Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

open access articleMotivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge-Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

On The Component Analysis and Transformation of An Explicit Fourth – Stage Fourth – Order Runge – Kutta Methods

This work is designed to transform the fourth stage – fourth order explicit Runge-Kutta method with the aim of projecting a new method of implementing it through tree diagram analysis. Efforts will be made to represent the equations derived from the y derivatives and x,y derivatives separately on Butcher’s rooted trees. This idea is derivable from general graphs and combinatorics. Key words: Rooted tree diagram, Transformation, Vertex, explicit, y partial derivatives, x,y partial derivatives, Runge-Kutta Methods, Linear and non- linear equations, Taylor series, Graphs. Keywords: key words, Runge - Kutta, Combinatorics, Derivatives, Transformation, Analysi

Efficient Computation of the Nonlinear Schrödinger Equation with Time-Dependent Coefficients

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and stability characteristics. Additionally, we develop a modified step size control algorithm, which increases the efficiency of our pair and all other pairs included in the numerical experiments. The numerical results on the nonlinear Schrödinger equation with a periodic solution verified the superiority of the new algorithm in terms of efficiency. The new method also presents a good behaviour of the maximum absolute error and the global norm in time, even after a high number of oscillations

A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This algorithm extends the classical Feynman-Kac formula to fully nonlinear partial differential equations, by using random trees that carry information on nonlinearities on their branches. It applies to functional, non-polynomial nonlinearities that are not treated by standard branching arguments, and deals with derivative terms of arbitrary orders. A Monte Carlo numerical implementation is provided

A Parametric Method Optimised for the Solution of the (2+1)-Dimensional Nonlinear Schrödinger Equation

open access articleWe investigate the numerical solution of the nonlinear Schrödinger equation in two spatial dimensions and one temporal dimension. We develop a parametric Runge–Kutta method with four of their coefficients considered as free parameters, and we provide the full process of constructing the method and the explicit formulas of all other coefficients. Consequently, we produce an adaptable method with four degrees of freedom, which permit further optimisation. In fact, with this methodology, we produce a family of methods, each of which can be tailored to a specific problem. We then optimise the new parametric method to obtain an optimal Runge–Kutta method that performs efficiently for the nonlinear Schrödinger equation. We perform a stability analysis, and utilise an exact dark soliton solution to measure the global error and mass error of the new method with and without the use of finite difference schemes for the spatial semi-discretisation. We also compare the efficiency of the new method and other numerical integrators, in terms of accuracy versus computational cost, revealing the superiority of the new method. The proposed methodology is general and can be applied to a variety of problems, without being limited to linear problems or problems with oscillatory/periodic solutions

A Rooted-Tree Based Derivation of ROW-Type Methods with Non-Exact Jacobian Entries for Index-One DAEs

