A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

In this paper, we describe how to construct a finite-difference shock-capturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domain Omega, possibly with moving boundary. The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls. Although the motion of the boundary could be coupled with the fluid, all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow. The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domain Omega(R) superset of Omega. We identify inner and ghost points. The inner points are the grid points located inside Omega, while the ghost points are the grid points that are outside W but have at least one neighbor inside Omega. The evolution equations for inner points data are obtained from the discretization of the governing equation, while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables (density, velocities and pressure). Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains. Several numerical experiments are conducted to illustrate the validity of the method. We demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.