Power Series Method for Time-Integration in Computational Acoustics

Abstract

A time-marching scheme based on power series is developed for accurate unsteady simulations in acoustics and fluid dynamics. Unlike the widely used time integration methods such as the Adam-Bashforth multi-step methods and the Runge-Kutta multi-stage methods that represent the solution on discrete points in time, the current method represents the solution as truncated power series in time, thus providing a continuous solution within each time step. The method is developed for arbitrary order of accuracy. Explicit time marching is enabled by recursive evaluation of the coecients of the power series. It was shown that the stability range scales up with the order of truncated power series. As a result, ecient high accuracy computation can be achieved by using very high order of power series and very large time steps. The order can be easily varied locally to adapt local grid resolution so that highly ecient multi-resolution simulations can be achieved under uniform time stepping. The method is general enough to be applicable to time dependent problems in other physical and engineering fields.