RADIATION BOUNDARY-CONDITION AND ANISOTROPY CORRECTION FOR FINITE-DIFFERENCE SOLUTIONS OF THE HELMHOLTZ-EQUATION

In this paper finite-difference solutions Of the Helmholtz equation in an open domain are considered. By using a second-order central differ ence scheme and the Bayliss-Turkel radiation boundary condition, reaso nably accurate solutions can be obtained when the number of grid point s per acoustic wavelength used is large. However, when a smaller numbe r of grid points per wavelength is used excessive reflections occur wh ich tend to overwhelm the computed solutions. Excessive reflections ar e due to the incompability between the governing finite difference equ ation and the Bayliss-Turkel radiation boundary condition. The Bayliss -Turkel radiation boundary condition was developed from the asymptotic solution of the partial differential equation. To obtain compatibilit y, the radiation boundary condition should be constructed from the asy mptotic solution of the finite difference equation instead. Examples a re provided using the improved radiation boundary condition based on t he asymptotic solution of the governing finite difference equation. Th e computed results are free of reflections even when only five grid po ints per wavelength are used. The improved radiation boundary conditio n has also been tested for problems with complex acoustic sources and sources embedded in a uniform mean flow. The present method of develop ing a radiation boundary condition is also applicable to higher order finite difference schemes. In all these cases no reflected waves could be detected. The use of finite difference approximation inevitably in troduces anisotropy into the governing field equation. The effect of a nisotropy is to distort the directional distribution of the amplitude and phase of the computed solution. It can be quite large when the num ber of grid points per wavelength used in the computation is small. A way to correct this effect is proposed. The correction factor develope d from the asymptotic solutions is source independent and, hence, can be determined once and for all. The effectiveness of the correction fa ctor in providing improvements to the computed solution is demonstrate d in this paper. (C) 1994 Academic Press, Inc.