Tension spline approach for the numerical solution of nonlinear Klein–Gordon equation

The nonlinear Klein–Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein–Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k2+k2h2+h2)O(k2+k2h2+h2) and O(k2+k2h2+h4)O(k2+k2h2+h4). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.