A new hybrid imbedded variable-step procedure for the numerical integration of the Schrödinger equation

AbstractA new imbedded variable-step procedure is developed for the numerical integration of the radial Schrödinger equation. The new imbedded method is based on P-stable methods of exponential order eight, ten, 12, and 14. Numerical results indicate that the new procedure is more efficient than similar variable-step procedures

A Runge-Kutta Fehlberg method with phase-lag of order infinity for initial-value problems with oscillating solution

AbstractA Runge-Kutta method is developed for the numerical solution of initial-value problems with oscillating solution. Based on the Runge-Kutta Fehlberg 2(3) method, a Runge-Kutta method with phase-lag of order infinity is developed. Based on these methods we produce a new embedded Runge-Kutta Fehlberg 2(3) method with phase-lag of order infinity. This method is called as Runge-Kutta Fehlberg Phase Fitted method (RKFPF). The numerical results indicate that this new method is much more efficient, compared with other well-known Runge-Kutta methods, for the numerical solution of differential equations with oscillating solution, using variable step size

Runge–Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics

AbstractIn this paper a new Runge–Kutta method with minimal dispersion and dissipation error is developed. The Chebyshev pseudospectral method is utilized using spatial discretization and a new fourth-order six-stage Runge–Kutta scheme is used for time advancing. The proposed scheme is more efficient than the existing ones for acoustic computations

A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde

A generator of high-order embedded P-stable methods for the numerical solution of the Schrödinger equation

AbstractA generator of new embedded P-stable methods of order 2n+2, where n is the number of layers used by the embedded methods, for the approximate numerical integration of the one-dimensional Schrödinger equation is developed in this paper. These new methods are called embedded methods because of a simple natural error control mechanism. Numerical results obtained for one-dimensional differential equations of the Schrödinger type show the validity of the developed theory

New modified Runge–Kutta–Nyström methods for the numerical integration of the Schrödinger equation

AbstractIn this work we construct new Runge–Kutta–Nyström methods especially designed to integrate exactly the test equation y″=−w2y. We modify two existing methods: the Runge–Kutta–Nyström methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrödinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential

On modified Runge–Kutta trees and methods

AbstractModified Runge–Kutta (mRK) methods can have interesting properties as their coefficients may depend on the step length. By a simple perturbation of very few coefficients we may produce various function-fitted methods and avoid the overload of evaluating all the coefficients in every step. It is known that, for Runge–Kutta methods, each order condition corresponds to a rooted tree. When we expand this theory to the case of mRK methods, some of the rooted trees produce additional trees, called mRK rooted trees, and so additional conditions of order. In this work we present the relative theory including a theorem for the generating function of these additional mRK trees and explain the procedure to determine the extra algebraic equations of condition generated for a major subcategory of these methods. Moreover, efficient symbolic codes are provided for the enumeration of the trees and the generation of the additional order conditions. Finally, phase-lag and phase-fitted properties are analyzed for this case and specific phase-fitted pairs of orders 8(6) and 6(5) are presented and tested

A Phase-Fitted Runge-Kutta-Nystr\"om method for the Numerical Solution of Initial Value Problems with Oscillating Solutions

A new Runge-Kutta-Nystr\"om method, with phase-lag of order infinity, for the integration of second-order periodic initial-value problems is developed in this paper. The new method is based on the Dormand and Prince Runge-Kutta-Nystr\"om method of algebraic order four\cite{pa}. Numerical illustrations indicate that the new method is much more efficient than the classical one.Comment: 10 page

Volume I. Introduction to DUNE

The preponderance of matter over antimatter in the early universe, the dynamics of the supernovae that produced the heavy elements necessary for life, and whether protons eventually decay—these mysteries at the forefront of particle physics and astrophysics are key to understanding the early evolution of our universe, its current state, and its eventual fate. The Deep Underground Neutrino Experiment (DUNE) is an international world-class experiment dedicated to addressing these questions as it searches for leptonic charge-parity symmetry violation, stands ready to capture supernova neutrino bursts, and seeks to observe nucleon decay as a signature of a grand unified theory underlying the standard model. The DUNE far detector technical design report (TDR) describes the DUNE physics program and the technical designs of the single- and dual-phase DUNE liquid argon TPC far detector modules. This TDR is intended to justify the technical choices for the far detector that flow down from the high-level physics goals through requirements at all levels of the Project. Volume I contains an executive summary that introduces the DUNE science program, the far detector and the strategy for its modular designs, and the organization and management of the Project. The remainder of Volume I provides more detail on the science program that drives the choice of detector technologies and on the technologies themselves. It also introduces the designs for the DUNE near detector and the DUNE computing model, for which DUNE is planning design reports. Volume II of this TDR describes DUNE\u27s physics program in detail. Volume III describes the technical coordination required for the far detector design, construction, installation, and integration, and its organizational structure. Volume IV describes the single-phase far detector technology. A planned Volume V will describe the dual-phase technology