A novel optimization technique for explicit finite‐difference schemes with application to AeroAcoustics

The present paper addresses the optimization of finite‐difference schemes when these are to be used for numerically approximating spatial derivatives in aeroacoustics evolution problems. With that view in mind, finite‐difference operators are firstly detailed from a theoretical point of view. Secondly, time, the way such operators can be optimized in a spectral‐like sense is recalled, before the main limitations of such an optimization are highlighted. This leads us to propose an alternative optimization approach of innovative character. Such a novel optimization technique consists of enhancing the scheme's formal accuracy through a minimization of its leading‐order truncation error. This so‐called intrinsic optimization procedure is first detailed, before it is thoroughly analyzed, from both a theoretical and a practical point of view. The second part of the paper focuses on two particular intrinsically optimized schemes, which are carefully assessed via a direct comparison against their standard and/or spectral‐like optimized counterparts, such a comparative exercise being conducted utilizing several academic test cases of increasing complexity. There, it is shown how intrinsically optimized schemes indeed constitute an advantageous alternative to either the standard or the spectral‐like optimized ones, being allotted with both (i) the better scalability of the former scheme with respect to grid convergence effects when the grid density increases and (ii) the higher accuracy of the latter scheme when the discretization level becomes marginal. Thanks to that, such intrinsically optimized schemes offer very good trade‐offs in terms of (i) accuracy; (ii) robustness; and (iii) numerical efficiency (CPU cost). Copyright © 2015 John Wiley & Sons, Ltd.