The collision process is crucial in the direct simulation Monte Carlo (DSMC) method as it considers the fundamental aspects of the Boltzmann or Kac stochastic equation. This article aims to facilitate the choice of collision pairs by using a symmetrization of the pair selection process of collisions. On the base of the recently created symmetrized simplified Bernoullitrials scheme, we applied our efforts to create a new approach called symmetrized and generalized Bernoulli trials (SGBT), which relies on the general Bernoulli-trials scheme theory. This new modified algorithm uses a combination of \\\\\\\\\\\\\\\"Symmetrized and Simplified Bernoulli-Trails\\\\\\\\\\\\\\\" (SSBT) and Generalized Bernoulli-Trials (GBT) schemes to select collision pairs. The DSMC traditional collision process in a collision cell is built on the assumption that both particles of a colliding pair are chosen randomly from the list of all particles N in the cell l . This list can include particles located before and after the first selected particle. In contrast to the basic Bernoulli-trials SBT and GBT scheme, which algorithms are built on the pair selection from the top triangle of the collision probability matrix, in the symmetrized algorithms, the whole matrix is used, thus the first particle is selected in strict order from the list, and the second one j is chosen at random from the others. The newly suggested SSBT and SGBT collision schemes symmetrize the selection process; both symmetrized algorithms are evaluated on two fundamental problems: the normal shock wave and the Fourier heat transfer problem. The convergence of heat flux and shear stress of the normal shock was investigated. In general, the new SSBT and SGBT results obtained for both cases showed the same accuracy of the basic SBT and GBT and an improved convergence to the final solution, even when the number of particles per cell is very small. The solutions obtained by the symmetrized collision schemes are compared to the standard NTC solution. The new modification SGBT results accurately replicate the correct solution when an appropriate number of selected pairs Nsel < N is chosen.

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