The 3rd order numerical acceleration method is presented. The basic assumption of this method is that the acceleration changes in the time interval Δt is a polynomial of the third degree. To quantitatively compare the error rate of this method with other numerical methods, two linear two-degree-of-freedom systems were investigated. The first and second floors of system 1 had the same mass and stiffness, but the mass of the first and second floors of system 2 were equal to each other, but the stiffness of the second floor of system 2 was one thousandth of the stiffness of the first floor. For systems 1 and 2, a suitable range of sinusoidal loading frequencies was selected according to their natural vibration frequencies. First, the exact (analytical) response of these two systems was calculated for each loading frequency, and then the approximate response of these two systems was calculated using the 8 numerical methods proposed in this article, and the average error values and the coefficient of error variation were calculated for these eight numerical methods. These eight methods include 2nd order acceleration method, 3rd order acceleration method, Newmark method (average acceleration), Newmark method (linear acceleration), Wilson method, central difference method, Jennings method and improved Jennings method. The most important results obtained from the comparison of the average errors of these methods were that in most methods, the average error increased with the increase of ∆t. Also, with the increase of ∆t, the average error of the 3rd order acceleration method significantly decreased compared to the average of the 2nd order acceleration method. In addition, the improved Jennings method had the lowest average error for both systems 1 and 2 for all ∆t, and for Wilson and central difference methods, the highest amount of error was observed for different ∆t.