In the present study, the convergence behavior of the direct simulation Monte Carlo (DSMC) method is extensively explored. The Simplified Bernoulli Trials (SBT) collision algorithm is applied to simulate a one-dimensional nano Fourier heat conduction problem, which consists of rarefied gas confined between two infinite parallel plates with unequal temperatures. The investigations compares the Sonine-polynomial coefficients ak calculated from the DSMC results with theoretical predictions of the Chapman-Enskog (CE) theory. In addition, the convergence behavior of the wall heat flux and the ratio of the DSMC-calculated bulk thermal conductivity (KDSMC) to the infinite-approximation of CE theoretical value (K) is studied. The numerical accuracy of the DSMC method is found to be restricted in regards to three parameters: time step, cell size, and number of computational particles per cell. The dependency of the SBT collision algorithm on these discretization errors has been investigated in comparison with the standard collision algorithm, i.e., no time counter (NTC). The results indicate that SBT can achieve analytical solutions of the Sonine polynomials using fewer particles per cell than NTC. Moreover, in the SBT algorithm, the effective parameter in the convergence is Δx/Δt ratio, which should be adjusted accurately. This study shows that by decreasing the number of particle per cell to even one particle in a constant Δx/Δt setting, the SBT algorithm accurately predicts solutions where the NTC algorithm fails.