Numerical Solutions of Klein-Gordon Equations

The Klein-Gordon equation is a hyperbolic partial differential equation which appears in various relativistic physics areas such as quantum field theory and quantum mechanics. Many numerical approaches have been suggested to approximate the analytical solutions of the Klein-Gordon equation. However, the arithmetic mean method has not been studied on the Klein-Gordon equation. In this study, the new schemes for approximating the solution of the Klein-Gordon equations by applying central finite difference formula in time and space (CTCS) incorporated with arithmetic mean averaging of functional values are proposed. Three-point and four-point arithmetic means are considered. The schemes are applied to a linear and a nonlinear inhomogeneous Klein-Gordon equations with initial conditions. The theoretical aspects of the numerical scheme such as stability, consistency and convergence for the Klein-Gordon equations are also considered and the stability of the proposed schemes is analysed by using von Neumann stability analysis and Miller Norm Lemma.