Semi-implicit Finite-Difference Methods for Compressible Gas Dynamics with Curved Boundaries: A Ghost-Point Approach

We present a finite-difference numerical method to solve systems of hyperbolic conservation laws in domains with arbitrary shapes. The curved boundary is immersed in a uniform Cartesian grid and implicitly defined by a level-set function. The method is based on a semi-implicit discretization of the differential equations coupled with a ghost-point approach to impose the boundary conditions. The method is designed to be straightforwardly extended to higher order accuracy. The semi-implicit approach alleviates the stability restriction on the time step that is associated with acoustic waves in explicit methods, while preventing the numerical dissipation introduced in fully implicit methods. Several numerical tests to solve the Euler equations of gas dynamics past steady obstacles with arbitrary shapes are presented to show the efficiency of the semi-implicit method and the efficacy of the ghost-point approach.