  The elliptic interface problems with discontinuous and high-contrast
coefficients appear in many applications and often lead to huge condition
numbers of the corresponding linear systems. Thus, it is highly desired to
construct high order schemes to solve the elliptic interface problems with
discontinuous and high-contrast coefficients. Let $\Gamma$ be a smooth curve
inside a rectangular region $\Omega$. In this paper, we consider the elliptic
interface problem $-\nabla\cdot (a \nabla u)=f$ in $\Omega\setminus \Gamma$
with Dirichlet boundary conditions, where the coefficient $a$ and the source
term $f$ are smooth in $\Omega\setminus \Gamma$ and the two nonzero jump
condition functions $[u]$ and $[a\nabla u\cdot \vec{n}]$ across $\Gamma$ are
smooth along $\Gamma$. To solve such elliptic interface problems, we propose a
high order compact finite difference scheme for numerically computing both the
solution $u$ and the gradient $\nabla u$ on uniform Cartesian grids without
changing coordinates into local coordinates. Our numerical experiments confirm
the fourth order accuracy for computing the solution $u$, the gradient $\nabla
u$ and the velocity $a \nabla u$ of the proposed compact finite difference
scheme on uniform meshes for the elliptic interface problems with discontinuous
and high-contrast coefficients.