Solving differential-algebraic equations (DAEs) efficiently by means of appropriate numerical schemes for time-integration is an ongoing topic in applied mathematics. In this context, especially when considering large systems that occur with respect to many fields of practical application effective computation becomes relevant. In particular, corresponding examples are given when having to simulate network structures that consider transport of fluid and gas or electrical circuits. Due to the stiffness properties of DAEs, time-integration of such problems generally demands for implicit strategies. Among the schemes that prove to be an adequate choice are linearly implicit Rung-Kutta methods in the form of Rosenbrock-Wanner (ROW) schemes. Compared to fully implicit methods, they are easy to implement and avoid the solution of non-linear equations by including Jacobian information within their formulation. However, Jacobian calculations are a costly operation. Hence, necessity of having to compute the exact Jacobian with every successful time-step proves to be a considerable drawback. To overcome this drawback, a ROW-type method is introduced that allows for non-exact Jacobian entries when solving semi-explicit DAEs of index one. The resulting scheme thus enables to exploit several strategies for saving computational effort. Examples include using partial explicit integration of non-stiff components, utilizing more advantageous sparse Jacobian structures or making use of time-lagged Jacobian information. In fact, due to the property of allowing for non-exact Jacobian expressions, the given scheme can be interpreted as a generalized ROW-type method for DAEs. This is because it covers many different ROW-type schemes known from literature. To derive the order conditions of the ROW-type method introduced, a theory is developed that allows to identify occurring differentials and coefficients graphically by means of rooted trees. Rooted trees for describing numerical methods were originally introduced by J.C. Butcher. They significantly simplify the determination and definition of relevant characteristics because they allow for applying straightforward procedures. In fact, the theory presented combines strategies used to represent ROW-type methods with exact Jacobian for DAEs and ROW-type methods with non-exact Jacobian for ODEs. For this purpose, new types of vertices are considered in order to describe occurring non-exact elementary differentials completely. The resulting theory thus automatically comprises relevant approaches known from literature. As a consequence, it allows to recognize order conditions of familiar methods covered and to identify new conditions. With the theory developed, new sets of coefficients are derived that allow to realize the ROW-type method introduced up to orders two and three. Some of them are constructed based on methods known from literature that satisfy additional conditions for the purpose of avoiding effects of order reduction. It is shown that these methods can be improved by means of the new order conditions derived without having to increase the number of internal stages. Convergence of the resulting methods is analyzed with respect to several academic test problems. Results verify the theory determined and the order conditions found as only schemes satisfying the order conditions predicted preserve their order when using non-exact Jacobian expressions.Die effiziente Lösung differential-algebraischer Gleichungen (DAEs) mittels geeigneter numerischer Verfahren zur zeitlichen Integration ist ein anhaltendes Thema in der angewandten Mathematik. In diesem Zusammenhang wird eine effektive Berechnung insbesondere im Fall zu betrachtender großer Systeme relevant, die in zahlreichen Feldern der praktischen Anwendung vorkommen. Beispiele hierfür ergeben sich vor allem bezüglich der Simulation von Netzwerk-Strukturen, die den Transport von Fluiden und Gasen oder elektrische Schaltungen betrachten. Bedingt durch die Steifheits-Eigenschaften von DAEs erfordert die Zeitintegration solcher Probleme grundsätzlich implizite Methoden. Zu den Verfahren die sich als eine geeignete Wahl erweisen zählen linear-implizite Runge-Kutta Methoden in der Form von Rosenbrock-Wanner (ROW) Verfahren. Im Vergleich zu voll-impliziten Methoden sind sie einfach zu implementieren und vermeiden eine Lösung nicht-linearer Gleichungen, indem sie die Jacobi-Matrix in ihrer Formulierung berücksichtigen. Allerdings ist die Berechnung der Jacobi-Matrix eine teure Operation. Daher erweist sich die Notwendigkeit der Ermittlung der exakten Jacobi-Matrix mit jedem erfolgreichen Zeitschritt als ein großer Nachteil. Um diesem Nachteil entgegen zu wirken wird ein ROW-Typ Verfahren vorgestellt, das für die Berechnung semi-expliziter DAEs vom Index eins die Verwendung nicht-exakter Einträge in der Jacobi-Matrix erlaubt. Das resultierende Verfahren ermöglicht es somit verschiedene Strategien zur Reduzierung des Rechenaufwands auszunutzen. Hierzu zählt unter anderem die Verwendung partieller expliziter Integration nicht-steifer Anteile, der Einsatz vorteilhafterer dünn besetzter Strukturen der Jacobi-Matrix oder die Nutzung zeitverzögerter Informationen. In der Tat kann das beschriebene Verfahren aufgrund der Eigenschaft einer Betrachtung nicht-exakter Jacobi-Matrizen als eine verallgemeinerte ROW-Typ Methode für DAEs interpretiert werden. Dies ist darauf zurück zu führen, dass es zahlreiche verschiedene, aus der Literatur bekannte ROW-Typ Verfahren beinhaltet. Um die Ordnungsbedingungen der vorgestellten ROW-Typ Methode herzuleiten wird eine Theorie entwickelt, die eine grafische Identifizierung auftretender Differentiale und Koeffizienten mittels Wurzelbäume erlaubt. Wurzelbäume zur Beschreibung numerischer Methoden wurden ursprünglich von J.C. Butcher eingeführt. Sie vereinfachen die Bestimmung und Definition relevanter Eigenschaften erheblich, weil sie die Anwendung unkomplizierter Prozeduren ermöglichen. In der Tat vereint die vorgestellte Theorie Strategien, die zur Darstellung von ROW-Typ Methoden mit exakter Jacobi-Matrix für DAEs und ROW-Typ Methoden mit nicht-exakter Jacobi-Matrix für ODEs geläufig sind. Zu diesem Zweck werden neue Knotentypen berücksichtigt um auftretende nicht-exakte Differentiale vollständig zu beschreiben. Die resultierende Theorie umfasst somit automatisch relevante, aus der Literatur bekannte Ansätze. In der Folge ermöglicht sie es Ordnungsbedingungen enthaltener bekannter Methoden zu erkennen und neue Bedingungen zu ermitteln. Mit der entwickelten Theorie werden neue Koeffizientensätze hergeleitet, die es erlauben die vorgestellte ROW-Typ Methode bis zur Ordnung zwei und drei zu realisieren. Einige von ihnen sind auf Basis von aus der Literatur bekannten Methoden konstruiert, die Zusatzbedingungen zum Zweck der Vermeidung von Effekten der Ordnungsreduktion erfüllen. Es wird gezeigt, dass diese Methoden mittels der neu hergeleiteten Ordnungsbedingungen verbessert werden können ohne die Anzahl interner Stufen erhöhen zu müssen. Die Konvergenz der resultierenden Methoden wird bezüglich verschiedener akademischer Testprobleme analysiert. Die Ergebnisse bestätigen die ermittelte Theorie und die gefundenen Ordnungsbedingungen, da nur jene Verfahren die Ordnung unter Betrachtung nicht-exakter Jacobi-Matrizen erhalten, welche die prognostizierten Ordnungsbedingungen erfüllen

