A finite difference scheme for the two-dimensional sine-Gordon equation

The sine-Gordon (SG) equation is a fundamental aspect of nonlinear physics. It models a wide range of phenomena in many scientific fields. While its mathematical structure allows analytical solutions under certain conditions, the complexity of real-world applications often requires numerical methods. Accurate and efficient numerical solutions enable a deeper understanding and advances applications in many fields. This study presents a finite difference scheme for the SG equation in two dimensions. Both the local truncation error and stability of the scheme are studied. We also present numerical simulations and error analysis to ensure the accuracy of the scheme.