For physical problems that involve accurate time-dependent wave propagation, as those arising in acoustics, the usual requirement of a high-order truncation error does not guarantee that a numerical method yields accurate results. Indeed, as has been pointed out mainly in [1], the dissipation and dispersion properties of the numerical method are very important for computing wave solutions of systems of partial differential equations. This is valid for both the spatial and the time discretization methods. The explicit Runge-Kutta (RK) methods are widely used to discretize the time derivative because of their advantages that include flexibility, large stability limits, and ease of programming. Hu and co-workers [2] showed that the dissipation and dispersion properties of the RK methods depend on their coefficients and optimized them for the convective wave equation, obtaining what they called low-dissipation and dispersion Runge-Kutta (LDDRK) methods. These methods are more efficient than classical ones, in terms of work required for a given accuracy, for wave propagation problems.

For large size physical problems, memory requirements may become exhaustive. They can be decreased using special RK schemes that can be written such that only 2N-storage is required, where N is the number of degrees of freedom of the system (ie, number of grid points× number of variables). To design such RK schemes, enough free coefficients must exist such that additional conditions hold between them. Williamson [3] first showed that all second-order and some third-order methods can be written in 2N-storage form. He also showed that fourth-order four-stage methods cannot be written in this way. By allowing additional stages and using the resulting new free coefficients to impose the 2N-storage constraints, Carpenter and Kennedy [4] devised a fourth-order, five-stages RK method that is compatible with the classical fourth-order method which however requires at least 3N storage.