On the use of reduced grids in conjunction with the Equator-Pole grid system

Figures & data

Fig. 1. A small part $0\le \lambda \le {45}^o$ and $\theta \ge 0$ of a reduced grid for n = 160 and K = 3.

Table 1. Reduced lat-lon grids with n = 360 and K segments on a hemisphere, using method I. Let N_K be the total number of discretisation points for given K, p_K is the decrease in per cent of N₁, that is $N_K=(1-p_K/100)N_1,$ and the quantity s_i is the number of parallels in the segment $S_i^d.$ The area of the polar regions as per cent of the total area of the sphere is given by $q_K=100(1-\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta }_K).$

K	θ1	θ2	θ3	${\theta }_{K-1}$	θK	s1	s2	s3	$s_{K-1}$	sK	σK	pK	qK
1	45.0					56					1.1892	0	29
2	37.4	50.8				43	17				1.1269	9.5	23
3	34.0	46.8	55.1			38	14	11			1.0988	14.2	18
4	31.5	43.6	52.0		57.5	35	13	9		8	1.0848	16.2	16
5	29.1	40.3	48.8	55.3	60.0	32	12	9	7	7	1.0718	17.9	13
6	27.5	38.8	46.4	57.8	61.6	30	12	8	6	6	1.0641	19.4	12

Table 2. Let N be the total number of discretisation points and s_i the number of parallels in the segment $S_i^d.$ The areas of the polar regions as per cent of the total area of the sphere is given by $q_{10}=100(1-\hspace{0.17em}\text{sin}\hspace{0.17em}{\theta }_{10}).$

Grid system optimised according to method I
θ1	θ2	θ3	θ4	θ10	s1	s2	s3	s4	s10	σ	q10	N
23.9	33.3	40.1	45.6	65.9	251	98	71	58	28	1.0456	8.7	$4.504·{10}^6$
Grid system with square net rectangles at $\theta =\pm \pi /4$ in $S_4^d$
${\overline{\theta }}_1$	${\overline{\theta }}_2$	${\overline{\theta }}_3$	${\overline{\theta }}_4$	${\overline{\theta }}_{10}$	${\overline{s}}_1$	${\overline{s}}_2$	${\overline{s}}_3$	${\overline{s}}_4$	${\overline{s}}_{10}$	$\overline{\sigma }$	${\overline{q}}_{10}$	$\overline{N}$
25.0	34.9	42.0	47.7	68.2	264	103	75	60	27	1.0508	7.2	$4.535·{10}^6$

Table 3. Normalised errors for solid rotation of the Cosine bell, n = 360, $\Delta t=600$ s, T = 12 days, $\alpha =\pi /3,$ and minimisation over c.

Minimum of	K	${10}^8·c$	2p	s	${\mathcal{l}}_2(H,12)$	${\overline{r}}_m(12)$	${\overline{r}}_e(12)$
${\mathcal{l}}_2(H,12)$	1	66.3	6	3	2.6832e-03	1.6530e-06	9.1873e-06
${\overline{r}}_m(12)$	1	717	6	3	5.1712e-03	3.6007e-07	6.9466e-05
${\overline{r}}_e(12)$	1	3.9	6	3	3.1638e-03	3.3152e-06	5.4354e-07
${\mathcal{l}}_2(H,12)$	3	57.6	6	3	2.8474e-03	8.6327e-08	8.8857e-06
${\overline{r}}_m(12)$	3	111	6	3	2.9961e-03	2.4432e-08	1.6017e-05
${\overline{r}}_e(12)$	3	2.08	6	3	3.4038e-03	4.4039e-06	5.3897e-07
${\mathcal{l}}_2(H,12)$	1	146	4	2	7.9403e-03	1.0634e-05	2.5847e-04
${\overline{r}}_m(12)$	1	39.5	4	2	8.3855e-03	7.6221e-06	7.3130e-05
${\overline{r}}_e(12)$	1	0	4	2	1.2667e-02	2.2503e-05	8.9115e-06
${\mathcal{l}}_2(H,12)$	3	130	4	2	8.4252e-03	1.4219e-07	2.6351e-04
${\overline{r}}_m(12)$	3	164	4	2	8.4670e-03	4.6614e-08	3.3125e-04
${\overline{r}}_e(12)$	3	0.475	4	2	9.6882e-03	1.2742e-05	4.4044e-06

Table 4. Normalised errors for time dependent flow(Läuter), ${\overline{\mathcal{\ell }}}_2({\Psi }^{*},15)$ minimised over c, n = 360, T = 15 days, $\alpha =\pi /4.$

