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

A two-step trigonometrically fitted semi-implicit hybrid method for solving special second order oscillatory differential equation

In this paper, we derived a semi-implicit hybrid method (SIHM) which is a two-step method to solve special second order ordinary differential equations (ODEs). The SIHM which is three-stage and fourth-order is then trigonometrically fitted and denoted by TF-SIHM3(4). The method is constructed using trigonometrically fitted properties instead of using phase-lag and amplification properties. Numerical integration show that TF-SIHM3(4) is more accurate in term of accuracy compared to the existing explicit and implicit methods of the same order

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

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

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

Two point block multistep methods with trigonometric−fitting for solving oscillatory problems

In this paper, we present the absolute stability of the existing 2-point implicit block multistep step methods of step number k = 3 and k = 5 and solving special second order ordinary differential equations (ODEs). The methods are then trigonometrically fitted so that they are suitable for solving highly oscillatory problems arising from the special second order ODEs. Their explicit counterparts are also trigonometrically fitted so that in the implementation the methods can act as a predictor-corrector pairs. The numerical results based on the integration over a large interval are given to show the performance of the proposed methods. From the numerical results we can conclude that the new trigonometrically-fitted methods are superior in terms of accuracy and execution time, compared to the existing methods in the scientific literature when used for solving problems which are oscillatory in nature

Exploring efficient: numerical methods for differential equations

Numerical analysis is a way to do higher mathematical problems on a computer, a technique widely used by scientists and engineers to solve their problems. A major advantage of numerical analysis is that a numerical answer can be obtained even when a problem has no “analytical” solution. Results from numerical analysis are an approximation, which can be made as accurate as desired. The analysis of errors in numerical methods is a critically important part of the study of numerical analysis. Hence, we will see in this research that computation of the error is a must as it is a way to measure the efficiency of the numerical methods developed. Numerical methods require highly tedious and repetitive computations that can only be done using the computer. Hence in this research, it is shown that computer programs must be written for the implementation of numerical methods. In the early part of related research the computer language used was Fortran. Subsequently more and more computer programs used the C programming language. Additionally, now computations can also be carried out using softwares like MATLAB, MATHEMATICA and MAPLE. Many physical problems that arise from ordinary differential equations (ODEs) have magnitudes of eigenvalues which vary greatly, and such systems are commonly known as stiff systems. Stiff systems usually consist of a transient solution, that is, a solution which varies rapidly at the beginning of the integration. This phase is referred to as the transient phase and during this phase, accuracy rather than stability restricts the stepsize of the numerical methods used. Thus the generally the structure of the solutions suggests application of specific methods for non-stiff equations in the transient phase and specific methods for stiff equations during the steady-state phase in a manner whereby computational costs can be reduced. Consequently, in this research we developed embedded Runge-Kutta methods for solving stiff differential equations so that variable stepsize codes can be used in its implementation. We have also included intervalwise partitioning, whereby the system is considered as non-stiff first, and solved using the method with simple iterations, and once stiffness is detected, the system is solved using the same method, but with Newton iterations. By using variable stepsize code and intervalwise partitioning, we have been able to reduce the computational costs. With the aim of increasing the computational efficiency of the Runge-Kutta methods, we have also developed methods of higher order with less number of stages or function evaluations. The method used is an extension of the classical Runge-Kutta method and the approximation at the current point is based on the information at the current internal stage as well as the previous internal stage. This is the idea underlying the construction of Improved Runge-Kutta methods, so that the resulting method will give better accuracy. Usually higher order ordinary differential equations are solved by converting them into a system of first order ODEs and using numerical methods suitable for first order ODEs. However it is more efficient, in terms of accuracy, number of function evaluations as well as computational time, if the higher order ODEs can be solved directly (without being converted to a system of first order ODEs), using numerical methods. In this research we developed numerical methods, particularly Runge-Kutta type methods, which can directly solve special third order and fourth order ODEs. Special second order ODE is an ODE which does not depend on the first derivative. The solution from this type of ODE often exhibits a pronounced oscillatory character. It is well known that it is difficult to obtain accurate numerical results if the ODEs are oscillatory in nature. In order to address this problem a lot of research has been focused on developing methods which have high algebraic order, reduced phase-lag or dispersion and reduced dissipation. Phaselag is the angle between the true and approximate solution, while dissipation is the difference between the approximate solution and the standard cyclic solution. If a method has high algebraic order, high order of dispersion and dissipation, then the numerical solutions obtained will be very accurate. Hence in this research we have developed numerical methods, specifically hybrid methods which have all the above mentioned properties. If the solutions are oscillatory in nature, it means that the solutions will have components which are trigonometric functions, that is, sine and cosine functions. In order to get accurate numerical solutions we thus phase-fitted the methods using trigonometric functions. In this research, it is proven that trigonometrically-fitting the hybrid methods and applying them to solve oscillatory delay differential equations result in better numerical results. These are the highlights of my research journey, though a lot of work has also been done in developing numerical methods which are multistep in nature, for solving higher order ODEs, as well as implementation of methods developed for solving fuzzy differential equations and partial differential equations, which are not covered here

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

The Use of Phase Lag and Amplification Error Derivatives for the Construction of a Modified Runge-Kutta-Nyström Method

A new modified Runge-Kutta-Nyström method of fourth algebraic order is developed. The new modified RKN method is based on the fitting of the coefficients, due to the nullification not only of the phase lag and of the amplification error, but also of their derivatives. Numerical results indicate that the new modified method is much more efficient than other methods derived for solving numerically the Schrödinger equation