Computations are presented for one-dimensional Shockwaves of diatomic nitrogen. The direct simulation Monte Carlo method (DSMC) is employed. A model which allows the use of the Sutherland viscosity model in the DSMC technique is further developed to include the modelling of rotational relaxation. This is achieved by employing a temperature-dependent expression for the rotational collision number to simulate the rate of relaxation, in contrast to earlier DSMC studies which employed a constant value for the collision number. The behaviour of the new model is first considered for rotational relaxation in a heat bath. Comparison is made with the more traditional DSMC collision model termed the variable hard sphere (VHS) model in which the viscosity has a fixed temperature exponent. These two models are then applied to a number of shock-wave cases for which experimental data exist. These have Mach numbers ranging from 1.5 to 26, yet the enthalpies involved are sufficiently low to allow omission of vibrational relaxation. It is found that both the VHS and Sutherland viscosity models give excellent agreement with the experimental data for shock-wave thickness, and for profiles of density, rotational temperature, and velocity. The most significant finding of the study is that the reciprocal shock thickness varies with the upstream temperature condition. This variation is observed in the experimental data, and is simulated numerically by employing the temperature-dependent expression for the rotational collision number.