Numerical inverse Laplace transform (NILT) methods are potential methods for time domain simulations, for instance the analysis of the transient phenomena in systems with lumped and/or distributed parameters. This paper proposes a numerical inverse Laplace transform method based originally on hyperbolic relations. The method is further enhanced by properly adapting several convergence acceleration techniques, namely, the epsilon algorithm of Wynn, the quotient-difference algorithm of Rutishauser and the Euler transform. The resulting accelerated models are compared as for their accuracy and computational efficiency. Moreover, an expansion to two dimensions is presented for the first time in the context of the accelerated hyperbolic NILT method, followed by the error analysis. The expansion is done by repeated application of one-dimensional partial numerical inverse Laplace transforms. A detailed static error analysis of the resulting 2D NILT is performed to prove the effectivness of the method. The work is followed by a practical application of the 2D NILT method to simulate voltage/current distributions along a transmission line. The method and application are programmed using the Matlab language.