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## Accuracy

Scheme (3.18) employs only one difference operator, a second-order accurate approximation to the operator . While it might be tempting to conclude that the scheme itself will generate a solution which converges to the solution of (3.1) with an error that depends on (which is in fact true in this case), the analysis of accuracy of an entire scheme is slightly more subtle than that applied to a single operator in isolation. It is again useful to consider the action of the operator on a continuous function . In this case, scheme (3.18) may be rewritten as

 (3.32)

or, using (2.6),

 (3.33)

The left hand side of the above equation would equal 0 for solving (3.1), but for approximation (3.18), there is a residual error on the order of the square of the time step .

In general, the accuracy of a given scheme will be at least that of the constituent operators; an interesting case study is presented in §3.3.4.

Next: Energy Analysis Up: A Simple Scheme Previous: Frequency Domain Analysis   Contents   Index
Stefan Bilbao 2006-11-15