499 research outputs found
Magnus-based geometric integrators for dynamical systems with time-dependent potentials
[ES] Esta tesis trata sobre la integración numérica de sistemas hamiltonianos con potenciales explícitamente dependientes del tiempo. Los problemas de este tipo son comunes en la física matemática, porque provienen de la mecánica cuántica, clásica y celestial.
La meta de la tesis es construir integradores para unos problemas relevantes no autónomos: la ecuación de Schrödinger, que es el fundamento de la mecánica cuántica; las ecuaciones de Hill y de onda, que describen sistemas oscilatorios; el problema de Kepler con la masa variante en el tiempo.
El Capítulo 1 describe la motivación y los objetivos de la obra en el contexto histórico de la integración numérica. En el Capítulo 2 se introducen los conceptos esenciales y unas herramientas fundamentales utilizadas a lo largo de la tesis.
El diseño de los integradores propuestos se basa en los métodos de composición y escisión y en el desarrollo de Magnus. En el Capítulo 3 se describe el primero. Su idea principal consta de una recombinación de unos integradores sencillos para obtener la solución del problema. El concepto importante de las condiciones de orden se describe en ese capítulo. En el Capítulo 4 se hace un resumen de las álgebras de Lie y del desarrollo de Magnus que son las herramientas algebraicas que permiten expresar la solución de ecuaciones diferenciales dependientes del tiempo.
La ecuación lineal de Schrödinger con potencial dependiente del tiempo está examinada en el Capítulo 5. Dado su estructura particular, nuevos métodos casi sin conmutadores, basados en el desarrollo de Magnus, son construidos. Su eficiencia es demostrada en unos experimentos numéricos con el modelo de Walker-Preston de una molécula dentro de un campo electromagnético.
En el Capítulo 6, se diseñan los métodos de Magnus-escisión para las ecuaciones de onda y de Hill. Su eficiencia está demostrada en los experimentos numéricos con varios sistemas oscilatorios: con la ecuación de Mathieu, la ec. de Hill matricial, las ecuaciones de onda y de Klein-Gordon-Fock.
El Capítulo 7 explica cómo el enfoque algebraico y el desarrollo de Magnus pueden generalizarse a los problemas no lineales. El ejemplo utilizado es el problema de Kepler con masa decreciente.
El Capítulo 8 concluye la tesis, reseña los resultados y traza las posibles direcciones de la investigación futura.[CA] Aquesta tesi tracta de la integració numèrica de sistemes hamiltonians amb potencials explícitament dependents del temps. Els problemes d'aquest tipus són comuns en la física matemàtica, perquè provenen de la mecànica quàntica, clàssica i celest.
L'objectiu de la tesi és construir integradors per a uns problemes rellevants no autònoms: l'equació de Schrödinger, que és el fonament de la mecànica quàntica; les equacions de Hill i d'ona, que descriuen sistemes oscil·latoris; el problema de Kepler amb la massa variant en el temps.
El Capítol 1 descriu la motivació i els objectius de l'obra en el context històric de la integració numèrica. En Capítol 2 s'introdueixen els conceptes essencials i unes ferramentes fonamentals utilitzades al llarg de la tesi.
El disseny dels integradors proposats es basa en els mètodes de composició i escissió i en el desenvolupament de Magnus. En el Capítol 3, es descriu el primer. La seua idea principal consta d'una recombinació d'uns integradors senzills per a obtenir la solució del problema. El concepte important de les condicions d'orde es descriu en eixe capítol. El Capítol 4 fa un resum de les àlgebres de Lie i del desenvolupament de Magnus que són les ferramentes algebraiques que permeten expressar la solució d'equacions diferencials dependents del temps.
L'equació lineal de Schrödinger amb potencial dependent del temps està examinada en el Capítol 5. Donat la seua estructura particular, nous mètodes quasi sense commutadors, basats en el desenvolupament de Magnus, són construïts. La seua eficiència és demostrada en uns experiments numèrics amb el model de Walker-Preston d'una molècula dins d'un camp electromagnètic.
