Energy-conserving successive multi-stage method for the linear wave equation with forcing terms

We propose a high-order time-discretized method for a non-homogeneous linear wave equation with a forcing term. The method conserves the accumulated discrete energy with the external term. We provide detailed proofs of unique solvability and unconditional energy conservation of the proposed successive multi-stage (SMS) method. We also present reduced order conditions up to the fourth order with aid of some important algebraic identities from the features of the SMS methods. We demonstrate the accuracy and stability of the SMS methods using numerical experiments. In addition, to show the applicability of the proposed method, we extend the method to solve quasi-linear wave equations and provide numerical simulations for sine-Gordon and Boussinesq-type equations.