A novel approach to solve PDE's involving boundary conditions on complex geometrical bounding surfaces using the Volume-Filtered Immersed Boundary (VF-IB) method.

We present a Novel approach to solve for partial differential equations that involve boundary conditions on complex geometrical bounding surfaces, termed the Volume-Filtered Immersed Boundary (VF-IB) method. This approach is derived by volume-filtering the point wise conservative equations. Through this process, the boundary conditions which apply on the interface are converted to body-forces that apply to the right-hand side of the filtered equation. Additionally, a feature of the volume-filtering process is the production of sub-filter scale terms. This approach is physically rigorous and does not depend on any numerical considerations. In this process the spatial resolution is controlled by two variables, namely the grid resolution and the filter width. The resolution of the bounding surface is controlled by the number of Lagrangian points utilized in defining the surface. In this study we take the case of a 2-D varying coefficient hyperbolic equation. Picking a pure advection case allows us to test the numerical accuracy of the method without any dissipation, which would normally hide any numerical errors. Through this we highlight three important points. First, we present a methodical approach to volume filtering any arbitrary PDE. Secondly, we show that the VF-IB method is fully capable in handling hyperbolic equations involving complex bounding surfaces while keeping numerical dissipation at a minimum. Thus, providing a path forward to solve compressible fluid flows and plasma physics using the VF-IB method. Lastly, we show a complete analysis of the impact of sub-filter scale terms on the solution.