Table 4.3. Example 2 results for the two partial derivatives of the solution (upper) and the adjoint solution (lower) at T = 3.9.

Δ t	true ∂1q	est. ∂1q	Error	IIIab	IIb	Estimate	Error Estimate	est. ∂1q+Est. true ∂1q
0.3	−7.89E + 00	−4.42E + 00	−3.47E + 00	−4.06E + 00	−1.49E + 00	−5.55E + 00	1.60	1.26
0.15	−7.89E + 00	−9.32E + 00	1.43E + 00	5.88E – 01	8.18E – 01	1.41E + 00	0.986	1.00
0.075	−7.89E + 00	−8.11E + 00	2.12E – 01	8.05E – 02	1.01E – 01	1.82E – 01	0.858	1.00
Δ t	true ∂2q	est. ∂2q	Error	IIIab	IIb	Estimate	Error Estimate	est. ∂2q+Est. true ∂2q
0.3	−1.94E – 01	2.26E – 01	−4.20E – 01	9.44E – 01	1.09E + 00	2.03E + 00	−4.83	−11.7
0.15	−1.93E – 01	5.62E – 02	−2.50E – 01	−1.91E – 01	−5.68E – 02	−2.48E – 01	0.992	0.990
0.075	−1.93E – 01	−1.78E – 01	−1.55E – 02	−4.55E – 03	−1.44E – 02	−1.90E – 02	1.22	1.02

Δ t	φ(0)	Φ(0)	Error	II	Error Estimate	Φ(0)+ Est. φ(0)
0.3	2.02E + 00	2.49E + 00	−4.64E – 01	1.19E + 00	−2.57	1.82
0.15	2.02E + 00	2.35E + 00	−3.31E – 01	−3.54E – 01	1.07	0.988
0.075	2.02E + 00	2.07E + 00	−5.03E – 02	−4.93E – 02	0.981	1.00