An energy-conserving difference scheme for nonlinear coupled transverse/longitudinal string vibration

In the quest for more realistic string sound synthesis, nonlinear (large-amplitude) effects have recently come under scrutiny. Though a mathematical description of the coupled longitudinal/transverse motion of such a string is straightforward, the development of numerical methods (and thus synthesis algorithms) is complicated due to stability considerations; frequency domain analysis cannot be fruitfully applied when nonlinearities are present. We present here a finite difference scheme for a nonlinear string whose stability can be guaranteed, not through frequency domain analysis, but through an exact discrete energy conservation property. Under certain simple conditions, the so-called energy method leads to bounds on the solution size in terms of initial conditions, and, thus, to a stability guarantee. Implementation details and numerical results are presented.