Michael Kraus

Michael Kraus
Max Planck Institute for Plasma Physics | IPP · Numerical Methods in Plasma Physics

PhD Mathematics

About

29
Publications
4,554
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340
Citations
Introduction
I am working on the development of geometrical discretization methods for dynamical systems from plasma physics.
Additional affiliations
October 2017 - present
Technische Universität München
Position
  • Lecturer
Description
  • Geometric Numerical Integration of Ordinary and Partial Differential Equations
April 2017 - present
Max-Planck-Institut für Plasmaphysik
Position
  • Research Associate
Description
  • Geometric and Structure Preserving Numerical Methods
April 2016 - March 2017
Waseda University
Position
  • Visiting Assistant Professor
Description
  • Discrete Dirac Mechanics and Geometric Numerical Integration Methods
Education
March 2010 - July 2013
Technische Universität München
Field of study
  • Mathematics
October 2007 - March 2010
April 2004 - September 2007

Publications

Publications (29)
Article
Full-text available
We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi-identity, as well as conservation of its Casimir invari...
Article
Full-text available
We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a finite element discretization for the velocity space, transform the infinite-dimensional system into a fin...
Article
Full-text available
We propose and compare several projection methods applied to variational integrators for degenerate Lagrangian systems, whose Lagrangian is of the form $L = \vartheta(\bq) \cdot \bv - H(\bq)$ and thus linear in velocities. While previous methods for such systems only work reliably in the case of $\vartheta$ being a linear function of $\bq$, our met...
Article
Full-text available
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence...
Thesis
Full-text available
Variational integrators are a special kind of geometric discretisation methods applicable to any system of differential equations that obeys a Lagrangian formulation. In this thesis, variational integrators are developed for several important models of plasma physics: guiding centre dynamics (particle dynamics), the Vlasov-Poisson system (kinetic t...
Preprint
Full-text available
This paper deals with the control of parasitism in variational integrators for degenerate Lagrangian systems by writing them as general linear methods. This enables us to calculate their parasitic growth parameters which are responsible for the loss of long-time energy conservation properties of these algorithms. As a remedy and to offset the effec...
Chapter
Full-text available
Many systems from fluid dynamics and plasma physics possess a so-called metriplectic structure, that is the equations are comprised of a conservative, Hamiltonian part, and a dissipative, metric part. Consequences of this structure are conservation of important quantities, such as mass, momentum and energy, and compatibility with the laws of thermo...
Article
Full-text available
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian, which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropria...
Preprint
Full-text available
Particle-based simulations of the Vlasov equation typically require a large number of particles, which leads to a high-dimensional system of ordinary differential equations. Solving such systems is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work we compare several m...
Preprint
In this note we propose a trilinear bracket formulation for the Hamiltonian extended Magnetohydrodynamics (XMHD) model with homogeneous mass density. The corresponding two-dimensional representation is derived by performing spatial reduction on the three-dimensional bracket, upon introducing a symmetric representation for the field variables. Subse...
Article
Full-text available
In this note we propose a trilinear bracket formulation for the Hamiltonian extended Magnetohydrodynamics (XMHD) model with homogeneous mass density. The corresponding two-dimensional representation is derived by performing spatial reduction on the three-dimensional bracket, upon introducing a symmetric representation for the field variables. Subse...
Preprint
Full-text available
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriat...
Preprint
Full-text available
Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriat...
Article
Full-text available
Metriplectic dynamics is applied to compute equilibria of fluid dynamical systems. The result is a relaxation method in which Hamiltonian dynamics (symplectic structure) is combined with dissipative mechanisms (metric structure) that relaxes the system to the desired equilibrium point. The specific metric operator, which is considered in this work,...
Preprint
Full-text available
Metriplectic dynamics is applied to compute equilibria of fluid dynamical systems. The result is a relaxation method in which Hamiltonian dynamics (symplectic structure) is combined with dissipative mechanisms (metric structure) that relaxes the system to the desired equilibrium point. The specific metric operator, which is considered in this work,...
Article
Full-text available
This paper explores energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator. We discuss two approaches, namely direct Galerkin formulations and discretizations of the underlying infinite-dimensional metriplectic structure of the collision integral. The spatial discretizatio...
Article
Full-text available
Recently, an extended version of magnetohydrodynamics that incorporates electron inertia, dubbed inertial magnetohydrodynamics, has been proposed. This model features a noncanonical Hamiltonian formulation with a number of conserved quantities, including the total energy and modified versions of magnetic and cross helicity. In this work, a variatio...
Article
Full-text available
In this paper, we present a new framework for addressing the nonlinear Landau collision operator in terms of particle-in-cell methods. We employ the underlying metriplectic structure of the collision operator and, using a macro particle discretization for the distribution function, we transform the infinite-dimensional system into a finite-dimensio...
Article
Full-text available
Symplectic integrators offer many advantages for the numerical solution of Hamiltonian differential equations, including bounded energy error and the preservation of invariant sets. Two of the central Hamiltonian systems encountered in plasma physics --- the flow of magnetic field lines and the guiding center motion of magnetized charged particles...
Article
Full-text available
A variational integrator for ideal magnetohydrodynamics is derived by applying a discrete action principle to a formal Lagrangian. Discrete exterior calculus is used for the discretisation of the field variables in order to preserve their geometrical character. The resulting numerical method is free of numerical resistivity, thus the magnetic field...
Presentation
Full-text available
Seminar on the Variational Integrators for Nonvariational Partial Differential Equations
Presentation
Full-text available
Seminar on the Structure-Preserving Discretisation of Poisson and Leibniz Brackets for the Vlasov-Maxwell System
Chapter
Full-text available
Zahlreiche Eigenschaften eines Plasmas, die experimentell nicht oder nicht im Detail zugänglich sind, können nur in Computersimulationen systematisch untersucht werden. Viele Codes nutzen aber numerische Methoden, die wichtige Eigenschaften der mathematischen Gleichungen nur unzureichend berücksichtigen. Die Folge ist, dass wichtige Phänomene in Si...
Article
Full-text available
Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle appl...
Presentation
Full-text available
In this talk we will describe two lines of research towards geometric particle-in-cell methods for the Vlasov-Maxwell system. In the first part, we present particle-in-cell finite-element methods for the Vlasov-Maxwell system based on a fully discrete action principle based on Low’s Lagrangian. These variational integrators approximately preserve t...
Presentation
Full-text available
In this talk we will describe two lines of research towards geometric particle-in-cell methods for the Vlasov-Maxwell system. In the first part, we present particle-in-cell finite-element methods for the Vlasov-Maxwell system based on a fully discrete action principle based on Low’s Lagrangian. These variational integrators approximately preserve t...
Poster
Full-text available
Variational discretisation of nonvariational partial differential equations by application of the variational integrator method to formal Lagrangians. Demonstration of excellent conservation properties for problems from plasma physics, namely the Vlasov-Maxwell system and ideal magnetohydrodynamics.
Technical Report
Full-text available
In these notes we give an overview of the derivation of the TRANSP current drive algorithms by Hirshman and Sigmar (refs. [1, 2, 3], NMCURB=1) and by Lin-Liu and Hinton as well as Kim, Callen and Hamnen, (refs. [5, 4], NMCURB=3). The latter one was derived independently by Lin-Liu et al., and earlier by Kim et al., whereby both take a somewhat diff...

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Projects

Projects (6)
Project
Extension and application of the variational integrator method to problems which do not have a natural variational formulation in terms of Eulerian variables such as many problems from plasma physics and fluid dynamics.
Project
Development of novel geometric integrators for Hamiltonian PDEs from plasma physics and fluid dynamics.