giang Huynh1, Robert Haber2

1University of Tennessee, United States of America; 2University of Illinois at Urbana-Champaign, United States of America

The causal Spacetime Discontinuous Galerkin (cSDG) method [1] can be used to solve hyperbolic problems. It has proven very effective in this role due to its local conservation properties, linear computational complexity, unconditional stability, powerful adaptive meshing capabilities, and other favorable properties. Here we focus on extensions of cSDG solution technology for solving elliptic PDEs. Two classes of elliptic PDEs are considered. First, we consider the steady-state limits of parabolic or hyperbolic PDEs. Second, we address long-term solutions of hyperbolic PDEs driven by harmonic forces, in particular, the Helmholtz equation.

We discuss two distinct schemes for steady-state limit problems. In the cSDG implementation of the viscous damping method [2], we solve a damped hyperbolic PDE for a sufficiently long time to reach the steady state limit. In the cSDG version of the kinetic damping method [3], we force the outflow velocities of spacetime elements to zero, a process that drains energy from the system. We use the wave-based WaveHoltz method [4] in a cSDG scheme for the Helmholtz equation. We seek initial conditions for which the solution of the wave equation is harmonic and from which real or complex solutions of the Helmholtz equation can be extracted. A novel projection operator proposed in [4] ensures the uniqueness of the solution as well as the method’s stability in problems that lack energy loss mechanisms, such as damping or transmitting boundary conditions. For both the steady-state and Helmholtz problems, the mapping from initial conditions to the solution at any later time can be viewed as an affine operator. Following [4], we use this fact to accelerate the convergence of hyperbolic solutions to their elliptic limits.

References:

[1] R. Abedi, B. Petracovici, and R.B. Haber, “A spacetime discontinuous Galerkin method for linearized elastodynamics with element–wise momentum balance”, Computer Methods in Applied Mechanics and Engineering, 195:3247 – 3273, 2006.

[2] Alireza Mazaheri and Hiroaki Nishikawa. Efficient high-order discontinuous Galerkin schemes with first-order hyperbolic advection-diffusion system approach. Journal of Computational Physics, 321:729-54, 2016.

[3] Jung, Samuel, Tae-Yun Kim, and Wan-Suk Yoo. "Dynamic Relaxation Using Continuous Kinetic Damping—Part I: Basic Algorithm." Journal of Computational and Nonlinear Dynamics 13.8 (2018).

[4] Appelo, Daniel, Fortino Garcia, and Olof Runborg. "WaveHoltz: iterative solution of the Helmholtz equation via the wave equation." arXiv preprint arXiv:1910.10148 (2019).



4:30pm - 5:00pm

A posteriori error estimate and mesh adaptation for the numerical solution of the Richards equation

Vit Dolejsi1, Filip Roskovec1, Miloslav Vlasak2

1Charles University, Prague, Czech Republic; 2Czech Technical University, Prague, Czech Republic

Unsaturated flows in porous media are frequently described the Richards equation which belong among the degenerate nonlinear parabolic problems. In [1], we discretized the Richards equation by the space-time discontinuous Galerkin method (STDGM) which, in combination with the anisotropic hp-mesh adaptation technique [2], exhibits a very efficient and accurate technique for the numerical solution of time dependent problems. However, in [1], we have used rather heuristic residual error estimates and a rigorous numerical analysis was missing.

Motivated by [3], we derive a posteriori error estimates which are based on the equilibrated flux space-time reconstruction techniques. These estimates are locally computable and they guarantee the upper bound of the error directly. Moreover, employing these estimates we propose an adaptive method, which adaptively chooses the size of the time step, the size and shape of mesh elements and the polynomial approximation degrees locally. Several numerical experiments demonstrate the potential of the proposed method.

[1] V. Dolejsí, M. Kuraz, P. Solin, Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows, Applied Mathematical Modelling 72: 276-305, 2019.

[2] V. Dolejsi, G. May, F. Roskovec, P. Solin: Anisotropic hp-mesh optimization technique based on the continuous mesh and error models, Comput. Math. Appl. 74:45-63, 2017.

[3] V. Dolejsi, F. Roskovec, M. Vlasak: A posteriori error estimates for higher order space-time Galerkin discretizations of nonlinear parabolic problems, SIAM J. Numer. Anal, accepted, 2021.



5:00pm - 5:30pm

PML applied to spacetime Trefftz-DG numerical formulation for the acoustic wave equation.

Hélène Barucq1, Henri Calandra2, Julien Diaz1, Vinduja Vasanthan1

1Project-Team Makutu, Inria, E2S UPPA, CNRS, France; 2Total R&D, France

The coupling between Trefftz methods and Discontinuous Galerkin (Trefftz-DG) formulation has been employed for solving harmonic wave equations, but it has only recently been applied to transient wave equations. Trefftz methods [Trefftz, journal, 1926] consist in deriving variational formulation of the considered problem by using exact solutions as trial functions. Consequently, the volumic integrals vanish, and the variational formulation reduces to surface integrals.

A Trefftz-DG variational formulation applied to wave problems is a Trefftz variational formulation applied on each cell of a mesh of the domain of interest with appropriate transmission conditions between the cells. The basis functions are exact solutions (or approximations of the exact solution), which strongly reduces the numerical dispersion. In time–domain, the main drawback of the approach is that, contrary to standard DG methods, which allow naturally for the use of explicit time-scheme, Trefftz-DG methods lead to an implicit scheme that requires the solution to a huge linear system. This prevents the use of the method for solving realistic and industrial problems without any additional treatment.

We have developed a Trefftz-DG framework for solving acoustic equations using Tent Pitcher algorithms. This algorithm consists in building a space-time mesh composed of cells satisfying a causality constraint, which is related to the speed of the waves inside the cells. Under this condition, the value of the solution inside one cell depends only on the solution on the inflow boundaries of this cell and it can be computed independently. Thus, tent-pitcher meshes allow for turning the previously described huge matrix into a collection of easily invertible small matrices. It is worth noting that, since Trefftz-DG formulations involve only surface meshes, we never have to consider 4D cells but only 3D elements. Moreover, this formulation handles naturally local time stepping and allows easily for coupled implicit/explicit time integration.

After having presented the computational framework we have implemented, we will address the natural question of truncating the computational domain. For that purpose, we have constructed Perfectly Matched Layers (PML). These boundary conditions were first introduced by Bérenger in 1994, as an artificial absorbing layer. To our knowledge, such conditions have not been implemented within the scope of space-time polynomial Trefftz spaces yet. The main difficulty of the implementation of PML within the Trefftz framework lies in the derivation of exact polynomial solutions to the PML formulation. However, analytical representation of the Green’s functions of PML formulations are well-known and can be derived using for instance the Cagniard de Hoop method. We introduce a new Trefftz-DG method for the acoustic wave equation based on the use of Green’s functions as basis functions, which involves a Trefftz-DG variational formulation inside the PML in a Tent-Pitcher framework.



 
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