Optimization of DRP Schemes for Non-Constant-Amplitude Oscillations

Abstract

It was recently observed that finite difference schemes optimized to perform well with few points per wavelength (commonly referred to as DRP schemes) currently perform poorly when applied to waves of non-constant amplitude. In this paper, attempts are described at optimizing explicit symmetric finite difference schemes to require relatively few points per wavelength for waves with a range of growth-rates and decay-rates. Several optimization criteria are proposed, and some advantage may be gained from using such optimized schemes if the growth-and decay-rates present are known a priori. In particular, for the test case considered here, the usual 7 point 4th order DRP schemes are found to be overly ambitious in their optimization, and better performing schemes are derived which require almost half the number of points per wavelength to achieve the same accuracy. Without a priori knowledge, however, the best choice of finite difference schemes for waves of varying amplitude remain the classical maximal order schemes.

Optimized Runge-Kutta (LDDRK) schemes with non-constant-amplitude waves

“…Inspired by previous such optimizations for spatial derivatives [17], we consider two generalizations of the optimization metric (3) to regions of the complex ω∆t plane, shown schematically in figure 5. For the Opt6 rectangular region in figure 5a, defined by α 1 ≥ 0, α 2 ≤ 0 and η > 0, the optimization is taken to be…”

“…All results are using the 7-point 4th order DRP spatial derivative of Tam and Shen [3] with PPW = 8 and the 7-point 6th order F 6 filter, giving an error of around 0.2 when used with a "perfect" time integration. The error against computational effort (17) is plotted in the bottom plot.…”

“…Recently [16], it was suggested for spatial derivatives that optimization of a metric such as (3) which assumes real k∆x results in a scheme which performs well for constant amplitude waves corresponding to real k, but which performs poorly for waves of non-constant amplitude corresponding to complex k. Unfortunately, non-constant amplitude waves are rather common in aeroacoustics, especially in the vicinity of acoustic linings, for high-order spinning modes which decay rapidly away from duct walls, close to nearsingularities such as sharp trailing edges or strongly localized sources, and for instabilities; non-constantamplitude oscillations are also common in other branches of physics. Attempts at reoptimizing spatial derivatives to perform well for both non-constant and constant amplitude waves [17] concluded that, with sufficient a priori knowledge of expected wavenumbers, optimized derivative schemes could be constructed to perform well, but that in general, and certainly for broadband excitation, maximal order schemes were more likely to be more accurate.…”