High-order solution of one-dimensional sine–Gordon equation using compact finite difference and DIRKN methods

In this work we propose a high-order and accurate method for solving the one-dimensional nonlinear sine–Gordon equation. The proposed method is based on applying a compact finite difference scheme and the diagonally implicit Runge–Kutta–Nyström (DIRKN) method for spatial and temporal components, res...

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Bibliographic Details
Published in:Mathematical and computer modelling Vol. 51; no. 5; pp. 537 - 549
Main Authors: Mohebbi, Akbar, Dehghan, Mehdi
Format: Journal Article
Language:English
Published: Kidlington Elsevier Ltd 01.03.2010
Elsevier
Subjects:
Compact finite difference
DIRKN methods
Exact sciences and technology
High accuracy
Mathematical analysis
Mathematics
Measure and integration
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Sciences and techniques of general use
Sine–Gordon equation
High accuracy
Sine–Gordon equation
DIRKN methods
Compact finite difference
Second order equation
Transcendental equation
Integration
Differential equation
Runge Kutta method
Non linear system
Algorithm
Equation system
Non linear equation
Extrapolation
Numerical analysis
Multistep method
Applied mathematics
Sine-Gordon equation
Algebraic equation
Mathematical model
Computer aided analysis
Finite difference method
ISSN:0895-7177, 1872-9479
Online Access:Get full text
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Summary:In this work we propose a high-order and accurate method for solving the one-dimensional nonlinear sine–Gordon equation. The proposed method is based on applying a compact finite difference scheme and the diagonally implicit Runge–Kutta–Nyström (DIRKN) method for spatial and temporal components, respectively. We apply a compact finite difference approximation of fourth order for discretizing the spatial derivative and a fourth-order A -stable DIRKN method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables and is unconditionally stable. The results of numerical experiments show that the combination of a compact finite difference approximation of fourth order and a fourth-order A -stable DIRKN method gives an efficient algorithm for solving the one-dimensional sine–Gordon equation.
ISSN:0895-7177
1872-9479
DOI:10.1016/j.mcm.2009.11.015