  Let $\Gamma$ be a smooth curve inside a two-dimensional rectangular region
$\Omega$. In this paper, we consider the Poisson interface problem $-\nabla^2
u=f$ in $\Omega\setminus \Gamma$ with Dirichlet boundary condition such that
$f$ is smooth in $\Omega\setminus \Gamma$ and the jump functions $[u]$ and
$[\nabla u\cdot \vec{n}]$ across $\Gamma$ are smooth along $\Gamma$. This
Poisson interface problem includes the weak solution of $-\nabla^2
u=f+g\delta_\Gamma$ in $\Omega$ as a special case. Because the source term $f$
is possibly discontinuous across the interface curve $\Gamma$ and contains a
delta function singularity along the curve $\Gamma$, both the solution $u$ of
the Poisson interface problem and its flux $\nabla u\cdot \vec{n}$ are often
discontinuous across the interface. To solve the Poisson interface problem with
singular sources, in this paper we propose a sixth order compact finite
difference scheme on uniform Cartesian grids. Our proposed compact finite
difference scheme with explicitly given stencils extends the immersed interface
method (IIM) to the highest possible accuracy order six for compact finite
difference schemes on uniform Cartesian grids, but without the need to change
coordinates into the local coordinates as in most papers on IIM in the
literature. Also in contrast with most published papers on IIM, we explicitly
provide the formulas for all involved stencils. The coefficient matrix $A$ in
the resulting linear system $Ax=b$, following from the proposed scheme, is
independent of any source term $f$, jump condition $g\delta_\Gamma$, interface
curve $\Gamma$ and Dirichlet boundary conditions. Our numerical experiments
confirm the sixth accuracy order of the proposed compact finite difference
scheme on uniform meshes for the Poisson interface problems with various
singular sources.
