Spatial discretization of partial differential equations with integrals

We consider the problem of constructing spatial finite difference approximations on a fixed, arbitrary grid, which have analogues of any number of integrals of the partial differential equation and of some of its symmetries. A basis for the space of of such difference operators is constructed; most cases of interest involve a single such basis element. (The ``Arakawa'' Jacobian is such an element.) We show how the topology of the grid affects the complexity of the operators.Comment: 24 pages, LaTeX sourc

Adaptive Geometric Numerical Integration of Mechanical Systems

This thesis is about structure preserving numerical integration of initial value problems, i.e., so called geometric numerical integrators. In particular, we are interested in how time-step adaptivity can be achieved in conjunction with structure preserving properties without destroying the good long time integration properties which are typical for geometric integration methods. As a specific application we consider dynamic simulations of rolling bearings and rotor dynamical problems. The work is part of a research collaboration between SKF (www.skf.com) and the Centre of Mathematical Sciences at Lund University

Structure Preserving Model Order Reduction of Shallow Water Equations

In this paper, we present two different approaches for constructing reduced-order models (ROMs) for the two-dimensional shallow water equation (SWE). The first one is based on the noncanonical Hamiltonian/Poisson form of the SWE. After integration in time by the fully implicit average vector field method, ROMs are constructed with proper orthogonal decomposition/discrete empirical interpolation method (POD/DEIM) that preserves the Hamiltonian structure. In the second approach, the SWE as a partial differential equation with quadratic nonlinearity is integrated in time by the linearly implicit Kahan's method and ROMs are constructed with the tensorial POD that preserves the linear-quadratic structure of the SWE. We show that in both approaches, the invariants of the SWE such as the energy, enstrophy, mass, and circulation are preserved over a long period of time, leading to stable solutions. We conclude by demonstrating the accuracy and the computational efficiency of the reduced solutions by a numerical test problem

Variational Integrators in Plasma Physics

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 theory), and ideal magnetohydrodynamics (plasma fluid theory). Special attention is given to physical conservation laws like conservation of energy and momentum. Most systems in plasma physics do not possess a Lagrangian formulation to which the variational integrator methodology is directly applicable. Therefore the theory is extended towards nonvariational differential equations by linking it to Ibragimov's theory of integrating factors and adjoint equations. It allows us to find a Lagrangian for all ordinary and partial differential equations and systems thereof. Consequently, the applicability of variational integrators is extended to a much larger family of systems than envisaged in the original theory. This approach allows for the application of Noether's theorem to analyse the conservation properties of the system, both at the continuous and the discrete level. In numerical examples, the conservation properties of the derived schemes are analysed. In case of guiding centre dynamics, momentum in the toroidal direction of a tokamak is preserved exactly. The particle energy exhibits an error, but the absolute value of this error stays constant during the entire simulation. Therefore numerical dissipation is absent. In case of the kinetic theory, the total number of particles, total linear momentum and total energy are preserved exactly, i.e., up to machine accuracy. In case of magnetohydrodynamics, the total energy, cross helicity and the divergence of the magnetic field are preserved up to machine precision.Comment: PhD Thesis, 222 page

