In wave propagation phenomena, time-advancing numerical methods must accurately represent the amplitude and phase of the propagating waves. The acoustic waves are non-dispersive and non-dissipative. However, the standard schemes both retain dissipation and dispersion errors. Thus, this paper aims to analyse the dissipation, dispersion, accuracy, and stability of the Runge–Kutta method and derive a new scheme and algorithm that preserves the symmetry property. The symmetrised method is introduced in the time-of-finite-difference method for solving problems in aeroacoustics. More efficient programming for solving acoustic problems in time and space, i.e. the IMR method for solving acoustic problems, an advection equation, compares the square-wave and step-wave Lax methods with symmetrised IMR (one-and two-step active). The results of conventional methods are usually unstable for hyperbolic problems. The forward time central space square equation is an unstable method with minimal usefulness, which can only study waves for short fractions of one oscillation period. Therefore, nonlinear instability and shock formation are controlled by numerical viscosities such as those discussed with the Lax method equation. The one- and two-step active symmetrised IMR methods are more efficient than the wave method.
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