the present invention is a particle position solver that allows particles to reach an end of a fixed time , time step in an invalid state and which are then pushed toward a valid state . this allows the solver to continue the simulation while the simulation results move toward a valid behavior . particle collision calculations are simplified by adopting a non - sequential model ignoring some collisions or combining them . the speed of the simulation is also improved by performing some operations outside the solver calculation loop and by using specialized data structures .