Discretization Methods:

There are a variety of approaches to solving continuous-time system equations using discrete-time methods. In general, a higher sampling rate will produce more accurate results. In this section, we overview two different Finite Differences approximations, as well as the Bilinear Transform.

Numerous other methods exist for solving systems of partial differential equations. The interested reader could start with the following general overview of finite difference schemes.

For those interested in applications of finite-difference time-domain methods in musical acoustics, a few good references include:

• Brian Hamilton, Tutorial on finite-difference time-domain (FDTD) methods for room acoustics simulation.
• Silvin Willemsen, The Emulated Ensemble: Real-Time Simulation of Musical Instruments using Finite-Difference Time-Domain Methods, PhD thesis, Aalborg University, Denmark, 2021.
• Stefan Bilbao, Numerical sound synthesis: finite difference schemes and simulation in musical acoustics. John Wiley and Sons, 2009.