En el Capítol 6 es dissenyen els mètodes de Magnus-escissió per a les equacions d'onda i de Hill. El seu rendiment està demostrat en els experiments numèrics amb diversos sistemes oscil·latoris: amb l'equació de Mathieu, l'ec. de Hill matricial, les equacions d'onda i de Klein-Gordon-Fock.
El Capítol 7 explica com l'enfocament algebraic i el desenvolupament de Magnus poden generalitzar-se als problemes no lineals. L'exemple utilitzat és el problema de Kepler amb massa decreixent.
El Capítol 8 conclou la tesi, ressenya els resultats i traça les possibles direccions de la investigació futura.[EN] The present thesis addresses the numerical integration of Hamiltonian systems with explicitly time-dependent potentials. These problems are common in mathematical physics because they come from quantum, classical and celestial mechanics.
The goal of the thesis is to construct integrators for several import ant non-autonomous problems: the Schrödinger equation, which is the cornerstone of quantum mechanics; the Hill and the wave equations, that describe oscillating systems; the Kepler problem with time-variant mass.
Chapter 1 describes the motivation and the aims of the work in the historical context of numerical integration. In Chapter 2 essential concepts and some fundamental tools used throughout the thesis are introduced.
The design of the proposed integrators is based on the composition and splitting methods and the Magnus expansion. In Chapter 3, the former is described. Their main idea is to recombine some simpler integrators to obtain the solution. The salient concept of order conditions is described in that chapter. Chapter 4 summarises Lie algebras and the Magnus expansion ¿ algebraic tools that help to express the solution of time-dependent differential equations.
The linear Schrödinger equation with time-dependent potential is considered in Chapter 5. Given its particular structure, new, Magnus-based quasi-commutator-free integrators are build. Their efficiency is shown in numerical experiments with the Walker-Preston model of a molecule in an electromagnetic field.
In Chapter 6, Magnus-splitting methods for the wave and the Hill equations are designed. Their performance is demonstrated in numerical experiments with various oscillatory systems: the Mathieu equation, the matrix Hill eq., the wave and the Klein-Gordon-Fock eq.
Chapter 7 shows how the algebraic approach and the Magnus expansion can be generalised to non-linear problems. The example used is the Kepler problem with decreasing mass.
The thesis is concluded by Chapter 8, in which the results are reviewed and possible directions of future work are outlined.Kopylov, N. (2019). Magnus-based geometric integrators for dynamical systems with time-dependent potentials [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/118798TESI
Explicit exponential Runge-Kutta methods for semilinear integro-differential equations
The aim of this paper is to construct and analyze explicit exponential
Runge-Kutta methods for the temporal discretization of linear and semilinear
integro-differential equations. By expanding the errors of the numerical method
in terms of the solution, we derive order conditions that form the basis of our
error bounds for integro-differential equations. The order conditions are
further used for constructing numerical methods. The convergence analysis is
performed in a Hilbert space setting, where the smoothing effect of the
resolvent family is heavily used. For the linear case, we derive the order
conditions for general order and prove convergence of order , whenever
these conditions are satisfied. In the semilinear case, we consider in addition
spatial discretization by a spectral Galerkin method, and we require locally
Lipschitz continuous nonlinearities. We derive the order conditions for orders
one and two, construct methods satisfying these conditions and prove their
convergence. Finally, some numerical experiments illustrating our theoretical
results are given
Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework
We analyze temporal approximation schemes based on overlapping domain
decompositions. As such schemes enable computations on parallel and distributed
hardware, they are commonly used when integrating large-scale parabolic
systems. Our analysis is conducted by first casting the domain decomposition
procedure into a variational framework based on weighted Sobolev spaces. The
time integration of a parabolic system can then be interpreted as an operator
splitting scheme applied to an abstract evolution equation governed by a
maximal dissipative vector field. By utilizing this abstract setting, we derive
an optimal temporal error analysis for the two most common choices of domain
decomposition based integrators. Namely, alternating direction implicit schemes
and additive splitting schemes of first and second order. For the standard
first-order additive splitting scheme we also extend the error analysis to
semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which
also contains numerical experiments. Version 3 and 4: Only comments added.
