**Convergence:**Since the approximations to the second derivatives we used were second order accurate (in and ), the scheme as a whole is accurate as .- Making an FDS more
*accurate*(i.e., converge faster) generally requires a recursion involving more grid variables. - An FDS for a higher order PDE also generally involves more grid variables.
- From a signal processing point of view, a more accurate
simulation of an LTI medium is obtained by increasing the
*order*of the filter. - Note that an optimal filter design yields FDS coefficients which
may be translated back to differential equation coefficients (which
may or may not have physical meaning).
- Stability becomes more difficult to ensure in general (need to check
eigenvalue magnitudes). The addition of boundary conditions makes this
even more difficult.
- A good finite difference scheme may not be
*explicit*, and hence may require matrix inversions (generally sparse).

For example, the dependence diagram below represents an *implicit* scheme: We cannot calculate the grid variables at the
current timestep as weighted sums of grid variables at previous
instants.

Download NumericalInt.pdf

Download NumericalInt_2up.pdf

Download NumericalInt_4up.pdf

Copyright ©

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

[Automatic-links disclaimer]