Some procedures for the construction of high-order exponentially fitted Runge–Kutta–Nyström methods of explicit type

Abstract

The construction of high-order exponentially fitted Runge–Kutta–Nyström (EFRKN) methods of explicit type for the numerical solution of oscillatory differential systems is analyzed. Based on two basic symmetric and symplectic EFRKN methods of reference we present two procedures for constructing high-order explicit methods. The first procedure is based on composition methods and it allows the construction of high-order explicit EFRKN methods which are symmetric and symplectic. The second procedure is based on combining different EFRKN methods in order to construct embedded pairs of explicit parallel EFRKN methods which can be implemented in variable-step codes without additional cost. The numerical experiments carried out show the qualitative behavior and the efficiency of the new EFRKN methods when they are compared with some standard methods proposed in the scientific literature for solving second-order nonstiff differential systems. Program summaryProgram title: SVI-IIEXPOreferee.for and SVI-IIvarreferee.forCatalogue identifier: AEOO_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEOO_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public LicenseNo. of lines in distributed program, including test data, etc.: 2527No. of bytes in distributed program, including test data, etc.: 107433Distribution format: tar.gzProgramming language: Fortran 77.Computer: Standard PC.Operating system: Windows. It might work with others. Successfully tested by CPC on Linux.RAM: For the test problems used less than 1 MB.Classification: 4.3, 4.12, 16.3, 17.17.Nature of problem:Some models in astronomy and astrophysics, quantum mechanics and nuclear physics lead to second-order oscillatory differential systems. The solution of these oscillatory models requires accurate and efficient numerical methods. The codes SVI-IIEXPOreferee.for and SVI-IIvarreferee.for were developed for this purpose.Solution method:We propose high-order exponentially fitted Runge–Kutta-Nyström (EFRKN) methods of explicit type for solving second-order oscillatory models. The code SVI-IIEXPOreferee.for contains symmetric and symplectic EFRKN methods with orders six and eight, and it runs with fixed step. The variable-step code SVI-IIvarreferee.for contains embedded pairs of EFRKN methods with orders 8(6), 10(8) and 12(10).Restrictions:The codes are designed for solving second-order oscillatory differential systems with 200 equations. If the number of differential equations is greater than 200, then the dimension must be increased in the program.Additional comments:The codes require the fitting frequency of the problem.Running time:The test problems used require only a few seconds to run.