Improvement of High-Order Finite-Difference Schemes at Solid Walls for the Linearized Euler Equations

An alternate description of stable discretizations at boundaries with explicit finite-difference stencils of arbitrary order for solid walls located on a node is presented to solve the linearized Euler equations in 1D without using a stabilizing filter or artificial damping. The key to this approach is incorporating additional boundary constraints besides the physical impermeability condition into Hermite-based finite-difference stencils in a prescribed region near the boundary. The application of our ansatz is equivalent to ghost point formulations for specific constellations, e.g. methods introduced by Tam & Dong (1994) and Gloerfelt (2001). A numerical reflection problem demonstrates the accuracy in 1D for high-order schemes of 6th-and 20th-order. Stability analysis proves the significance of using multiple boundary constraints to improve the numerical stability of a boundary scheme. Our new formalism for boundary methods allows the characterization of propagation features of modified boundary stencils of first derivatives by spatial Fourier analysis. Likewise, incorporating multiple boundary constraints significantly improves the modified wavenumber signature of the boundary stencils.