This work focuses on the numerical performance of HHT-alpha and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche's method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter alpha for the HHT-alpha method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used.