Table 4.

Convergence of the reduced gradient computed via the SL method to the gradient computed via the RK2A method. We evaluate the reference gradient on a grid of size n_x= (512, 512)^⊤ via the RK2A method with a CFL number of 0.2. For the SL method the reduced gradient is computed on a grid of size n_x = (256, 256)^⊤ and n_x = (512, 512)^⊤ with a varying number of time steps n_t. We report the CFL number c, the associated number of time steps n_t, the relative ℓ²-error between numerical approximations to the reduced gradient g^h, and the wall-clock time for the evaluation of g^h. We consider the test problems SMOOTH A and SMOOTH B in Fig. 2 as input data. As a reference, we also provide relative errors for the RK2A scheme.

		SMOOTH A						SMOOTH B
$n_x^i$	c	run	nt	SL	time	RK2A	time	run	nt	SL	time	RK2A	time
256	10	#1	3	2.54E−3	4.51E−1	–	–	#2	9	2.28E−2	1.29E0	–	–
	5	#3	5	9.08E−4	1.02E0	–	–	#4	17	2.22E−2	2.19E0	–	–
	2	#5	11	2.19E−4	1.77E0	–	–	#6	41	2.21E−2	3.87E0	–	–
	1	#7	21	1.31E−4	2.59E0	–	–	#8	82	2.20E−2	9.55E0	–	–
	0.2	#9	102	1.21E−4	1.17E+1	1.21E−4	8.57E0	#10	408	2.20E−2	3.00E+1	2.19E−2	2.74E+1
512	10	#11	5	9.00E−4	3.66E0	–	–	#12	17	1.42E−3	7.59E0	–	–
	5	#13	9	2.74E−4	3.74E0	–	–	#14	33	3.94E−4	1.19E+1	–	–
	2	#15	21	4.93E−5	8.72E0	–	–	#16	82	7.89E−5	2.86E+1	–	–
	1	#17	41	1.23E−5	1.49E+1	–	–	#18	163	3.60E−5	6.13E+1	–	–
	0.2	#19	204	4.87E−7	7.32E+1	0	4.96E+1	#20	815	2.90E−5	2.65E+2	0	2.04E+2