A finite difference scheme for the two-dimensional sine-Gordon equation | ||
| Delta Journal of Science | ||
| Articles in Press, Accepted Manuscript, Available Online from 27 April 2025 | ||
| Document Type: Research and Reference | ||
| DOI: 10.21608/djs.2025.354889.1206 | ||
| Authors | ||
| A. A. Soliman; Manar M. Dahshan* ; Ahmed Saad Elgazzar | ||
| Mathematics Department, Faculty of Science, Arish University, Egypt | ||
| Abstract | ||
| The two-dimensional sine-Gordon equation is a fundamental aspect of nonlinear physics. It describes a wide range of phenomena in many fields. While its mathematical structure allows analytical solutions under certain conditions, the complexity of real-world applications often requires numerical methods. An accurate and efficient numerical solution enables a deeper understanding and advances applications in many fields. The finite difference method is a powerful numerical technique for solving partial differential equations. We present a finite difference scheme for the two-dimensional sine-Gordon equation. The stability and truncation error of the scheme are studied. We also present numerical simulations and error analysis to ensure the accuracy of the scheme. Numerical simulations show excellent agreement between the analytical and numerical solutions. The results of the error analysis show the convergence of the numerical solution to the analytical solution. These results are important for understanding the properties of the finite difference method, the two-dimensional sine-Gordon equation and its numerous applications. | ||
| Keywords | ||
| Sine-Gordon equation; Finite difference; Von Neumann stability; Local truncation error; Error analysis | ||
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