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The combined advection-diffusion-reaction (ADR) equation, which describe the transport problem of a contaminant in porous medium, does not generally admit an analytical solution. In general, when solving the ADR equation, the numerical methods (such as finite differences, finite elements, splitting), for most practical problems, the ADR equation is too difficult to solve analytically. The finite difference method is the oldest and most commonly used method for the numerical solution of this kind of equation. Although newer techniques, such as those based on finite elements and splitting are appropriate for the solution of equilibrium-type problems, the finite difference remains the most appropriate for the solution of time-dependent phenomena. The transport of a contaminant can also be written by the ADR equation; hence, our objective is to choose the most efficient method to study the 1D transport problem of a contaminant and its evolution in a porous medium. In this work, we will simulate the ADR equation using two different methods: those of finite difference and splitting ones. The numerical result will be compared with the analytical solution in order to discuss the stability and the convergence of each of them using those two different methods. In the end, we will show that the splitting technical method is more efficient for solving this kind of problems in comparison with the finite difference method despite the fact that the latter is the most widely used by researchers. The validation of the efficiency of this method, implemented in this simulation, is tested on a 1D-transport problem of contaminant in a saturated porous medium.