A particle-in-cell (PIC) simulator, using parallel computing architecture (CUDA) and execution on a graphical processing unit (GPU), was developed and used to calculate the spectral and temporal evolution of a femtosecond laser pulse during its propagation in a plasma medium. Validation of the numerical kernel in the case of relativistic self-focusing and filamentation is demonstrated. The calculated spectrum and temporal shape for propagation of a 45 fs, 800 nm laser pulse through a 300 μm long plasma channel is presented and compared to the experimental results of Couairon and co-workers (Opt. Expr. 18 , 2010) showing good agreement. The features of this simulator open up novel opportunities for studying ultrafast laser pulse propagation under different conditions