## 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

 (D.1)

where denotes the time sampling interval and denotes the spatial sampling interval. Other types of finite-difference schemes were derived in Chapter 77.3.1), 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):

 (D.2)

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 (i.e., for all and ). Solving for in terms of displacement samples at earlier times yields an explicit finite-difference scheme for string displacement:

 (D.3)

The FDS is called explicit because it was possible to solve for the state at time as a function of the state at earlier times (and any other positions ). This allows it to be implemented as a time recursion (or digital filter'') which computes a solution at time from solution samples at earlier times (and any spatial positions). When an explicit FDS is not possible (e.g., a non-causal filter is derived), the discretized differential equation is said to define an implicit FDS. An implicit FDS can often be converted to an explicit FDS by a rotation of coordinates [55,481].

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