Small-scale inhibiting characteristics of residual and solution filtering

TL;DR

This paper compares residual and solution filtering techniques for inhibiting small-scale, high-wavenumber content in fluid simulations, revealing fundamental differences in their mechanisms and effectiveness through theoretical and numerical analyses.

Contribution

It provides a detailed analysis of how residual and solution filtering methods differ in controlling small-scale features, supported by theoretical and numerical evaluations.

Findings

Residual filtering constrains scale generation via dispersive effects.

Solution filtering and artificial dissipation attenuate high wavenumber content through dissipative mechanisms.

Discrete filters' response characteristics influence their effectiveness in small-scale suppression.

Abstract

Residual and solution filtering procedures are studied with respect to inhibiting the accumulation of small-scale (i.e., high wavenumber) content. Assessing each method in terms of an ``equivalent residual equation" reveals fundamental differences in their behaviors, such as how the underlying solution can be constrained to a target filter width. The residual filtering (RF) approach paired with a dissipative filter kernel is shown to restrict scale generation in the fluid equations via dispersive effects; meanwhile, solution filtering (SF) -- and artificial dissipation (AD), by extension -- operates through dissipative mechanisms and actively attenuates high wavenumber content. Discrete filters (i.e., the Top-hat and implicit Tangent schemes) are analyzed in terms of their response characteristics and their associated effects on reducing small-scale activity when paired with the RF versus SF approaches. Linear theoretical assessments (e.g., von Neumann analysis) are shown to successfully characterize the fundamental behaviors of the methods in non-linear settings, as observed through canonical test cases based on 1D viscous Burgers, 2D Euler, 3D Navier-Stokes equations.