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

We propose a new high-order multi-stage method to solve the linear wave equation in an unconditionally energy stable manner. This Successive Multi-Stage (SMS) method is extended from the Crank–Nicolson method and unconditional energy conservation is guaranteed. We develop up to the sixth-order SMS method using the order conditions for Runge–Kutta methods and provide mathematical arguments showing that the SMS method is a different branch from well-known high order energy preserving methods for Hamiltonian systems. We present a proof of the unique solvability and numerically demonstrate the accuracy and stability of the proposed methods compared with comparisons.