Version 2, page 2: Clarified statement on stability issues for ADI schemes
with more than two operator
Explicit Runge-Kutta algorithm to solve non-local equations with memory effects: case of the Maxey-Riley-Gatignol equation
A standard approach to solve ordinary differential equations, when they
describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such
schemes, however, are not applicable to the large class of equations which do
not constitute dynamical systems. In several physical systems, we encounter
integro-differential equations with memory terms where the time derivative of a
state variable at a given time depends on all past states of the system.
Secondly, there are equations whose solutions do not have well-defined Taylor
series expansion. The Maxey-Riley-Gatignol equation, which describes the
dynamics of an inertial particle in nonuniform and unsteady flow, displays both
challenges. We use it as a test bed to address the questions we raise, but our
method may be applied to all equations of this class. We show that the
Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system
which is constructed by introducing a new dynamical co-evolving state variable
that encodes memory of past states. We develop a Runge-Kutta algorithm for the
resultant Markovian system. The form of the kernels involved in deriving the
Runge-Kutta scheme necessitates the use of an expansion in powers of .
Our approach naturally inherits the benefits of standard time-integrators,
namely a constant memory storage cost, a linear growth of operational effort
with simulation time, and the ability to restart a simulation with the final
state as the new initial condition.Comment: 26 pages, 5 figures, 1 table (v2) Typos correcte
Effect of Random Parameter Switching on Commensurate Fractional Order Chaotic Systems
This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.The paper explores the effect of random parameter switching in a fractional order (FO) unified chaotic system which captures the dynamics of three popular sub-classes of chaotic systems i.e. Lorenz, Lu and Chen's family of attractors. The disappearance of chaos in such systems which rapidly switch from one family to the other has been investigated here for the commensurate FO scenario. Our simulation study show that a noise-like random variation in the key parameter of the unified chaotic system along with a gradual decrease in the commensurate FO is capable of suppressing the chaotic fluctuations much earlier than that with the fixed parameter one. The chaotic time series produced by such random parameter switching in nonlinear dynamical systems have been characterized using the largest Lyapunov exponent (LLE) and Shannon entropy. The effect of choosing different simulation techniques for random parameter FO switched chaotic systems have also been explored through two frequency domain and three time domain methods. Such a noise-like random switching mechanism could be useful for stabilization and control of chaotic oscillation in many real-world applications
Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations
We propose and study a class of numerical schemes to approximate time
fractional differential equations. The methods are based on the approximation
of the Caputo fractional derivative by continuous piecewise polynomials, which
is strongly related to the backward differentiation formulae for the
integer-order case. We investigate their theoretical properties, such as the
local truncation error and global error analyses with respect to a sufficiently
smooth solution, and the numerical stability in terms of the stability region
and -stability by refining the technique proposed in
\cite{LubichC:1986b}. Numerical experiments are given to verify the theoretical
investigations.Comment: 34 pages, 3 figure
Effect of spatial configuration of an extended nonlinear Kierstead-Slobodkin reaction-transport model with adaptive numerical scheme
In this paper, we consider the numerical simulations of an extended nonlinear form
of Kierstead-Slobodkin reaction-transport system in one and two dimensions. We
employ the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4)
schemes proposed by Cox and Matthew (J Comput Phys 176:430-455,
2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214-1233,
2005), for the time integration of spatially discretized partial differential equations. We demonstrate
the supremacy of ETDRK4 over the existing exponential time differencing integrators
that are of standard approaches and provide timings and error comparison. Numerical
results obtained in this paper have granted further insight to the question "What is the
minimal size of the spatial domain so that the population persists?" posed by Kierstead
and Slobodkin (J Mar Res 12:141-147,
1953
), with a conclusive remark that the popula-
tion size increases with the size of the domain. In attempt to examine the biological
wave phenomena of the solutions, we present the numerical results in both one- and
two-dimensional space, which have interesting ecological implications. Initial data and
parameter values were chosen to mimic some existing patternsScopus 201
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