$\Delta t$	K	${10}^8·c$	2p	${\mathcal{l}}_2({\Psi }^{*},5)$	${\mathcal{l}}_2({\Psi }^{*},10)$	${\mathcal{l}}_2({\Psi }^{*},15)$
300	1	0.647	6	2.0990e-9	4.1368e-9	6.2114e-9
300	2	4.55	6	2.1000e-9	4.1388e-9	6.2162e-9
300	3	8.34	6	2.1009e-9	4.1406e-9	6.2194e-9
300	4	19.4	6	2.1017e-9	4.1425e-9	6.2246e-9
200	1	19.4	6	4.1770e-10	8.2351e-10	1.2441e-9
200	2	33.4	6	4.2473e-10	8.3782e-10	1.3045e-9
200	3	41.5	6	4.3385e-10	8.5626e-10	1.3103e-9
200	4	53.2	6	4.4641e-10	8.8151e-10	1.3570e-9
300	1	29.8	4	4.2311e-8	8.5435e-8	2.5160e-7
300	3	49.8	4	9.2164e-8	1.8420e-7	4.6289e-7
200	1	57.4	4	7.7793e-8	1.5523e-7	3.9484e-7
200	3	98.6	4	1.7989e-7	3.5752e-7	7.8727e-7

Table 5. Normalised errors for time dependent flow(Läuter), ${\overline{r}}_e(15)$ minimised over c, n = 360, $2p=6,\Delta t=300$ s, T = 15 days, $\alpha =\pi /4.$

K	${10}^8·c$	${\mathcal{\ell }}_2({\Psi }^{*},5)$	${\mathcal{\ell }}_2({\Psi }^{*},10)$	${\mathcal{\ell }}_2({\Psi }^{*},15)$	${\overline{r}}_m(15)$	${\overline{r}}_e(15)$
1	0.686	2.0990e-9	4.1368e-9	6.2114e-9	4.9431e-12	1.8120e-11
2	2.58	2.1000e-9	4.1388e-9	6.2175e-9	1.4348e-12	2.3018e-11
3	1.13	2.1008e-9	4.1418e-9	1.8382e-8	9.2212e-12	2.0840e-11
4	4.31	2.1011e-9	4.1610e-9	9.1909e-8	3.3190e-11	2.6892e-11

Fig. 2. Contour curves for the total height H of the fluid for the mountain problem, no. 5 from Williamson et al. (1992): n = 360, K = 3, 2p = 6, $c={10}^{-5},$ T = 15 days. Method I.

Table 6. Zonal flow over an isolated mountain, ${\theta }_c={30}^o,$ n = 360, K = 3, $\Delta t=300,$ T = 15 days, 6 or 4 before ${\overline{r}}_m(15)$ or ${\overline{r}}_e(15)$ denotes the approximation order.

Method	${10}^6·c$	6, ${\overline{r}}_m(15)$	6, ${\overline{r}}_e(15)$	${10}^6·c$	4, ${\overline{r}}_m(15)$	4, ${\overline{r}}_e(15)$
I	10	1.0986e-06	2.3144e-03	3	1.0277e-06	2.3085e-03
I	12	1.0977e-06	2.3143e-03	5	9.9984e-07	2.3039e-03
I	15	1.0985e-06	2.3142e-03	10	1.0358e-06	2.2925e-03
II, q = 6	0.4	3.4746e-07	2.3150e-03	1	5.5676e-07	2.3132e-03
II, q = 6	0.5	3.4208e-07	2.3150e-03	3	4.8095e-07	2.3086e-03
II, q = 6	1	3.5403e-07	2.3150e-03	5	4.8333e-07	2.3039e-03

Table 7. The mountain problem, ${\theta }_c={30}^o,$ n = 360, K = 3, $\Delta t=300,$ T = 5 years.

Method	2p	${10}^6·c$	$r_m(T)$	$r_e(T)$
I	6	12	$1.49·{10}^{-5}$	$1.12·{10}^{-2}$
II, q = 6	6	0.5	$0.87·{10}^{-5}$	$1.04·{10}^{-2}$

Williamson, D. L., Drake, J. B., Hack, J. J., Jakob, R. and Swarztrauber, P. N. 1992. A standard test set for numerical approximation to the shallow water equations in spherical geometry. J. Comput. Phys. 102, 211–224. doi:10.1016/S0021-9991(05)80016-6

 Google Scholar