Tomasz M. Tyranowski, Michael Kraus
Symplectic model reduction for the Vlasov equation
PDF, arXiv:1910.06026

Abstract. Particle-based simulations of the Vlasov equation typically require a large number of particles, which leads to ordinary differential equations of a very high dimension. Solving such equations is computationally very expensive, especially when simulations for many different values of input parameters are desired. In this work we compare several model reduction techniques and demonstrate their applicability to numerical simulations of the Vlasov equation. The necessity of symplectic model reduction algorithms is illustrated with a simple numerical experiment.

Eero Hirvijoki, Joshua W. Burby, Michael Kraus
Energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator with exact H-theorems
PDF, arXiv:1804.08546

Abstract. 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 discretizations are chosen to reproduce the time-continuous conservation laws that correspond to Casimir invariants and to guarantee the positivity of the distribution function. Both the direct and the metriplectic discretization are demonstrated to have exact H-theorems and unique, physically exact equilibrium states. Most importantly, the two approaches are shown to coincide, given the chosen Galerkin method. A temporal discretization, preserving all of the mentioned properties, is achieved with so-called discrete gradients. Hence the proposed algorithm successfully translates all properties of the infinite-dimensional time-continuous Landau collision operator to time- and space-discrete sparse-matrix equations suitable for numerical simulation.

Eero Hirvijoki, Michael Kraus, Joshua W. Burby
Metriplectic Particle-in-Cell Integrators for the Landau Collision Operator
PDF, arXiv:1802.05263

Abstract. 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-dimensional time-continuous metriplectic system for advancing the macro particle weights. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of density, momentum, and energy, as well as the positive semi-definite production of entropy in both the time-continuous and the fully discrete system is demonstrated algebraically. The new algorithm is fully compatible with the existing particle-in-cell Poisson integrators for the Vlasov-Maxwell system.

Michael Kraus
Projected Variational Integrators for Degenerate Lagrangian Systems.
PDF, arXiv:1708.07356

Abstract. We propose and compare several projection methods applied to variational integrators for degenerate Lagrangian systems, whose Lagrangian is of the form \(L = \vartheta(q) \cdot \dot{q} - H(q)\) and thus linear in velocities. While previous methods for such systems only work reliably in the case of \(\vartheta\) being a linear function of \(q\), our methods are long-time stable also for systems where \(\vartheta\) is a nonlinear function of \(q\). We analyse the properties of the resulting algorithms, in particular with respect to the conservation of energy, momentum maps and symplecticity. In numerical experiments, we verify the favourable properties of the projected integrators and demonstrate their excellent long-time fidelity. In particular, we consider a two-dimensional Lotka-Volterra system, planar point vortices with position-dependent circulation and guiding centre dynamics.

Michael Kraus, Omar Maj
Variational Integrators for Ideal Magnetohydrodynamics.
PDF, arXiv:1707.03227

Abstract. 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 line topology is preserved and unphysical reconnection is absent. In 2D numerical examples we find that important conservation laws like total energy, magnetic helicity and cross helicity are satisfied within machine accuracy.

Papers in Refereed Journals

Michael Kraus, Tomasz M. Tyranowski
Variational integrators for stochastic dissipative Hamiltonian systems. IMA Journal of Numerical Analysis, 2020, draa022.
PDF, arXiv:1804.08546, Journal

Abstract. 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 appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d'Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange-d'Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether's theorem. Furthermore, mean-square and weak Lagrange-d'Alembert Runge-Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to non-geometric methods. The Vlasov-Fokker-Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.

Dimitrios A. Kaltsas, Michael Kraus, George Throumoulopoulos
A discrete Nambu bracket for 2D extended Magnetohydrodynamics. Journal of Physics: Conference Series, Volume 1391, 2019, 012037.
PDF, arXiv:1909.07202, Journal

Abstract. 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. Subsequently, the trilinear bracket of the resulting two-dimensional, four-field model is discretized using a finite difference scheme, which results in semi-discrete dynamics that involve the Arakawa Jacobian. Simulations of planar dynamics show that this scheme respects the desired conservation properties to high precision.

