Tension spline solution of nonlinear sine-Gordon equation

Authors:
Jalil Rashidinia;Reza Mohammadi
Affiliations:
School of Mathematics, Iran University of Science and Technology, Tehran, Iran 1684613114;School of Mathematics, Iran University of Science and Technology, Tehran, Iran 1684613114 and Department of Mathematics, University of Neyshabour, Neyshabour, Iran
Venue:
Numerical Algorithms
Year:
2011

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Abstract

The sine-Gordon equation plays an important role in modern physics. By using spline function approximation, two implicit finite difference schemes are developed for the numerical solution of one-dimensional sine-Gordon equation. Stability analysis of the method has been given. It has been shown that by choosing the parameters suitably, we can obtain two schemes of orders $\mathcal{O}(k^{2}+k^{2}h^{2}+h^{2})$ and $\mathcal{O}(k^{2}+k^{2}h^{2}+h^{4})$ . At the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.