Introduction of the Adding and Doubling Method for Solving Bateman Equations for Nuclear Fuel Depletion

Figures & data

TABLE I Initial Number Densities ${\mathbf{N}}_0$ for All Test Cases Representing a Homogenized PWR Fuel Pellet with Light Water Coolant

Nuclide	Density$\left(\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}\right)$	Nuclide	Density$\left(\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}\right)$
Hydrogen-1	$0.049457000000000$	Uranium-234	$8.4512220988817\times {10}^{-6}$
Oxygen-16	$0.046688131274253$	Uranium-235	$0.0010518948891610$
Oxygen-17	$1.7748234213217\times {10}^{-5}$	Uranium-238	$0.022292593642974$

TABLE II The 297-Nuclide 1.0-s Test Case Comparison of Different CRAM Approximation Orders to the CRAM-48 Solution

Order	Time (ms)	MNARD	ANARD	NNND
4	$1.190\pm 0.046$	$9.87\times {10}^7$	$3.58\times {10}^5$	38
8	$1.861\pm 0.059$	$2.65\times {10}^4$	$1.53\times {10}^2$	25
12	$2.536\pm 0.070$	$7.02\times {10}^2$	$3.18\times {10}^0$	10
16	$3.228\pm 0.094$	$4.08\times {10}^0$	$1.73\times {10}^{-2}$	1
20	$3.868\pm 0.087$	$1.16\times {10}^{-2}$	$4.70\times {10}^{-5}$	0
24	$4.534\pm 0.094$	$6.67\times {10}^{-5}$	$3.88\times {10}^{-5}$	0
28	$5.208\pm 0.106$	$2.42\times {10}^{-8}$	$9.32\times {10}^{-11}$	0
32	$5.872\pm 0.112$	$2.23\times {10}^{-11}$	$8.90\times {10}^{-14}$	0
36	$6.535\pm 0.142$	$1.46\times {10}^{-14}$	$7.32\times {10}^{-16}$	0
40	$7.175\pm 0.148$	$7.63\times {10}^{-15}$	$7.04\times {10}^{-16}$	0
44	$7.863\pm 0.145$	$9.84\times {10}^{-15}$	$2.61\times {10}^{-15}$	0
48	$8.499\pm 0.162$	—	—	0

Fig. 1. The 297-nuclide 1.0-s test case relative differences computed for various CRAM approximation orders relative to the CRAM-48 solution.

TABLE III The 297-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM to the CRAM-48 Solution

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$28.102\pm 3.781$	$7.31\times {10}^4$	$3.47\times {10}^2$	11	$209.533\pm 4.456$	$5.42\times {10}^{-5}$	$1.69\times {10}^{-6}$
2	$49.388\pm 5.535$	$3.37\times {10}^2$	$2.67\times {10}^0$	12	$225.029\pm 3.715$	$1.36\times {10}^{-5}$	$4.22\times {10}^{-7}$
3	$66.955\pm 4.822$	$1.01\times {10}^1$	$1.74\times {10}^{-1}$	13	$239.717\pm 5.471$	$3.39\times {10}^{-6}$	$1.06\times {10}^{-7}$
4	$88.117\pm 6.205$	$1.18\times {10}^0$	$3.09\times {10}^{-2}$	14	$256.733\pm 6.293$	$8.48\times {10}^{-7}$	$2.64\times {10}^{-8}$
5	$106.092\pm 6.144$	$2.39\times {10}^{-1}$	$7.11\times {10}^{-3}$	15	$272.353\pm 6.443$	$2.12\times {10}^{-7}$	$6.59\times {10}^{-9}$
6	$123.549\pm 6.542$	$5.66\times {10}^{-2}$	$1.74\times {10}^{-3}$	16	$288.142\pm 7.339$	$5.30\times {10}^{-8}$	$1.65\times {10}^{-9}$
7	$142.001\pm 6.557$	$1.39\times {10}^{-2}$	$4.33\times {10}^{-4}$	17	$301.846\pm 7.313$	$1.32\times {10}^{-8}$	$4.16\times {10}^{-10}$
8	$158.481\pm 6.102$	$3.48\times {10}^{-3}$	$1.08\times {10}^{-4}$	18	$315.530\pm 7.611$	$3.31\times {10}^{-9}$	$1.10\times {10}^{-10}$
9	$172.409\pm 4.459$	$8.68\times {10}^{-4}$	$2.70\times {10}^{-5}$	19	$332.502\pm 7.635$	$8.32\times {10}^{-10}$	$4.24\times {10}^{-11}$
10	$193.101\pm 5.045$	$2.17\times {10}^{-4}$	$6.75\times {10}^{-6}$	20	$346.539\pm 8.824$	$1.75\times {10}^{-10}$	$3.71\times {10}^{-11}$

Fig. 2. The 297-nuclide 1.0-s test case relative differences computed for various ADM approximation powers to the CRAM-48 solution.