Camilla Bressan, Michael Kraus, Philip J. Morrison, Omar Maj
Relaxation to magnetohydrodynamics equilibria via collision brackets. Journal of Physics: Conference Series, Volume 1125, 2018, 012002.
PDF, arXiv:1809.03949, Journal

Abstract. 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, is formally analogous to the Landau collision operator. These ideas are illustrated by means of case studies. The considered physical models are the Euler equations in vorticity form, the Grad-Shafranov equation, and force-free MHD equilibria.

Michael Kraus
Variational Integrators for Inertial Magnetohydrodynamics. Physics of Plasmas, Volume 25, 082307, 2018. Editor's Pick.
PDF arXiv:1802.09676, Journal

Abstract. Recently, an extended version of magnetohydrodynamics that incorporates electron inertia, dubbed inertial magnetohydrodynamics, has been proposed. This model features a Hamiltonian formulation with various conserved quantities, including the total energy and a modified cross helicity. In this work, a variational integrator is presented which preserves these conservation laws to machine accuracy. As long as effects due to finite electron mass are neglected, the scheme preserves the magnetic field line topology so that unphysical reconnection is absent. Only when effects of finite electron mass are added, magnetic reconnection takes place. The excellent conservation properties of the method are illustrated by numerical examples in 2D.

C. Leland Ellison, John M. Finn, Joshua W. Burby, Michael Kraus, Hong Qin, William M. Tang
Degenerate Variational Integrators for Magnetic Field Line Flow and Guiding Center Trajectories. Physics of Plasmas, Volume 25, 052502, 2018.
PDF, arXiv:1801.07240, Journal

Abstract. 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 - resist symplectic integration by conventional means because the dynamics are most naturally formulated in non-canonical coordinates, i.e., coordinates lacking the familiar (q,p) partitioning. Recent efforts made progress toward non-canonical symplectic integration of these systems by appealing to the variational integration framework; however, those integrators were multistep methods and later found to be numerically unstable due to parasitic mode instabilities. This work eliminates the multistep character and, therefore, the parasitic mode instabilities via an adaptation of the variational integration formalism that we deem "degenerate variational integration". Both the magnetic field line and guiding center Lagrangians are degenerate in the sense that their resultant Euler-Lagrange equations are systems of first-order ODEs. We show that retaining the same degree of degeneracy when constructing a discrete Lagrangian yields one-step variational integrators preserving a non-canonical symplectic structure on the original Hamiltonian phase space. The advantages of the new algorithms are demonstrated via numerical examples, demonstrating superior stability compared to existing variational integrators for these systems and superior qualitative behavior compared to non-conservative algorithms.

Michael Kraus, Eero Hirvijoki
Metriplectic Integrators for the Landau Collision Operator. Physics of Plasmas, Volume 24, 102311, 2017.
PDF, arXiv:1707.01801, Journal

Abstract. 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, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the monotonic behaviour of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the entropy dissipation are guaranteed for finite-element discretizations in general, independently of the mesh configuration.

Michael Kraus, Katharina Kormann, Philip J. Morrison, Eric Sonnendrücker
GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods. Journal of Plasmas Physics, Volume 83, 905830401, 2017. Featured Article.
PDF, arXiv:1609.03053, Journal

Abstract. 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 invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge-invariant, feature exact charge conservation and show excellent long-time energy and momentum behavior. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy some compatibility condition.

Michael Kraus, Emanuele Tassi, Daniela Grasso
Variational Integrators for Reduced Magnetohydrodynamics. Journal of Computational Physics, Volume 321, Pages 435-458, 2016.
PDF, arXiv:1511.09314, Journal

Abstract. 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 applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws are described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature. In the case of reduced magnetohydrodynamics with electron inertia effects, simulations of magnetic reconnection are performed and compared also with those of a pseudo-spectral code.

