Numerical simulation of the N-dimensional sine-Gordon equation via operational matrices |
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| Authors: | Francisco de la Hoz Fernando Vadillo |
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| Affiliation: | 1. University of the Basque Country, Departamento de Matemática Aplicada, Plaza de La Casilla 3, 48012 Bilbao, Spain;2. University of the Basque Country, Departamento de Matemática Aplicada y Estadística e Investigación Operativa, Apdo. 644, 48080 Bilbao, Spain;1. Atomistic Modelling and Simulation, ICAMS, Ruhr-Universität Bochum, D-44801 Bochum, Germany;2. Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom;3. High-Performance Computing in Materials Science, ICAMS, Ruhr-Universität Bochum, D-44801 Bochum, Germany;4. Department of Physics, Applied Physics and Astronomy, Binghamton University, State University of New York, Vestal, NY 13850, USA;1. DEN-Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France;2. Université Paris-Est, Cermics (ENPC), F-77455 Marne-la-Vallée Cedex 2, France;3. Laboratoire Navier, Ecole des Ponts ParisTech, F-77455 Marne-la-Vallée Cedex 2, France;4. MATHERIALS project-team, INRIA Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France;1. Supercomputing Center, Computer Network Information Center, Chinese Academy of Sciences, Beijing, China;2. University of Chinese Academy of Sciences, Beijing, China;3. Material Science Division, Lawrence Berkeley National Laboratory, Berkeley, USA;1. Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia;2. Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, 3 Nobel St., Moscow 143026, Russia;3. Northwestern Polytechnical University, School of Materials Science and Engineering, Xi’an 710072, China;4. International Center for Materials Design, Northwestern Polytechnical University, Xi’an, 710072, China |
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| Abstract: | In this paper, we develop a numerical method for the N-dimensional sine-Gordon equation using differentiation matrices, in the theoretical frame of matrix differential equations.Our method avoids calculating exponential matrices, is very intuitive and easy to express, and can be implemented without toil in any number of spatial dimensions. Although there is currently a vast literature on the numerical treatment of the one-dimensional sine-Gordon equation, the references for the two-dimensional case are much sparser, and virtually nonexistent for higher dimensions.We apply it to a battery of two-dimensional problems taken from the literature, showing that it largely outperforms the previously existing algorithms; while for three-dimensional problems, the results seem very promising. |
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