Table 3.

Convergence of the SL method to a reference solution computed via the RK2A scheme. We compute the reference solution on a grid of size n_x = (512, 512)^⊤ using the RK2A scheme with a CFL number of 0.2. We report results for varying discretization sizes. We report the CFL number c, the associated number of time steps n_t, and the relative ℓ²-error between the solution for the SL scheme and the reference solution computed via the RK2A scheme. We also report errors for the RK2A method as a reference (self convergence). We report results for the state equation (two blocks on the left; see (7a)) and the adjoint equation (two blocks on the right; see (8a)). We consider the test problems SMOOTH A and SMOOTH B in Fig. 2 for the velocity field and to set up the initial and terminal conditions, respectively.

		STATE EQ										ADJOINT EQ
		SMOOTH A					SMOOTH B					SMOOTH A					SMOOTH B
$n_x^i$	c	nt	run	SL	run	RK2A	nt	run	SL	run	RK2A	nt	run	SL	run	RK2A	nt	run	SL	run	RK2A
64	10	2	#1	5.84E−4		–	3	#2	8.11E−2		–	2	#3	1.57E−2		–	3	#4	4.18E−1		–
	5	2	#5	5.84E−4		—	5	#6	7.58E−2		–	2	#7	1.57E−2		–	5	#8	4.41E−1		–
	2	3	#9	2.67E−4		–	11	#10	7.99E−2		–	3	#11	7.44E−3		–	11	#12	4.90E−1		–
	1	6	#13	7.04E−5		–	21	#14	8.57E−2		–	6	#15	2.01E−3		–	21	#16	5.31E−1		–
	0.2	26	#17	5.10E−6	#18	1.66E−5	102	#19	9.81E−2	#20	7.66E−2	26	#21	1.14E−4	#22	1.71E−4	102	#23	6.06E−1	#24	4.19E−1
128	10	2	#25	5.84E−4		–	5	#26	1.78E−2		–	2	#27	1.57E−2		–	5	#28	7.10E−2		–
	5	3	#29	2.67E−4		–	9	#30	1.08E−2		–	3	#31	7.43E−3		–	9	#32	5.91E−2		–
	2	6	#33	6.90E−5		–	21	#34	9.55E−3		–	6	#35	2.01E−3		–	21	#36	5.90E−2		–
	1	11	#37	2.11E−5		–	41	#38	9.84E−3		–	11	#39	6.22E−4		–	41	#40	6.14E−2		–
	0.2	51	#41	1.33E−6	#42	4.10E−6	204	#43	1.07E−2	#44	9.83E−3	51	#45	3.08E−5	#46	4.25E−5	204	#47	6.65E−2	#48	5.94E−2
256	10	3	#49	2.67E−4		–	9	#50	5.06E−3		–	3	#51	7.43E−3		–	9	#52	1.55E−2		–
	5	5	#53	9.86E−5		–	17	#54	1.60E−3		–	5	#55	2.84E−3		–	17	#56	5.21E−3		–
	2	11	#57	2.10E−5		–	41	#58	5.02E−4		–	11	#59	6.21E−4		–	41	#60	2.09E−3		–
	1	21	#61	5.96E−6		–	82	#62	3.93E−4		–	21	#63	1.75E−4		–	82	#64	2.12E−3		–
	0.2	102	#65	4.92E−7	#66	8.19E−7	408	#67	4.21E−4	#68	1.20E−4	102	#69	8.61E−6	#70	8.52E−6	408	#71	2.47E−3	#72	7.31E−4
512	10	5	#73	9.86E−5		–	17	#74	1.47E−3		–	5	#75	2.84E−3		–	17	#76	4.60E−3		–
	5	9	#77	3.11E−5		–	33	#78	4.15E−4		–	9	#79	9.18E−4		–	33	#80	1.35E−3		–
	2	21	#81	5.94E−6		–	82	#82	8.61E−5		–	21	#83	1.75E−4		–	82	#84	2.85E−4		–
	1	41	#85	1.71E−6		–	163	#86	3.59E−5		–	41	#87	4.70E−5		–	163	#88	1.29E−4		–
	0.2	204	#89	3.19E−7	#90	0	815	#91	2.09E−5	#92	0	204	#93	3.73E−6	#94	0	815	#95	1.05E−4	#96	0