Compact difference schemes for Klein–Gordon equation with variable coefficients

In this paper, we consider the compact difference approximation of the fourth and second-order schemes on a three-point stencil for Klein–Gordon equations with variable coefficients. Despite the linearity of the differential and difference problems, it is not possible in this case to apply the well-known results on the theory of stability of three-layer operator-difference schemes by A. A. Samarskii. The main purpose is to prove the stability with respect to the initial data and the right-hand side of compact difference schemes in the grid norms L 2 (W h ), W 1 2  (W h ), C (W h ). Using the method of energy inequalities, the corresponding a priori estimates, expressing the stability and convergence of the solution to the difference problem with the assumption h ≤ = h 0,   h 0  = const, τ≥h is obtained. The conducted numerical experiment shows how Runge rule is used to determine the different orders of the convergence rate of the difference scheme in the case of two independent variables.