Stability of θθ-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments

A finite difference method, namely the θθ-scheme, is used to solve a partial differential equation with piecewise continuous arguments. First, an example is given to show that the results for original PDEs do not hold for PDEs with piecewise continuous arguments. Next, for the mesh ratio satisfying certain conditions, the stability of this scheme is obtained.