We introduce two new computer models in which the influence of the finite potential mesh is, for most practical purposes, eliminated. The Quiet Particle-Mesh model (QPM) uses a Gaussian-shaped charge cloud and careful shaping of the potential solution in either real or transform space. The QPM model is measured to be 150 times less noisy than the conventional CIC model and is suitable for large collisionless plasma simulations with 10 5 or more particles. This is achieved at little or no cost in computing time. The hybrid Particle-Particle/Particle-Mesh model (PPPM), on the other hand, adds to the above mesh calculation contributions from nearby particles by direct summation. This model has a spatial resolution which is independent of the mesh spacing and may be a small fraction of a mesh cell. It is more time-consuming and is suitable for high-resolution sub-Debye length investigations with a smaller number of particles (∼10 4). Applied to molecular dynamics, the PPPM model can simulate systems with approximately 100 times the number of particles than conventional methods.