Fifth-order Hermitian schemes for computational linear aeroacoustics
We develop a class of fifth-order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth-order accuracy. A one-dimensional (ID) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy-Kovaleskaya (the 'Delta-P5 scheme'); the second one relies on a semi-discrete form and evolves in time the discrete unknowns by using a five-stage Runge-Kutta method (the 'RGK-P5 scheme'). Although the RGK-P5 scheme shares the same local spatial interpolator with the Delta-P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi-dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210:133-170; J. Comput. Phys. 2006; 217:530-562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the 'reconstruction stage', we define a fifth-order local spatial interpolator based on an upwind stencil. In the 'decomposition stage', we decompose the time derivatives into simple wave contributions. In the 'evolution stage', we use these fluctuations to update either by a Cauchy-Kovaleskaya procedure or by a five-stage Runge-Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the 'evolution stage', two fifth-order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK-P5 appears as the best compromise between simplicity and accuracy, while the Delta-P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright (c) 2007 John Wiley & Sons, Ltd.
hermitian schemes aeroacoustics acoustics upwind discretization linear wave modelling non-reflective boundary conditions linear hyperbolic systems