The aim of this article is to propose a generalization of the finite difference scheme suitable with solutions of Dirac distribution type. This type of solution is for example encountered in earthquake or explosion simulations. In such problems, the difficulty is to catch sharply a moving singular front modeled by a Dirac type distribution. We give a general framework to deal with numerical methods, and use it to build finite difference methods in distribution spaces. Numerical examples are provided for a one-dimensional wave equation.