Design, stability analysis and application of solid-wall immersed boundary schemes for the linearized Euler equations

This paper presents a stability-focused investigation of a high-order immersed boundary method for solid-wall treatment in acoustic simulations. Unlike conventional approaches that entirely rely on spatial filtering to suppress instabilities arising from stencil modifications near boundaries, our method aims to minimize them by incorporating a suitable set of boundary constraints into the finite-difference stencil design. Using eigenvalue analysis of the spatial discretization matrix, we identify critical stencil parameters that significantly influence numerical stability. These include the number and spacing of surface enforcement points, the enforcement of multiple boundary constraints per variable, and the imposition of vanishing vorticity at surface and selected volume enforcement points. By systematically evaluating both interior and exterior acoustic scenarios, we demonstrate that the stability of hard-wall boundary treatments is sensitive to the stencil configuration, stencil order, and relative surface curvature. Sharp corners and pointed tips might require special treatment, which is not addressed here. The approach enables stable and accurate simulations with minimal artificial damping at nodes within half a stencil width of solid boundaries. The required nondimensional damping strength σ (obtained via normalization by the mesh-based frequency f 0 = c 0 / h 0 involving the speed of sound c 0 and the mesh-width h 0 ) is reduced from O ( 1 ) to as low as σ ≈ 0.1, thereby minimizing distortions of the physical solution. Test cases validate the method’s accuracy for planar wave scattering and flow-induced sound from a circular cylinder and other geometries (2D and 3D).