On finite-difference methods for simulating linear acoustic wave propagation

Several high-accuracy finite-difference methods are compared with respect to their grid resolution requirements for accurate simulation of long-range propagation of linear acoustic waves. The spatial finite-difference operators considered include various optimized compact and noncompact schemes introduced over the past few years. These are combined with high-order Runge-Kutta and Adams-Bashforth time-marching methods. In addition, the following fourth-order fully-discrete finite-difference methods are included in the comparisons: a one-step implicit scheme with a three-point spatial stencil, a one-step explicit scheme with a five-point spatial stencil, and a two-step explicit scheme with a five-point spatial stencil. The compact spatial operators are capable of propagating waves accurately over two hundred wavelengths with less than six grid points per wavelength. The noncompact spatial operators require eight to ten grid points per wavelength for this distance of travel while the fully-discrete schemes require twenty to thirty. Based on the results presented, specific methods are identified as suitable for efficient simulation of linear acoustic wave propagation. Introduction In order to avoid the need for excessively fine grids, the simulation of linear acoustic wave propagation over long distances requires the use of highly accurate numerical methods. In a previous study, * we compared several finite-difference schemes on the basis of the number of grid points per wavelength (PPW) required to achieve a given standard of accuracy for propagation distances expressed in terms of the number of wavelengths travelled. The schemes compared include staggered schemes, * Associate Professor, Member AIAA. Copyright ©1996 by David W. Zingg. Published by the American Institute of Aeronautics and Astronautics Inc. with permission. f\ upwind leapfrog schemes, z and the schemes of Zingg at al., *• ^ which combine a seven-point spatial stencil with a low-storage six-stage time-marching method of RungeKutta type. Two second-order algorithms were considered: 1) the combination of second-order staggered spatial differencing with staggered leapfrog time marching and 2) the second-order upwind leapfrog scheme. Although these schemes are appealing because of their simplicity, their grid requirements are excessive for waves propagating at arbitrary angles with respect to the grid. The two-dimensional implementation of the fourth-order upwind leapfrog scheme studied in Ref. 1 is no better than the second-order scheme, despite excellent accuracy in one dimension. The combination of fourth-order staggered spatial differencing with staggered leapfrog time marching is much more effective, requiring less than 18 PPW to propagate a wave accurately over 200 wavelengths in an arbitrary direction. However, this accuracy is achieved only for mean flows at low Mach num3 A ber. The maximum-order scheme of Zingg at al., ' requires less than 14 PPW for 200 wavelengths of travel. In addition, an optimized scheme is presented by Zingg at al. 3> 4 which is formally of lower order but provides improved accuracy for practical levels of grid refinement. For distances of travel up to 300 wavelengths, the optimized scheme requires less than 10 PPW. In this paper, several further high-accuracy finitedifference schemes are compared on the same basis. The Q A seven-point, six-stage schemes of Zingg et al. '*• are again included. The new spatial differencing schemes considered are the optimized compact schemes of Haras and Ta'asan **, the optimized upwind-biased scheme of Lockard et al., ^ which uses an eight-point stencil, and the optimized scheme of Tarn, which uses a sevenpoint stencil. In addition to the six-stage method of Zingg et al., *>• ^ the time-marching methods compared include high-order Runge-Kutta and Adams-Bashforth methods and the five-stage methods of Haras and Ta'asan. 5 FiAmerican Institute of Aeronautics and Astronautics nally, three fourth-order methods which discretize time and space simultaneously are evaluated. These are a one-step implicit method with a three-point stencil in space proposed by Davis, ' a one-step explicit method with a five-point spatial stencil developed by Gottlieb and Q Turkel, ° and a two-step explicit scheme with a five-point spatial stencil. The purpose of this paper is to evaluate and compare the above schemes, many of which are relatively new. The comparisons are useful in choosing a scheme for a given application and in selecting schemes for further development. Furthermore, the information provided is useful in choosing a grid density and a time step for a specific problem. In the next section, we present the finite difference schemes in the context of the linear advection equation. This is followed by a brief review of Fourier error analysis, which is used to evaluate the schemes. Next the PPW requirements of each scheme are presented and compared. Finite-Difference Methods Compared In this section, we present the various finite-difference methods in the context of the linear advection equation given by du du ,,. -dtdx= > where u is a scalar quantity propagating with speed a, which is real and positive. The spatial difference operators are thus approximations to d/dx. The time-marching methods are presented as applied to a scalar ordinary differential equation of the form