An implicit Lie-group iterative scheme for solving the nonlinear Klein-Gordon and sine-Gordon equations. (English) Zbl 1446.65125
Summary: In this article, the nonlinear Klein-Gordon and sine-Gordon equations are solved by pondering the semi-discretization numerical schemes and then, the resulting ordinary differential equations at the discretized spaces are numerically integrated toward the time direction by using the implicit Lie-group iterative method to find the unknown physical quantity. When six numerical experiments are examined, we reveal that the present implicit Lie-group iterative scheme is applicable to the nonlinear Klein-Gordon and sine-Gordon equations and convergent very fast at each time marching step, and the accuracy is raised several orders, of which the numerical results are rather accurate, effective and stable.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 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 |
Keywords:
nonlinear Klein-Gordon equation; sine-Gordon equation; nonlinear wave problem; implicit Lie-group iterative scheme; group preserving scheme (GPS)References:
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