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

Table 2

Specification of a sixth-order (D2V81) and eighth-order (D2V141) lattice-Boltzmann stencil: 𝑎_𝑠=0.970008498739395 and 𝑎_𝑠=0.8369204054303525, respectively. The second column indicates the number of velocity vectors, obtained by permutating the vector components, sharing the given weight.

𝐜^⋆𝑖	𝑝	𝑤𝑖 (D2V81)	𝑤𝑖 (D2V141)
(0, 0)	1	1.5381368561993594×10^−1	1.1508949125706189×10^−1
(±1, 0)	4	9.0384577224028734×10^−2	7.5595334899625166×10^−2
(±1, ±1)	4	6.0915016936039915×10^−2	5.7734363121950370×10^−2
(±2, 0)	4	2.4220944974511140×10^−2	2.9033095241565582×10^−2
(±2, ±1)	8	1.3146486542343966×10^−2	1.8074672747613353×10^−2
(±2, ±2)	4	3.9625069477690709×10^−3	7.4435479269746455×10^−3
(±3, 0)	4	1.8542634450574055×10^−3	4.2549928887228462×10^−3
(±3, ±1)	8	1.6035095499280431×10^−3	3.7759145451527918×10^−3
(±3, ±2)	8	2.2249142091156890×10^−4	9.5510661516062627×10^−4
(±4, 0)	4	1.0005397521769796×10^−4	4.9926973701248283×10^−4
(±4, ±1)	8	3.4398327658312326×10^−5	2.1766901272228135×10^−4
(±3, ±3)	4	5.4494804327695976×10^−5	2.7427360033189315×10^−4
(±4, ±2)	8	1.8837067344235765×10^−5	1.3852638332696371×10^−4
(±5, 0)	4	2.4044498210968849×10^−6	1.2905073342509215×10^−5
(±4, ±3)	8	—	5.9323648903820310×10^−6
(±5, ±1)	8	—	1.6161185887309810×10^−5
(±5, ±2)	8	4.3501093550344725×10^−7	2.4159493337948245×10^−6
(±4, ±4)	4	—	3.3101853527875648×10^−6
(±5, ±3)	8	—	1.3086410701892049×10^−6
(±6, 0)	4	—	1.1118416570950374×10^−7
(±6, ±1)	8	—	4.6121305137932601×10^−7
(±6, ±3)	8	—	3.6530518727364592×10^−8
(±7, 0)	4	—	8.3273853753395782×10^−9
(±7, ±2)	8	—	2.3109247814347261×10^−9