High-order finite-difference schemes and their application to computational acoustics

The primary focus of this study is upon the numerical stability of high-order finite-difference schemes and their application to duct acoustics. Since acoustic waves are known to be non-dissipative and non-dispersive, high-order schemes are favored for their low dissipation and low dispersion relative to the low-order schemes. The primary obstacle to the the development of explicit high-order finite-difference schemes is the construction of boundary closures which simultaneously maintain the formal order of accuracy and the numerical stability of the overall scheme. In this thesis a hybrid seven-point, fourth-order stencil for computing spatial derivatives is presented and the time stability is analyzed in conjunction with a family of optimized low-dissipation and low-dispersion Runge-Kutta time-marching schemes. Using an eigenvalue analysis it is found that the combined spatial and temporal discretization is weakly unstable when applied to the linearized Euler equations. Two methods of achieving numerical stability were investigated. The first method involved the addition of artificial dissipation and the second involved a filtering procedure. Both methods resulted in significant improvements in stability and the eigenvalue analysis demonstrates that the hybrid stencil is numerically stable when used with either damping or filtering. Numerical examples are presented.; A secondary objective is to develop a time-domain implementation for lined-wall boundary conditions given in the frequency domain. Lined-wall boundary conditions are often formulated in the frequency domain since acoustic response is a function of wave frequency. In order to compute the acoustic modes directly from the linearized Euler equations, however, it is necessary the impedance boundary condition be reformulated in the time domain. In the past this has been accomplished by rewriting the impedance as an ordinary-differential equation in time and by using the z-transform. In this document it is showing that a specific impedance condition used frequently in the literature as a model for point-reacting liners may in fact be rewritten in the time domain in a simple algebraic form involving the values of pressure and normal velocity at previous times. Numerical examples of duct acoustic applications are presented.