The bracket geometry of statistics

In this thesis we build a geometric theory of Hamiltonian Monte Carlo, with an emphasis on symmetries and its bracket generalisations, construct the canonical geometry of smooth measures and Stein operators, and derive the complete recipe of measure-constraints preserving dynamics and diffusions on arbitrary manifolds. Specifically, we will explain the central role played by mechanics with symmetries to obtain efficient numerical integrators, and provide a general method to construct explicit integrators for HMC on geodesic orbit manifolds via symplectic reduction. Following ideas developed by Maxwell, Volterra, Poincaré, de Rham, Koszul, Dufour, Weinstein, and others, we will then show that any smooth distribution generates considerable geometric content, including ``musical" isomorphisms between multi-vector fields and twisted differential forms, and a boundary operator - the rotationnel, which, in particular, engenders the canonical Stein operator. We then introduce the ``bracket formalism" and its induced mechanics, a generalisation of Poisson mechanics and gradient flows that provides a general mechanism to associate unnormalised probability densities to flows depending on the score pointwise. Most importantly, we will characterise all measure-constraints preserving flows on arbitrary manifolds, showing the intimate relation between measure-preserving Nambu mechanics and closed twisted forms. Our results are canonical. As a special case we obtain the characterisation of measure-preserving bracket mechanical systems and measure-preserving diffusions, thus explaining and extending to manifolds the complete recipe of SGMCMC. We will discuss the geometry of Stein operators and extend the density approach by showing these are simply a reformulation of the exterior derivative on twisted forms satisfying Stokes' theorem. Combining the canonical Stein operator with brackets allows us to naturally recover the Riemannian and diffusion Stein operators as special cases. Finally, we shall introduce the minimum Stein discrepancy estimators, which provide a unifying perspective of parameter inference based on score matching, contrastive divergence, and minimum probability flow.Open Acces

Symmetry methods in the atmospheric sciences

Zahlreiche Symmetriemethoden werden auf Differentialgleichungen der Atmosphärendynamik angewandt. Die Lie-Punktsymmetrien der barotropen Vorticitygleichung, der barotropen potentiellen Vorticitygleichung und des baroklinen Zweischichtmodels werden berechnet. Ein- und zweidimensionale inäquivalente Subalgebren der jeweiligen maximalen Lie-Invarianzalgebren werden klassifiziert und dazu verwendet, exakte Lösungen der jeweiligen Gleichungen zu bestimmen. Die physikalische Bedeutung dieser Lösungen wird untersucht und diskutiert. Mittels der Symmetrien der barotropen potentiellen Vorticitygleichung auf der beta-Ebene und der barotropen Vorticitygleichung auf der rotierenden Kugel können Punkttransformationen gefunden werden, die beide Gleichungen in die jeweiligen Gleichungen im Inertialsystem transformieren. Zwei erweiterte Techniken zur Berechnung der gesamten Punktsymmetriegruppe von Differentialgleichungen werden vorgestellt, die im Rahmen der direkten Methode angewandt werden können. Die erste Technik basiert auf der Invarianz von Megaidealen der maximalen Lie-Invarianzalgebra unter von Punktsymmetrien erzeugten Automorphismen. Die zweite Technik verwendet Kenntnisse über admissible transformations von Klassen von Differentialgleichungen, die die untersuchte Gleichung enthalten. Weiters wird gezeigt wie Symmetrien dazu verwendet werden können, Schließungen im Zuge des Parameterisierungsproblems zu definieren. Für diesen Zweck werden Verfahren der direkten und inversen Gruppenklassifikation benützt. Als Beispiel werden verschiedene Parameterisierungen für den Eddy-Vorticityfluß in der Reynolds-gemittelten Vorticitygleichung konstruiert, die unterschiedliche Symmetrieeigenschaften besitzen. In einem weiteren Schritt werden die Symmetrien der barotropen Vorticitygleichung und der Saltzman'schen Konvektionsgleichungen dazu verwendet um spektrale, niedrigdimensionale Approximationen dieser Gleichungen zu erzeugen. Dazu werden Lie-Punkt- und diskrete Symmetrien als Kriterium zur Auswahl der Fouriermoden verwendet. Es wird bewiesen dass das Lorenz--1960 Modell systematisch unter Zuhilfenahme der Punktsymmetrien der Vorticitygleichung ableitbar ist. Auf ähnliche Weise wird demonstriert dass die Wahl der Moden des Lorenz--1963 Modells der thermischen Konvektion nicht mittels Symmetrien begründbar ist. Zudem wird gezeigt dass sowohl die Hamiltonsche als auch die Nambu Form des Lorenz--1963 Modells nicht mit der entsprechenden Hamiltonschen bzw. Nambu-Darstellung der Saltzman'schen Konvektionsgleichungen zusammenhängen. Aus diesem Grund wird ein sechskomponentiges Modell der Konvektionsgleichungen abgeleitet. Die Modenwahl dieses neuen Modells basiert vollständig auf Punktsymmetrien der Saltzman'schen Gleichungen. Durch geeignetes Skalieren dieser Moden ist es möglich eine Hamiltonsche bwz. Nambu-Darstellung dieses sechskomponentigen Modells zu finden, die der Hamilton- bzw. Nambuformulierung der kontinuierlichen Konvektionsgleichungen vollständig analog ist.Wide ranges of symmetry methods are applied to several differential equations arising in the atmospheric sciences. Lie point symmetries of the barotropic vorticity equation, the barotropic potential vorticity equation and the two-layer baroclinic model are computed. One- and two-dimensional inequivalent subalgebras of the respective maximal Lie invariance algebras are classified. Based on this classification, we determine various group-invariant solutions of the investigated differential equations. The physical relevance of these particular solutions is evaluated. Symmetries are used to find point transformations that map the barotropic potential vorticity equation on the beta-plane and the barotropic vorticity equation on the rotating sphere to the respective equations in the inertial frame. Two refined techniques for the computation of the complete point symmetry group of differential equations are proposed within the framework of the direct method. The first technique is based on the invariance of megaideals of the maximal Lie invariance algebra under automorphisms generated by point symmetries. The second technique involves knowledge on the admissible transformations of classes of differential equations containing the given equation. It is shown how symmetries can be employed to determine closure schemes in the course of the parameterization problem. The methods we apply rest on techniques of direct and inverse group classifications. These methods are exemplified by parameterizing the eddy vorticity flux in the Reynolds averaged vorticity equation. This leads to several invariant parameterization schemes possessing different degrees of symmetry. The symmetries of the barotropic vorticity equation and the Saltzman convection equations are used to derive spectral finite-mode approximations. This is done using both Lie and discrete point symmetries as a criterion for the selection of Fourier modes. It is proved that the Lorenz--1960 model can be systematically re-derived with the aid of point symmetries of the vorticity equation. In a similar manner, it is demonstrated that the selection of modes for the Lorenz--1963 convection model is not compatible with the symmetries of the Saltzman equations. It is shown that the Hamiltonian and Nambu structures of the Lorenz--1963 model are not related to the Hamiltonian and Nambu forms of the Saltzman convection equations. A new six-component truncation of the convection equation is proposed. The selection of modes for this model is based on point symmetries of the convection equations. These modes are suitably scaled to allow the six-component model to be of Hamiltonian and Nambu forms analog to those of the original Saltzman equations

