An iterative starting method to control parasitism for the Leapfrog method |
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| Institution: | 1. Department of Physics, School of Basic and Applied Sciences, Central University of Tamilnadu, Thiruvarur 610 005, India;2. The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy;1. Department of Mathematics, Penza State University, 40, Krasnaya Str., Penza, 440026, Russia;2. Department of Engineering Science and Mechanics, The Pennsylvania State University, 203 B Earth and Engineering Science Building, University Park, PA 16802-1401, USA;3. St-Petersburg State University, 198504, St-Petersburg, 1 Ulianovskaya Str., Russia;1. CMCC, Universidad de La Frontera, Temuco, Chile;2. Department of Mathematics and Natural Sciences, Institute of Mathematics, University of Kassel, Germany;1. Departamento de Matemáticas, ESFM, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos Edificio 9, 07738 Cd. de México, Mexico;2. Posgrado en Ciencias Fisicomatemáticas, ESFM, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos Edificio 9, 07738 Cd. de México, Mexico |
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| Abstract: | The Leapfrog method is a time-symmetric multistep method, widely used to solve the Euler equations and other Hamiltonian systems, due to its low cost and geometric properties. A drawback with Leapfrog is that it suffers from parasitism. This paper describes an iterative starting method, which may be used to reduce to machine precision the size of the parasitic components in the numerical solution at the start of the computation. The severity of parasitic growth is also a function of the differential equation, the main method and the time-step. When the tendency to parasitic growth is relatively mild, computational results indicate that using this iterative starting method may significantly increase the time-scale over which parasitic effects remain acceptably small. Using an iterative starting method, Leapfrog is applied to the cubic Schrödinger equation. The computational results show that the Hamiltonian and soliton behaviour are well-preserved over long time-scales. |
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| Keywords: | Parasitism Starting method Leapfrog Cubic Schrödinger equation |
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