  We present a characterization of the forcing and the sub-filter scale terms
produced in the volume-filtering immersed boundary (VF-IB) method by Dave et
al, JCP, 2023. The process of volume-filtering produces bodyforces in the form
of surface integrals to describe the boundary conditions at the interface.
Furthermore, the approach also produces unclosed subfilter scale (SFS) terms.
The level of contribution from SFS terms on the numerical solution depends on
the filter width. In order to understand these terms better, we take a 2
dimensional, varying coefficient hyperbolic equation shown by Brady & Liverscu,
JCP, 2021. This case is chosen for two reasons. First, the case involves 2
distinct regions seperated by an interface, making it an ideal case for the
VF-IB method. Second, an existing analytical solution allows us to properly
investigate the contribution from SFS term for varying filter sizes. The latter
controls how well resolved the interface is. The smaller the filter size, the
more resolved the interface will be. A thorough numerical analysis of the
method is presented, as well as the effect of the SFS term on the numerical
solution. In order to perform a direct comparison, the numerical solution is
compared to the filtered analytical solution. Through this, we highlight three
important points. First, we present a methodical approach to volume filtering a
hyperbolic PDE. Second, we show that the VF-IB method exhibits second order
convergence with respect to decreasing filter size (i.e. making the interface
sharper). Finally, we show that the SFS term scales with square the filter
size. Large filter sizes require modeling the SFS term. However, for
sufficiently finer filters, the SFS term can be ignored without any significant
reduction in the accuracy of solution.