Mathematical and physical ideas for climate science

The climate is a forced and dissipative nonlinear system featuring nontrivial dynamics on a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the nonequilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate, characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on nonequilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models

The Hamiltonian structure and Euler-Poincar\'{e} formulation of the Vlasov-Maxwell and gyrokinetic systems

We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincar\'{e} theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. [1] H. Cendra et. al., Journal of Mathematical Physics 39, 3138 (1998)Comment: 36 pages, 1 figur

Lotka-Volterra predator-prey models analytic and numerical methods.

The Lotka-Volterra equations are a classical model of the populations of interacting species. In the case of two interacting species, we present a closed parametric solution to a particular case of the Lotka-Volterra model. We also determine closed expressions for the branch points, bounds on the parameter, amplitude of the oscillation of the prey and predator populations, and period of this model in terms of the Lambert W function. In the case of three interacting species, under certain conditions solutions are again periodic. However, standard numerical methods often fail to preserve this periodicity, as well as other important properties of the model. The underlying geometry of the three-species predator-prey model is developed through the framework of Poisson dynamics. It is shown that the system is bi-Poisson and possesses two independent first integrals. Numerical methods for approximating solutions to the model are constructed which incorporate the underlying Poisson geometry of the continuous system. These methods preserve the periodicity of solutions, and the error in the first integrals remains bounded. Simulations are used to show that these methods produce more accurate results than standard numerical methods which do not consider the Poisson structure of the equations.Master of Science (MSc) in Computational Science