High order finite difference approximations are derived for a one-dimensional model of the shifted wave equation written in second-order form. The domain is discretized using fully compatible summation by parts operators and the boundary conditions are imposed using a penalty method, leading to fully explicit time integration. This discretization yields a strictly stable and efficient scheme. The analysis is verified by numerical simulations in one-dimension. The present study is the first step towards a strictly stable simulation of the second-order formulation of Einstein's equations in three spatial dimensions.