Multi-Value Numerical Modeling for Special Di erential Problems

2013 - 2014The subject of this thesis is the analysis and development of new numerical methods for Ordinary Di erential Equations (ODEs). This studies are motivated by the fundamental role that ODEs play in applied mathematics and applied sciences in general. In particular, as is well known, ODEs are successfully used to describe phenomena evolving in time, but it is often very di cult or even impossible to nd a solution in closed form, since a general formula for the exact solution has never been found, apart from special cases. The most important cases in the applications are systems of ODEs, whose exact solution is even harder to nd; then the role played by numerical integrators for ODEs is fundamental to many applied scientists. It is probably impossible to count all the scienti c papers that made use of numerical integrators during the last century and this is enough to recognize the importance of them in the progress of modern science. Moreover, in modern research, models keep getting more complicated, in order to catch more and more peculiarities of the physical systems they describe, thus it is crucial to keep improving numerical integrator's e ciency and accuracy. The rst, simpler and most famous numerical integrator was introduced by Euler in 1768 and it is nowadays still used very often in many situations, especially in educational settings because of its immediacy, but also in the practical integration of simple and well-behaved systems of ODEs. Since that time, many mathematicians and applied scientists devoted their time to the research of new and more e cient methods (in terms of accuracy and computational cost). The development of numerical integrators followed both the scienti c interests and the technological progress of the ages during whom they were developed. In XIX century, when most of the calculations were executed by hand or at most with mechanical calculators, Adams and Bashfort introduced the rst linear multistep methods (1855) and the rst Runge- Kutta methods appeared (1895-1905) due to the early works of Carl Runge and Martin Kutta. Both multistep and Runge-Kutta methods generated an incredible amount of research and of great results, providing a great understanding of them and making them very reliable in the numerical integration of a large number of practical problems. It was only with the advent of the rst electronic computers that the computational cost started to be a less crucial problem and the research e orts started to move towards the development of problem-oriented methods. It is probably possible to say that the rst class of problems that needed an ad-hoc numerical treatment was that of sti problems. These problems require highly stable numerical integrators (see Section ??) or, in the worst cases, a reformulation of the problem itself. Crucial contributions to the theory of numerical integrators for ODEs were given in the XX century by J.C. Butcher, who developed a theory of order for Runge-Kutta methods based on rooted trees and introduced the family of General Linear Methods together with K. Burrage, that uni ed all the known families of methods for rst order ODEs under a single formulation. General Linear Methods are multistagemultivalue methods that combine the characteristics of Runge-Kutta and Linear Multistep integrators... [edited by Author]XIII n.s

Analysis of Hamiltonian boundary value problems and symplectic integration: a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatu, New Zealand

Listed in 2020 Dean's List of Exceptional ThesesCopyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author.Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for $A$-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers' equation, KdV, fluid equations,...) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed