Optimized finite-difference (DRP) schemes perform poorly for decaying or growing oscillations

Abstract

Computational aeroacoustics often use finite difference schemes optimized to require relatively few points per wavelength; such optimized schemes are often called Dispersion Relation Preserving (DRP). Similar techniques are also used outside aeroacoustics. Here the question is posed: what is the equivalent of points per wavelength for growing or decaying waves, and how well are such waves resolved numerically? Such nonconstant-amplitude waves are common in aeroacoustics, such as the exponential decay caused by acoustic linings, the O(1/r) decay of an expanding spherical wave, and the decay of high-azimuthal-order modes in the radial direction towards the centre of a cylindrical duct. It is shown that optimized spatial derivatives perform poorly for waves that are not of constant amplitude, under performing maximal-order schemes. An equivalent criterion to points per wavelength is proposed for non-constant-amplitude oscillations, reducing to the standard definition for constant-amplitude oscillations and valid even for pure growth or decay with no oscillation. Using this definition, coherent statements about points per wavelength necessary for a given accuracy can be made for maximal-order schemes applied to non-constant-amplitude oscillations. These features are illustrated through a numerical example of a one-dimensional wave propagating through a damping region.

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

“…The theory was illustrated by solving an example 1D wave equation in section 4 taken from Ref. 16. The results suggest that optimized timestepping schemes have been over-ambitiously optimized with target errors of around δ = 10 −3 per timestep, where as significantly smaller errors are required to get an overall simulation error of around 10 −2 or 10 −3 .…”

“…Equation (15) is solved on a periodic x-domain [0, 24), with initial conditions v(x, 0) = p(x, 0) and damping k p (x) = k v (x) as specified in Ref. 16 consisting of a wave packet with wavelength 1 propagating across a damping region of length 2 and decaying by a factor of e −6 . By comparing with the analytic solution p a (x, t), v a (x, t), given in [16], the numerical error is then given by Figure 7 compares various timestepping schemes for a "perfect" 15-point 14th order maximal order spatial derivative with 32 points per wavelength (PPW), using a "perfect" spatial filter F 16,4 at each time step, as described in Ref.…”

“…By comparing with the analytic solution p a (x, t), v a (x, t), given in [16], the numerical error is then given by Figure 7 compares various timestepping schemes for a "perfect" 15-point 14th order maximal order spatial derivative with 32 points per wavelength (PPW), using a "perfect" spatial filter F 16,4 at each time step, as described in Ref. 16. A "perfect" time integration would then result in an error of approximately 5 × 10 −11 using this scheme, giving a noise floor due to the spatial discretization used.…”

“…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.…”

Recent developments in accuracy and stability improvement of nonlinear filter methods for DNS and LES of compressible flows

“…Low dispersive temporal discretization and special treatments for IBVPs of the different CAA applications are also needed. See Tam [44,45] , Brambley [6] , Haras and Taasan [11] , and Linders and Nordström [24] , Linders et al [25] for formulations and overviews. Some of the DRP schemes might perform poorly for decaying or growing oscillations.…”