The dynamics of planetesimals plays an important role in planet formation because their velocity distribution sets the growth rate to larger bodies. When planetesimals form in the gaseous environment of protoplanetary discs, their orbits are nearly circular and planar due to the effect of gas drag. However, mutual close encounters of the planetesimals increase eccentricities and inclinations until an equilibrium between stirring and damping is reached. After disc dissipation there is no more gas that damps the motion and mutual close encounters as well as encounters with planets stir the orbits again. After disc dissipation there is no gas that can damp the motion, and mutual close encounters and encounters with planets can stir the orbits. The large number of planetesimals in protoplanetary discs makes it difficult to simulate their dynamics by means of direct N-body simulations of planet formation. Therefore, we developed a novel method for the dynamical evolution of planetesimals that is based on following close encounters between planetesimal-mass bodies and gravitational stirring by planet-mass bodies. To separate the orbital motion from the close encounters we employ a Hamiltonian splitting scheme, as used in symplectic N-body integrators. Close encounters are identified using a cell algorithm with linear scaling in the number of bodies. A grouping algorithm is used to create small groups of interacting bodies which are integrated separately. Our method can simulate a large number of planetesimals interacting through gravity and collisions at low computational cost. The typical computational time is of the order of minutes or hours, up to a few days for more complex simulations, compared to several hours or even weeks for the same setup with full N-body. The dynamical evolution of the bodies is sufficiently well reproduced. This will make it possible to study the growth of planetesimals through collisions and pebble accretion coupled to their dynamics for a much larger number of bodies than previously accessible with full N-body simulations.