Finite-difference schemes for nonlinear wave equation that inherit energy conservation property

We propose two general finite-difference schemes that inherit energy conservation property from nonlinear wave equations, such as the nonlinear Klein–Gordon equation (NLKGE). One of proposed schemes is implicit and another is explicit. Many studies exist on FDSs that inherit energy conservation property from NLKGE and we can derive all of their schemes from the proposed general schemes in this paper. The most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts, which implies that the inherited properties are satisfied exactly. Because of this the derived schemes are expected to be numerically stable and yield solutions converging to PDE solutions. We make new FDSs for Fermi–Pasta–Ulam equation, string vibration equation, Shimoji–Kawai equation (SKE) and Ebihara equation and verify numerically the inheritance of the energy conservation property for NLKGE and SKE.