Because the range of human hearing is bounded (nominally between 20 and 20 kHz), spectral components of a signal outside this range are not audible. Therefore, when the solution to a differential equation is to be considered an audio signal, there are frequency regions over which convergence is not a requirement.
Instead of pointwise convergence, we may ask for the following two properties:
- Superposition holds.
- Convergence occurs within the frequency band of human hearing.
In many cases, such as in digital waveguide modeling of vibrating strings, we can do better than convergence. We can construct finite difference schemes which agree with the corresponding continuous solutions exactly at the sample points. (See §C.4.1.)
Finite Difference Time Domain (FDTD) Scheme