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Correction terms in the Crank-Nicolson scheme (1)
[Dtu=−a¯ut]n+12,
Definition of the truncation error R and correction terms C:
[Dtue+a¯uet=C+R]n+12.
Must Taylor expand
- the derivative
- the arithmetic mean
Cn+12+Rn+12=124ue‴
Let C^{n+\half} cancel the \Delta t^2 terms:
C^{n+\half} =
\frac{1}{24}\uex'''(t_{n+\half})\Delta t^2
+ \frac{a}{8}\uex''(t_{n})\Delta t^2\tp