Table 4.4. Example 2 results for the two partial derivatives of the solution (upper) and the adjoint solution (lower) at T = 10.

Δ t	∂1q (true)	∂1q (est)	∂1q (est) ∂1q (true)	True Error	Term IIIab	Term IIb	Error Estimate	Error Ratio	∂1q (est)+Error Estimate ∂1q (true)
0.04	−1.52E + 04	−1.91E + 04	1.25290	3.85E + 03	5.24E + 02	7.08E + 03	7.60E + 03	1.97601	0.75317
0.02	−1.52E + 04	−1.56E + 04	1.02488	3.78E + 02	4.93E + 01	2.60E + 02	3.10E + 02	0.81829	1.00452
0.01	−1.52E + 04	−1.53E + 04	1.00369	5.61E + 01	5.98E + 00	3.61E + 01	4.21E + 01	0.75009	1.00092
Δ t	∂2q (true)	∂2q (est)	∂2q (est) ∂2q (true)	True Error	Term IIIab	Term IIb	Error Estimate	Error Ratio	∂2q (est)+Error Estimate ∂2q (true)
0.04	6.66E + 02	7.03E + 02	1.05505	−3.67E + 01	1.17E + 03	−1.12E + 03	4.73E + 01	−1.29028	1.12609
0.02	6.66E + 02	6.79E + 02	1.01928	−1.28E + 01	8.71E + 01	−9.40E + 01	−6.94E + 00	0.54057	1.00886
0.01	6.66E + 02	6.68E + 02	1.00328	−2.19E + 00	1.02E + 01	−1.17E + 01	−1.48E + 00	0.67836	1.00106

Δ t	φ(0)	Φ(0)	Φ(0) φ(0)	True Error	Term II	Error Ratio	Φ(0)+Error Estimate φ(0)
0.04	1.52E + 03	1.90E + 03	1.25043	−3.81E + 02	−7.63E + 02	2.00188	0.74909
0.02	1.52E + 03	1.56E + 03	1.02465	−3.75E + 01	−3.11E + 01	0.83007	1.00419
0.01	1.52E + 03	1.53E + 03	1.00366	−5.57E + 00	−4.23E + 00	0.75982	1.00088