We develop a numerical simulator to solve the time-dependent Schrödinger equation (TDSE) to simulate strong field ionization of a single electron under the influence of a one-dimensional separable potential. We use the split-step method to solve TDSE, obtain the ground state using the imaginary time method and benchmark it with the exact diagonalization of Hamiltonian. We perform the stability analysis in both real-time and imaginary-time propagation and obtain the fact that the usual model of separable orbital is not suitable in this method and demonstrate how a Gaussian model can solve this issue. Finally, we perform a comparative analysis between the analytical ground state and an adapted version of ground state adjusted according to the time-step of TDSE to justify the origin of the numerical error.