Progress towards a new computational scheme for aeroacoustics

Accurate schemes for advection and aeroacoustics

“…If the advection direction is oblique, a combination of scheme(a) and (b) is used to transit from one to another continuously. To blend two schemes will introduce slight dissipation and the dissipation rate is proportional to d<f> 2 , where d<j> is the phase difference of two updated scalars, i.e. it is eighth-order for the fourthorder schemes.…”

“…In this paper we continue a sequence of developments [1,2,3] of Iserles' 'generalised' or 'upwind' leapfrog schemes [4]. The philosophy on which these schemes are based is to avoid numerical dissipation by constructing them to be time-reversible, but the symmetry that ensures this is a skew symmetry with regard to the centroid of the stencil, rather than (as with conventional leapfrog schemes) a strict symmetry in both space and time.…”

Accurate multi‐level schemes for advection

“…They primarily depend on the computational grid type as mentioned by Roe and Thomas [4,5]. Two di erent mesh arrangements appear in Figure 8.…”

“…If the advection speed is variable or the grid spacing is not uniform, a more careful approach is demanded to maintain the accuracy of the upwind leapfrog scheme [4]. The governing equation of this case can be simply written as…”

Higher‐order upwind leapfrog methods for multi‐dimensional acoustic equations

“…Roe and Thomas [4] developed the fourth-order scheme by extending the stencil in space and time domain as well as multi-dimensional second-order scheme for linear wave system, especially for acoustics by using the bicharacteristic theory. They devised the staggered grid technique storing values at the cell edge not to introduce any numerical dissipation error [5]. Meanwhile, Nguyen and Roe compared phase properties of multi-dimensional upwind leapfrog method and Yee's standard leapfrog method for acoustics and electromagnetics [6].…”