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**A family of parametric finite-difference methods for the solution of the sine-Gordon equation.**
*(English)*
Zbl 0943.65102

Summary: A family of finite difference methods is used to transform the initial-boundary value problem associated with the nonlinear hyperbolic sine-Gordon equation, into a linear algebraic system. Numerical methods are developed by replacing the time and space derivatives by finite difference approximants. The resulting finite difference methods are analyzed for local truncation errors, stability and convergence. The results of a number of numerical experiments are given.

### MSC:

65M06 | Finite difference methods 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 |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |

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

### Keywords:

error bounds; soliton wave; finite difference methods; nonlinear hyperbolic sine-Gordon equation; stability; convergence; numerical experiments
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\textit{A. G. Bratsos} and \textit{E. H. Twizell}, Appl. Math. Comput. 93, No. 2--3, 117--137 (1998; Zbl 0943.65102)

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### References:

[1] | Bratsos, A. G., Numerical Solutions of Nonlinear Partial Differential Equations, (Ph.D. Thesis (1993), Brunel University) · Zbl 1136.65078 |

[2] | Bratsos, A. G.; Twizell, E. H., The numerical solution of the sine Gordon equation using the method of lines, Int. J. Comput. Math., 61, 271-292 (1998) · Zbl 1001.65512 |

[3] | Argyris, J.; Haase, M., An engineer’s guide to solution phenomena: application of the finite element method, (Computer Method in Applied Mechanics and Engineering, vol. 61 (1987), North-Holland: North-Holland Amsterdam), 71-122 · Zbl 0624.76020 |

[4] | Evans, D. J.; Roomi, A. S., The numerical solution of the sine Gordon partial differential equation by the AGE method, Int. J. Comput. Math., 37, 79-88 (1990) · Zbl 0732.65106 |

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