Splitting and composition methods in the numerical integration of differential equations

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.Comment: Review paper; 56 pages, 6 figures, 8 table

Numerical Integration of Damped Maxwell Equations

We study the numerical time integration of Maxwell's equations from electromagnetism. Following the method of lines approach we start from a general semi-discrete Maxwell system for which a number of time-integration methods are considered. These methods have in common an explicit treatment of the curl terms. Central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two, in particular in the presence of an explicitly or implicitly treated damping term which models conduction

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

A Magneto-Gravitational Neutron Trap for the Measurement of the Neutron Lifetime

Thesis (Ph.D.) - Indiana University, Physics, 2015Neutron decay is the simplest example of nuclear beta-decay. The mean decay lifetime is a key input for predicting the abundance of light elements in the early universe. A precise measurement of the neutron lifetime, when combined with other neutron decay observables, can test for physics beyond the standard model in a way that is complimentary to, and potentially competitive with, results from high energy collider experiments. Many previous measurements of the neutron lifetime used ultracold neutrons (UCN) confined in material bottles. In a material bottle experiment, UCN are loaded into the apparatus, stored for varying times, and the surviving UCN are emptied and counted. These measurements are in poor agreement with experiments that use neutron beams, and new experiments are needed to resolve the discrepancy and precisely determine the lifetime. Here we present an experiment that uses a bowl-shaped array of NdFeB magnets to confine neutrons without material wall interactions. The trap shape is designed to rapidly remove higher energy UCN that might slowly leak from the top of the trap, and can facilitate new techniques to count surviving UCN within the trap. We review the scientific motivation for a precise measurement of the neutron lifetime, and present the commissioning of the trap. Data are presented using a vanadium activation technique to count UCN within the trap, providing an alternative method to emptying neutrons from the trap and into a counter. Potential systematic effects in the experiment are then discussed and estimated using analytical and numerical techniques. We also investigate solid nitrogen-15 as a source of UCN using neutron time-of-flight spectroscopy. We conclude with a discussion of forthcoming research and development for UCN detection and UCN sources

Numerical scalar curvature deformation and a gluing construction

In this work a new numerical technique to prepare Cauchy data for the initial value problem (IVP) formulation of Einstein's field equations (EFE) is presented. Our method is directly inspired by the exterior asymptotic gluing (EAG) result of Corvino (2000). The argument assumes a moment in time symmetry and allows for a composite, initial data set to be assembled from (a finite subdomain of) a known asymptotically Euclidean initial data set which is glued (in a controlled manner) over a compact spatial region to an exterior Schwarzschildean representative. We demonstrate how (Corvino, 2000) may be directly adapted to a numerical scheme and under the assumption of axisymmetry construct composite Hamiltonian constraint satisfying initial data featuring internal binary black holes (BBH) glued to exterior Schwarzschild initial data in isotropic form. The generality of the method is shown in a comparison of properties of EAG composite initial data sets featuring internal BBHs as modelled by Brill-Lindquist and Misner data. The underlying geometric analysis character of gluing methods requires work within suitably weighted function spaces, which, together with a technical impediment preventing (Corvino, 2000) from being fully constructive, is the principal difficulty in devising a numerical technique. Thus the single previous attempt by Giulini and Holzegel (2005) (recently implemented by Doulis and Rinne (2016)) sought to avoid this by embedding the result within the well known Lichnerowicz-York conformal framework which required ad-hoc assumptions on solution form and a formal perturbative argument to show that EAG may proceed. In (Giulini and Holzegel, 2005) it was further claimed that judicious engineering of EAG can serve to reduce the presence of spurious gravitational radiation - unfortunately, in line with the general conclusion of (Doulis and Rinne, 2016) our numerical investigation does not appear to indicate that this is the case. Concretising the sought initial data to be specified with respect to a spatial manifold with underlying topology R×S² our method exploits a variety of pseudo-spectral (PS) techniques. A combination of the eth-formalism and spin-weighted spherical harmonics together with a novel complex-analytic based numerical approach is utilised. This is enabled by our Python 3 based numerical toolkit allowing for unified just-in-time compiled, distributed calculations with seamless extension to arbitrary precision for problems involving generic, geometric partial differential equations (PDE) as specified by tensorial expressions. Additional features include a layer of abstraction that allows for automatic reduction of indicial (i.e., tensorial) expressions together with grid remapping based on chart specification - hence straight-forward implementation of IVP formulations of the EFE such as ADM-York or ADM-York-NOR is possible. Code-base verification is performed by evolving the polarised Gowdy T³ space-time with the above formulations utilising high order, explicit time-integrators in the method of lines approach as combined with PS techniques. As the initial data we prepare has a precise (Schwarzschild) exterior this may be of interest to global evolution schemes that incorporate information from spatial-infinity. Furthermore, our approach may shed light on how more general gluing techniques could potentially be adapted for numerical work. The code-base we have developed may also be of interest in application to other problems involving geometric PDEs

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282