We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrödinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove second-order error estimates in space and time in the
l
2
{l^2}
norm.