Oberseminar Numerik

Das Oberseminar richtet sich an Studenten und Forscher mit Interesse an aktuellen Fragestellung in der Numerik, wissenschaftlichem Rechnen und Optimierung. Es finden in unregelmäßigen Abständen Vorträge zu aktuellen Themen statt.

Wintersemester 2023/24

Mittwoch, 13.12.2023 (13:00 Uhr in IA 02/105): Tuan Ahn Dao (Uppsala University)

Title: Viscous regularization and structure-preserving schemes for ideal MHD

 

Abstract: This talk discusses viscosity methods for the ideal MHD equations. For the choice of the viscous flux, I will talk about the monolithic parabolic flux and motivate why it is a good choice. To be more precise, at the PDE level, the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. The artificial viscosity is constructed using an entropy viscosity method. Several numerical examples will be demonstrated.

The second part of the talk is about our new structure-preserving method to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler’s equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. The method provably preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. In our approach, the CFL condition does not depend on magnetosonic wavespeeds, but only on the usual maximum wavespeed from Euler’s system.

Sommersemester 2023

Mittwoch, 20.09.2023 (11:00 Uhr): Tileuzhan Mukhamet (Master student at RUB)

Titel: An arbitrary Lagrangian Eulerian Discontinuous Galerkin method for the Vlasov equation with a strong magnetic field

 

Freitag, 11.08.2023 (11:00 Uhr, IA 1/53): Mahima Yadav (PhD student at RUB)

Titel: On discrete ground states of rotating Bose-Einstein condensates

Abstract: The talk focuses on the study of ground states of Bose-Einstein condensates in a rotating frame. The ground states are described as the constrained minimizers of the Gross-Pitaevskii energy functional with an angular momentum term. The problem is discretized using Lagrange finite element spaces of arbitrary polynomial order and the approximation properties of the corresponding numerical approximations are presented, taking into account the missing uniqueness of ground states which is mainly caused by the invariance of the energy functional under complex phase shifts. Error estimates of optimal order are shown for the L2- and H1-norm, as well as for the ground state energy and chemical potential.

 

Donnerstag, 11.05.2023 (11:15 Uhr): Paul Wilhelm (RWTH Aachen)

Titel: NuFI: An implicit Lagrangian Flow reconstruction scheme for the Vlasov equation

Abstract: The Vlasov equation is a high-dimensional partial differential equation arising from kinetic theory and used to model the behaviour of plasma flows in collision-less and strongly non-equilibrium regimes. We present a novel approach, the numerical flow iteration (NuFI), which evaluates the numerical solution via storing the three-dimensional electric potentials, using these to iteratively reconstruct the Lagrangian flow and directly evaluating the initial data via the method of characteristics. This reduces the total memory-requirement by several orders of magnitude and allows to shift workload from frequent memory access to computations on the fly, i.e., yielding high flop/Byte-rates, which is favourable on modern compute-architectures. Furthermore using the Lagrangian formulation one conserves desired properties like $L^p$-norms and kinetic entropy exactly, as well as total energy up to time-integration error. We demonstrate the accuracy and scalability of the new approach on several test-cases in up to six dimensions showing computations done on a workstation as well as a GPU-cluster.

 

Donnerstag, 27.04.2023 (11:00 Uhr): Ivo Dravins (RUB)

Title: Preconditioning for block matrices with square blocks

Abstract: Linear systems of equations appear in one way or another in almost every scientific and engineering problem. They are so ubiquitous that, in addition to solving linear problems, also non-linear problems are typically reduced to a sequence of linear ones. The availability of modern large-scale computational resources motivates the development and the use of well parallelizable efficient solvers with a limited memory footprint. For many problems, these properties can be achieved by the employment of iterative solution methods combined with preconditioning techniques. We explore the design of preconditioners for block-matrices with square blocks. This form of matrices occurs in many applications, encountered for instance when numerically solving partial differential equations, ordinary differential equations and others.

We focus on two classes of problems, one being optimal control problems within the PDE-constrained optimization framework, and the other being fully implicit Runge-Kutta time-stepping schemes. Both necessitate the solution of large and sparse linear systems, for which we employ preconditioned Krylov subspace methods. The main topic of the talk is on the design of preconditioners, although the entire solution procedure is explored.

 

 

 

Wintersemester 2022/23

Donnerstag, 10.11.2022 (13 Uhr): Fado Philo (Universität Duisburg-Essen)

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