A study on the efficiency and stability of high-order numerical methods for Form-II and Form-III of the nonlinear Klein–Gordon equations

A. H. Encinas (), V. Gayoso-Martínez (), A. Martín del Rey (), J. Martín-Vaquero () and A. Queiruga-Dios ()
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A. H. Encinas: University of Salamanca, Salamanca, E37008, Spain
V. Gayoso-Martínez: Institute of Physical and Information Technologies (ITEFI), Spanish National Research Council (CSIC), Madrid, Spain
A. Martín del Rey: University of Salamanca, Salamanca, E37008, Spain
J. Martín-Vaquero: University of Salamanca, Salamanca, E37008, Spain
A. Queiruga-Dios: University of Salamanca, Salamanca, E37008, Spain

International Journal of Modern Physics C (IJMPC), 2016, vol. 27, issue 09, 1-18

Abstract: In this paper, we discuss the problem of solving nonlinear Klein–Gordon equations (KGEs), which are especially useful to model nonlinear phenomena. In order to obtain more exact solutions, we have derived different fourth- and sixth-order, stable explicit and implicit finite difference schemes for some of the best known nonlinear KGEs. These new higher-order methods allow a reduction in the number of nodes, which is necessary to solve multi-dimensional KGEs. Moreover, we describe how higher-order stable algorithms can be constructed in a similar way following the proposed procedures. For the considered equations, the stability and consistency of the proposed schemes are studied under certain smoothness conditions of the solutions. In addition to that, we present experimental results obtained from numerical methods that illustrate the efficiency of the new algorithms, their stability, and their convergence rate.

Keywords: Klein–Gordon equations; fast convergence; finite difference methods; numerical methods; stability (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1142/S0129183116500972

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