Small-scale inhibiting characteristics of residual and solution filtering

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.