Finite-Difference Schemes

*Finite-Difference Schemes* (FDSs) aim to solve differential
equations by means of finite differences. For example, as discussed
in §C.2, if
denotes the displacement in meters of a vibrating
string at time
seconds and position
meters, we may approximate
the first- and second-order partial derivatives by

where denotes the time sampling interval and denotes the spatial sampling interval. Other types of finite-difference schemes were derived in Chapter 7 (§7.3.1), including a look at frequency-domain properties. These finite-difference approximations to the partial derivatives may be used to compute solutions of differential equations on a discrete grid:

Let us define an abbreviated notation for the grid variables

and consider the ideal string wave equation (cf, §C.1):

where is a positive real constant (which turns out to be wave propagation speed). Then, as derived in §C.2, setting and substituting the finite-difference approximations into the ideal wave equation leads to the relation

everywhere on the time-space grid (

The FDS is called

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University