ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
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: 9th Dec 2022, 12:00:59am CET

Session Overview 
Session  
MS 11a: recent developments in highorder methods for timedependent problems
 
Session Abstract  
Both hyperbolic and parabolic timedependent 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 parallelintime and spacetime methods, among many others. In this minisymposium we aim to present the stateoftheart 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, spacetime discontinuous Galerkin, spacetime nite element, implicitexplicit methods, asynchronous, parallelintime and adaptive mesh renement. Presentations regarding exascale or highly parallel implementations (such as application of GPU platforms) are also welcomed.  
Presentations  
4:00pm  4:30pm
The solution of elliptic PDEs by using a hyperbolic PDE solver ^{1}University of Tennessee, United States of America; ^{2}University of Illinois at UrbanaChampaign, 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 steadystate limits of parabolic or hyperbolic PDEs. Second, we address longterm solutions of hyperbolic PDEs driven by harmonic forces, in particular, the Helmholtz equation. We discuss two distinct schemes for steadystate 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 wavebased 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 steadystate 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 highorder discontinuous Galerkin schemes with firstorder hyperbolic advectiondiffusion system approach. Journal of Computational Physics, 321:72954, 2016. [3] Jung, Samuel, TaeYun Kim, and WanSuk 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 ^{1}Charles University, Prague, Czech Republic; ^{2}Czech 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 spacetime discontinuous Galerkin method (STDGM) which, in combination with the anisotropic hpmesh 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 spacetime 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 higherorder spacetime discontinuous Galerkin method for the computer simulation of variablysaturated porous media flows, Applied Mathematical Modelling 72: 276305, 2019. [2] V. Dolejsi, G. May, F. Roskovec, P. Solin: Anisotropic hpmesh optimization technique based on the continuous mesh and error models, Comput. Math. Appl. 74:4563, 2017. [3] V. Dolejsi, F. Roskovec, M. Vlasak: A posteriori error estimates for higher order spacetime Galerkin discretizations of nonlinear parabolic problems, SIAM J. Numer. Anal, accepted, 2021. 5:00pm  5:30pm
PML applied to spacetime TrefftzDG numerical formulation for the acoustic wave equation. ^{1}ProjectTeam Makutu, Inria, E2S UPPA, CNRS, France; ^{2}Total R&D, France The coupling between Trefftz methods and Discontinuous Galerkin (TrefftzDG) 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 TrefftzDG 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 timescheme, TrefftzDG 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 TrefftzDG framework for solving acoustic equations using Tent Pitcher algorithms. This algorithm consists in building a spacetime 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, tentpitcher meshes allow for turning the previously described huge matrix into a collection of easily invertible small matrices. It is worth noting that, since TrefftzDG 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 spacetime 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 wellknown and can be derived using for instance the Cagniard de Hoop method. We introduce a new TrefftzDG method for the acoustic wave equation based on the use of Green’s functions as basis functions, which involves a TrefftzDG variational formulation inside the PML in a TentPitcher framework. 
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