Second-Order Accuracy in Time of Finite Difference Methods for Computational Aeroacoustics

Abstract

The recently developed second-order accuracy in time finite difference method suitable for computational aeroacoustics (CAA) is introduced. Although, it is straight forward to compute the coefficients for finite-difference method of any order of accuracy using the Taylor series and to then further optimize them to enhance their wavenumber preserving properties, there are difficult questions concerning their numerical stability The goal of this work is to develop an effective numerical technique that includes both linear and nonlinear wave propagation in order to solve acoustics problems in time and space. It also aims to evaluate the accuracy, effectiveness, and stability of the new technique. In 1-D linear and nonlinear computational aeroacoustics, the novel techniques were used. The findings of the conventional methods (square wave (FTCS) technique and step wave lax approach) are presented in this paper, and it is shown that the FTCS method is typically unstable for hyperbolic situations and cannot be employed. Unfortunately, the FTCS equation has very little practical application. It is an unstable method, which can be used only (if at all) to study waves for a short fraction of one oscillation period. Nonlinear instability and shock formation are thus somewhat controlled by numerical viscosity such as that discussed in connection with Lax method equation. The second-order accuracy in time finite difference method is more efficient than the (square wave (FTCS), step wave lax) methods and is faster than the step wave lax method.