A predictor-corrector scheme for the sine-Gordon equation.

*(English)*Zbl 0951.65089The authors consider an initial-boundary value problem involving a second-order nonlinear hyperbolic partial differential equation, known as sine-Gordon equation.

With reference to the numerical solution for this problem the authors use the method of lines approach to obtain an approximating system of second-order ordinary differential equations, use PadĂ© approximants for the recurrence relation satisfied by the solution of this approximating system and design a predictor-corrector algorithm. They apply their algorithm to a test equation to do a stability analysis and use their numerical results to demonstrate the efficiency and accuracy of the predictor-corrector method over some earlier methods.

With reference to the numerical solution for this problem the authors use the method of lines approach to obtain an approximating system of second-order ordinary differential equations, use PadĂ© approximants for the recurrence relation satisfied by the solution of this approximating system and design a predictor-corrector algorithm. They apply their algorithm to a test equation to do a stability analysis and use their numerical results to demonstrate the efficiency and accuracy of the predictor-corrector method over some earlier methods.

Reviewer: V.P.Tyagi (Bombay)

##### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

35Q53 | KdV equations (Korteweg-de Vries equations) |

##### Keywords:

numerical examples; initial-boundary value problem; second-order nonlinear hyperbolic partial differential equation; sine-Gordon equation; method of lines; predictor-corrector algorithm; stability
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\textit{A. Q. M. Khaliq} et al., Numer. Methods Partial Differ. Equations 16, No. 2, 133--146 (2000; Zbl 0951.65089)

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