Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

## Finite Difference Schemes

Finite Difference Schemes (FDSs) aim to solve differential equations by means of finite differences. For example, as discussed in §G.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

 (L.1)

where denotes the time sampling interval and denotes the spatial sampling interval. 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, §G.1):

 (L.2)

where is a positive real constant (which turns out to be wave propagation speed). Then, as derived in §G.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:

 (L.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 [50,458].

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite and copy this work]