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

An Optimized Runge-Kutta Method for the Numerical Solution of the Radial Schrödinger Equation

An optimized explicit modified Runge-Kutta (RK) method for the numerical integration of the radial Schrödinger equation is presented in this paper. This method has frequency-depending coefficients with vanishing dispersion, dissipation, and the first derivative of dispersion. Stability and phase analysis of the new method are examined. The numerical results in the integration of the radial Schrödinger equation with the Woods-Saxon potential are reported to show the high efficiency of the new method

A trigonometrically adapted embedded pair of explicit Runge-Kutta-Nyström methods to solve periodic systems.

[EN]In this paper a 5(3) pair of explicit trigonometrically adapted Runge-Kutta-Nystr om methods with four stages is derived based on an explicit pair appeared in the literature. The new adapted method is able to integrate exactly the usual test equation: y′′ = -w^2y. The local truncation error of the new method is obtained, proving that the algebraic order of convergence is maintained. The stability interval of the new method is obtained, showing that the proposed method is absolutely stable. The numerical experiments performed demonstrate the robustness of the new embedded pair in comparison with some standard codes available in the literature

A New Two Derivative FSAL Runge-Kutta Method of Order Five in Four Stages

              المشتقة الثانية طريقة رنك-كوتا الفعالة الجديدة من الرتبة الخامسة  (TDRK) قد تم تطويرها من أجل الحل العددي للمعادلات التفاضلية الاعتيادية من الرتبة الأولى (ODEs). تم اشتقاق الطريقة الجديدة باستخدام خاصية الأول  نفس الأخير  (FSAL) . قمنا بتحليل استقرار الطريقة. تم عرض النتائج العددية لتوضيح كفاءة الطريقة الجديدة بالمقارنة مع بعض طرق رنك-كوتا (RK) المعروفة.A new efficient Two Derivative Runge-Kutta method (TDRK) of order five is developed for the numerical solution of the special first order ordinary differential equations (ODEs). The new method is derived using the property of First Same As Last (FSAL). We analyzed the stability of our method. The numerical results are presented to illustrate the efficiency of the new method in comparison with some well-known RK methods

Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods

High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation

In this work we introduce a new family of twelve-step linear multistep methods for the integration of the Schr\"odinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications.Comment: 36 pages, 6 figure