On the application of explicit spatial filtering to the variables or fluxes of linear equations

The need for filtering high-frequency waves is a recurrent issue in numerical simulations. These waves might indeed lead to instability, and they are in general not calculated accurately by the discretization algorithms. Selective filters have therefore been designed in order to damp high-frequency waves without affecting significantly low-frequency disturbances [1–6]. These filters are particularly used in computational aeroacoustics, but they appear also suitable for Large-Eddy Simulations (LES), in which only the scales larger than the grid size are computed, and whose equations are derived formally by applying a filter operator to the Navier– Stokes equations [7]. Moreover LES based specially on explicit filtering have been also developed [8–10]. In practice, the flow variables are usually filtered explicitly after each time step. Consider for example the time integration of the following differential equation: