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

Explicit Almost P-Stable Runge-Kutta-Nyström Methods for the Numerical Solution of the Two-Body Problem

The Publisher's final version can be found by following the DOI link.In this paper, three families of explicit Runge–Kutta–Nyström methods with three stages and third algebraic order are presented. Each family consists of one method with constant coefficients and one corresponding optimized “almost” P-stable method with variable coefficients, zero phase-lag and zero amplification error. The firstmethod with constant coefficients is new, while the second and third have been constructed by Chawla and Sharma. The newmethod with constant coefficients, constructed in this paper has larger interval of stability than the two methods of Chawla and Sharma. Furthermore, the optimized methods possess an infinite interval of periodicity, excluding some discrete values, while being explicit, which is a very desired combination. The preservation of the algebraic order is examined, local truncation error and stability/periodicity analysis are performed and the efficiency of the new methods is measured via the integration of the two-body problem

A 6(4) optimized embedded Runge–Kutta–Nyström pair for the numerical solution of periodic problems

The Publisher's final version can be found by following the DOI link.In this paper an optimization of the non-FSAL embedded RKN 6(4) pair with six stages of Moawwad El-Mikkawy, El-Desouky Rahmo is presented. The new method is derived after applying phase-fitting and amplification-fitting and has variable coefficients. The preservation of the algebraic order is verified and the principal term of the local truncation error is evaluated. Furthermore, periodicity analysis is performed, which reveals that the new method is ‘‘almost’’ P-stable. The efficiency of the new method is measured via the integration of several initial value problems