Problems with FDS

• Convergence: Since the approximations to the second derivatives we used were second order accurate (in $T$ and $\Delta$ ), the scheme as a whole is accurate as $O(T^{2}, \Delta^{2})$ .

• 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.