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Fourth-order compact scheme for the one-dimensional sine-Gordon equation. (English) Zbl 1175.65093

The author discusses a finite difference scheme for a generalized one-dimensional sine-Gordon equation. After in the space direction, the sine-Gordon equation is transformed using the central difference quotient into an initial-value problem of a second order system of ordinary differential equation. In the time direction the Padé approximant is used. The resulting fully discrete nonlinear finite difference equation in solved by a predictor corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in the proposed method. A stability analysis and an error estimate are given for the homogeneous Dirichlet boundary value problem using the energy method. The effectiveness of this approach is illustrated by a numerical example.

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

Software:

PDE2D
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References:

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