Michael Kraus, Omar Maj
Variational Integrators for Nonvariational Partial Differential Equations. Physica D: Nonlinear Phenomena, Volume 310, Pages 37-71, 2015.
PDF, arXiv:1412.2011, Journal

Abstract. 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 of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics.
On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems.
The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.

Various Contributions

Michael Kraus
Strukturerhaltende Numerik in der Plasmaphysik, Jahrbuch der Max-Planck-Gesellschaft, 2016.
HTML, MPG Jahrbuch

Abstract. 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 Simulationen nicht reproduziert werden können. Abhilfe könnten sogenannten strukturerhaltenden Integrationsmethoden schaffen. Sie kombinieren Ideen aus Numerik, Physik und Geometrie und ermöglichen realistischere Simulationen als klassische Verfahren.

In Preparation

Michael Kraus
Variational Integrators for Noncanonical Hamiltonian Systems.

Abstract. We propose a new approach to the derivation of variational integrators for noncanonical Hamiltonian systems of the form \(\dot{q} = \bar{\Omega}^{-1} (q) \, \nabla H (q)\). Such systems originate from degenerate Lagrangians of the form \(L (q, \dot{q}) = \left< \vartheta(q) , \dot{q} \right> - H(q)\), where \(\vartheta\) is a general, possibly nonlinear function of \(q\) and \(\bar{\Omega} = \nabla \vartheta^{T} - \nabla \vartheta\).
We do not directly discretise the degenerate Lagrangian as usually done with variational integrators, but instead we construct a formal Lagrangian and derive discrete Euler-Lagrange equations for the corresponding extended system. After a simple reduction procedure, we obtain variational Runge-Kutta integrators of the dynamics of the original system. We discuss gauge invariance and symplecticity and show how a generalised discrete Noether theorem can be used in order to prove conservation of momentum maps of the discrete system.

Michael Kraus, Omar Maj, Eric Sonnendrücker
Variational Integrators for the Vlasov-Poisson System

Abstract. Recently, an extension of the variational integrator framework has been proposed, which allows the application of the method even to nonvariational partial differential equations. In this work, this new method is applied to the Vlasov-Poisson system in Eulerian coordinates, in order to obtain grid-based numerical schemes, which respect important conservation laws on the discrete level. More precisely, the resulting integrators preserve mass, momentum, energy and the \(L^{2}\) norm of the distribution function exactly. Due to the absence of dissipation, small scale structures appear in the solution, which eventually cannot be resolved and render the simulation unstable. To mitigate this problem, we apply a collision operator, which adds a minimum amount of diffusion in velocity space, thereby dissipating the \(L^{2}\) norm of the distribution function, but retaining exact conservation of mass, momentum and energy. As most of the variational integrators obtained in this work are implicit, suitable preconditioning strategies are needed in order to solve the discrete Vlasov-Poisson system efficiently. We devise a new preconditioner based on a tensor product decomposition of the discrete Poisson bracket in the Vlasov equation, which is very effective so that convergence of the linear solver is achieve after only few iterations. Several numerical examples corroborate the favourable properties of the variational integrators.

PhD Thesis

PhD Thesis: Variational Integrators in Plasma Physics

Diploma Thesis

Diplomarbeit: Heiz- und Stromtriebprofile bei Neutralteilcheninjektion in Tokamakplasmen

Über den Effekt von Fehlern in den Ionisationsquerschnitten auf die berechneten Heiz- und Stromprofile bei Neutralteilcheninjektion, Auswirkungen verschiedener Stromtriebmodelle und Konsequenzen für aktuelle Transportuntersuchungen

IPP Report 5/122: Heiz- und Stromtriebprofile bei Neutralteilcheninjektion in Tokamakplasmen
(same as Diplomarbeit, with some minor changes and corrections)