Low-Dissipation and Low-Dispersion Runge–Kutta Schemes for Computational Acoustics

In this paper, we investigate accurate and efficient time advancing methods for computational acoustics, where nondissipative and nondispersive properties are of critical importance. Our analysis pertains to the application of Runge–Kutta methods to high-order finite difference discretization. In many CFD applications, multistage Runge–Kutta schemes have often been favored for their low storage requirements and relatively large stability limits. For computing acoustic waves, however, the stability consideration alone is not sufficient, since the Runge–Kutta schemes entail both dissipation and dispersion errors. The time step is now limited by the tolerable dissipation and dispersion errors in the computation. In the present paper, it is shown that if the traditional Runge–Kutta schemes are used for time advancing in acoustic problems, time steps greatly smaller than those allowed by the stability limit are necessary. Low-dissipation and low-dispersion Runge–Kutta (LDDRK) schemes are proposed, based on an optimization that minimizes the dissipation and dispersion errors for wave propagation. Optimizations of both single-step and two-step alternating schemes are considered. The proposed LDDRK schemes are remarkably more efficient than the classical Runge–Kutta schemes for acoustic computations. Moreover, low storage implementations of the optimized schemes are discussed. Special issues of implementing numerical boundary conditions in the LDDRK schemes are also addressed.