Ghost-point methods for Elliptic and Hyperbolic Equations

Abstract

In this thesis we have presented a finite-difference ghost-point method to solve elliptic and hyperbolic equations on arbitrary domains. The equations are discretized on a uniform Cartesian grid. At first we applied the Coco-Russo method, which represents a generalization of the finite-difference method for the elliptic equations on arbitrary domains, at the resolution of the Poisson equation. This method proposes a polynomial interpolation technique to impose boundary conditions and therefore the interpolation error can influence the accuracy order of the method itself. We have proposed linear and bilinear interpolation techniques. These conditions are imposed on the projections of the ghost points on the border of the domain. The numerical tests performed on the behaviors of the inverse matrix of the method, of the error and of the consistency error confirm the stability and convergence of the Coco-Russo method in 1D, 2D and 3D, in the case of Dirichlet problems and in the case of mixed problems. We have also presented a rigorous proof of the stability and convergence of the numerical method in the one-dimensional case. Once we certain of the convergence and stability of the Coco-Russo method, our interest it has moved to the study of the Euler equations of the gas dynamic. The Coco-Russo method was applied for the development of a semi-implicit method for Euler equations on domains that have obstacles, to impose boundary conditions in a manner similar to elliptic equations. This method being semi-implicit overcomes the problem of spatial restriction to guarantee the stability of the method typical of explicit methods.