String synthesis has been the subject of a lot of research. In its simplest form, one can model the displacement of a taut string by means of the 1D wave equation, and cheap algorithms such as digital waveguides may be employed. For more realistic simulations, though, one can try to simulate the inherent nonlinearities of string dynamics. These may be in the form of contact forces (bows, picks, fingers and fretboards), or geometric (i.e. arising from the large displacements, in some sense similar to what happens in plates). See also this paper. In the video, a colliding mass (a finger, or a pick) sets a string into motion. Notice the presence of the rigid curved barrier underneath.

BASS GUITAR SYNTHESIS
Synthesising the sounds of the bass guitar in a convincing manner is a challenging task. Thanks to the collision model described here though, coupled with a nonlinear string dynamics, one is able to reproduce in a faithful manner all the salient features, including pitch bends, slap and the typical "twang" that the string makes against the frets. Pretty impressive!

Open E (linear dynamics, single pluck)
Open E (nonlinear dynamics, multiple plucks)
Open D (nonlinear dynamics, multiple plucks and slap)