  In this paper a fourth order finite difference ghost point method for the
Poisson equation on regular Cartesian mesh is presented. The method can be
considered the high order extension of the second ghost method introduced
earlier by the authors. Three different discretizations are considered, which
differ in the stencil that discretizes the Laplacian and the source term. It is
shown that only two of them provide a stable method. The accuracy of such
stable methods are numerically verified on several test problems.
