A FOURTH-ORDER ACCURATE SCHEME FOR SOLVING ONE-DIMENSIONAL HIGHLY NONLINEAR STANDING WAVE EQUATION IN DIFFERENT THERMOVISCOUS FLUIDS

Abstract

Combination of a fourth-order Padé compact finite difference discretization in space and a fourth-order Runge–Kutta time stepping scheme is shown to yield an effective method for solving highly nonlinear standing waves in a thermoviscous medium. This accurate and fast-solver numerical scheme can predict the pressure, particle velocity, and density along the standing wave resonator filled with a thermoviscous fluid from linear to strongly nonlinear levels of the excitation amplitude. The stability analysis is performed to determine the stability region of the scheme. Beside the fourth-order accuracy in both time and space, another advantage of the given numerical scheme is that no additional attenuation is required to get numerical stability. As it is well known, the results show that the pressure and particle velocity waveforms for highly nonlinear waves are significantly different from that of the linear waves, in both time and space. For highly nonlinear waves, the results also indicate the presence of a wavefront that travels along the resonator with very high pressure and velocity gradients. Two gases, air and CO 2, are considered. It is observed that the slopes of the traveling velocity and pressure gradients are higher for CO 2 than those for air. For highly nonlinear waves, the results also indicate the higher asymmetry in pressure for CO 2 than that for air.