High-Accuracy Approximation of High-Rank Derivatives: Isotropic Finite Differences Based on Lattice-Boltzmann Stencils

Table 3

Comparison of various approximations for low- and moderate-rank derivatives with second- and fourth-order accuracy. The computational times are given in milliseconds and the Empty load refers to the Mehrstellen case where only the actual approximation is skipped: the relative computational times are defined with respect to the empty load. References for the below compared approximations are as follows: Lee and Lin [11], Lee and Fischer [12], Patra and Karttunen [2], and Philippi et al. [13].

	Nodes	Span	C. time	Rel. time	Rel. 𝐿2-error
Empty load	—	—	7984	0.0	—
𝜕^2 (2nd-order approximation)
Mehrstellen	9	1	8435	0.0565	4.124×10^−5
Equation (41), D2V17	17	3	8813	0.1038	9.160×10^−5
𝜕𝑥 (4th-order approximation)
Lee, E^(6)	24	4	8978	0.1245	1.632×10^−8
Lee, D2V17	32	6	9341	0.1699	1.678×10^−8
Equation (40), D2V37	36	3	9547	0.1958	1.491×10^−8
𝜕𝑥𝜕^2 (2nd-order approximation)
Patra, 𝑐1=1/24	12	2	8641	0.0823	1.237×10^−4
Patra, 𝑐1=0	16	2	8844	0.1077	1.237×10^−4
Equation (39), D2V37	36	3	9688	0.2134	1.727×10^−4
𝜕^2 (4th-order approximation)
Patra, 𝑐1=−1/120	17	2	8803	0.1026	2.721×10^−9
Patra, 𝑐1=0	21	2	8900	0.1147	2.721×10^−9
Philippi, D2V37	37	3	9503	0.1903	4.969×10^−9
Equation (43), D2V49	49	5	10109	0.2662	1.758×10^−8
𝜕^2𝜕^2 (2nd-order approximation)
Patra, 𝑐1=1/12	17	2	8781	0.0998	8.247×10^−5
Patra, 𝑐1=0	21	2	8891	0.1136	8.248×10^−5
Equation (42), D2V49	49	5	10294	0.2893	1.875×10^−4
𝜕𝑥𝜕^2 (4th-order approximation)
Patra, 𝑐1=−17/2160	26	3	9006	0.1281	1.428×10^−8
Patra, 𝑐1=0	30	3	9435	0.1817	1.428×10^−8
Section 5, D2V81	80	5	11253	0.4095	3.456×10^−8