TABLE IV The 297-Nuclide 1.0-s Test Case Number of Nuclides with CRAM-48 Calculated Number Densities Below the Given Numerical Cutoffs

Cutoff	Nuclides Below Cutoff	Cutoff	Nuclides Below Cutoff
${10}^{-200}$	0	${10}^{-50}$	52
${10}^{-150}$	11	${10}^{-30}$	79
${10}^{-100}$	29	—	—

TABLE V The 297-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-200}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-200}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$12.805\pm 0.585$	$7.31\times {10}^4$	$3.47\times {10}^2$	11	$71.406\pm 0.769$	$5.42\times {10}^{-5}$	$1.69\times {10}^{-6}$
2	$19.527\pm 0.295$	$3.37\times {10}^2$	$2.67\times {10}^0$	12	$76.391\pm 0.780$	$1.36\times {10}^{-5}$	$4.22\times {10}^{-7}$
3	$26.225\pm 0.453$	$1.01\times {10}^1$	$1.74\times {10}^{-1}$	13	$81.803\pm 0.893$	$3.39\times {10}^{-6}$	$1.06\times {10}^{-7}$
4	$32.257\pm 0.438$	$1.18\times {10}^0$	$3.09\times {10}^{-2}$	14	$86.383\pm 0.980$	$8.48\times {10}^{-7}$	$2.64\times {10}^{-8}$
5	$38.442\pm 0.588$	$2.39\times {10}^{-1}$	$7.11\times {10}^{-3}$	15	$91.313\pm 0.805$	$2.12\times {10}^{-7}$	$6.59\times {10}^{-9}$
6	$44.211\pm 0.509$	$5.66\times {10}^{-2}$	$1.74\times {10}^{-3}$	16	$96.251\pm 0.803$	$5.30\times {10}^{-8}$	$1.65\times {10}^{-9}$
7	$50.078\pm 0.621$	$1.39\times {10}^{-2}$	$4.33\times {10}^{-4}$	17	$102.088\pm 1.673$	$1.32\times {10}^{-8}$	$4.16\times {10}^{-10}$
8	$55.918\pm 0.650$	$3.48\times {10}^{-3}$	$1.08\times {10}^{-4}$	18	$106.656\pm 1.916$	$3.31\times {10}^{-9}$	$1.10\times {10}^{-10}$
9	$61.595\pm 0.783$	$8.68\times {10}^{-4}$	$2.70\times {10}^{-5}$	19	$109.628\pm 0.770$	$8.32\times {10}^{-10}$	$4.24\times {10}^{-11}$
10	$65.964\pm 0.636$	$2.17\times {10}^{-4}$	$6.75\times {10}^{-6}$	20	$113.387\pm 1.009$	$1.75\times {10}^{-10}$	$3.71\times {10}^{-11}$

TABLE VI The 297-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-150}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-150}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$7.963\pm 0.446$	$7.31\times {10}^4$	$3.49\times {10}^2$	11	$28.045\pm 0.547$	$5.42\times {10}^{-5}$	$1.16\times {10}^{-6}$
2	$10.170\pm 0.515$	$3.37\times {10}^2$	$2.35\times {10}^0$	12	$30.439\pm 0.783$	$1.36\times {10}^{-5}$	$2.91\times {10}^{-7}$
3	$12.274\pm 0.542$	$1.01\times {10}^1$	$1.30\times {10}^{-1}$	13	$31.418\pm 0.632$	$3.39\times {10}^{-6}$	$7.28\times {10}^{-8}$
4	$14.815\pm 0.541$	$1.18\times {10}^0$	$2.17\times {10}^{-2}$	14	$33.176\pm 0.699$	$8.48\times {10}^{-7}$	$1.82\times {10}^{-8}$
5	$16.672\pm 0.467$	$2.39\times {10}^{-1}$	$4.93\times {10}^{-3}$	15	$35.028\pm 0.695$	$2.12\times {10}^{-7}$	$4.55\times {10}^{-9}$
6	$18.972\pm 0.572$	$5.66\times {10}^{-2}$	$1.20\times {10}^{-3}$	16	$36.596\pm 0.746$	$5.30\times {10}^{-8}$	$1.14\times {10}^{-9}$
7	$20.752\pm 0.572$	$1.39\times {10}^{-2}$	$2.99\times {10}^{-4}$	17	$38.026\pm 0.614$	$1.32\times {10}^{-8}$	$2.88\times {10}^{-10}$
8	$22.644\pm 0.487$	$3.48\times {10}^{-3}$	$7.46\times {10}^{-5}$	18	$39.563\pm 0.746$	$3.31\times {10}^{-9}$	$7.75\times {10}^{-11}$
9	$24.624\pm 0.726$	$8.68\times {10}^{-4}$	$1.86\times {10}^{-5}$	19	$40.754\pm 0.856$	$8.32\times {10}^{-10}$	$3.45\times {10}^{-11}$
10	$26.567\pm 0.805$	$2.17\times {10}^{-4}$	$4.66\times {10}^{-6}$	20	$42.389\pm 0.779$	$1.75\times {10}^{-10}$	$3.50\times {10}^{-11}$

TABLE VII The 297-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-100}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-100}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$6.978\pm 0.517$	$4.18\times {10}^2$	$2.53\times {10}^0$	11	$20.694\pm 0.558$	$1.87\times {10}^{-5}$	$3.30\times {10}^{-7}$
2	$8.526\pm 0.505$	$1.58\times {10}^1$	$1.56\times {10}^{-1}$	12	$21.761\pm 0.615$	$3.96\times {10}^{-6}$	$7.98\times {10}^{-8}$
3	$10.050\pm 0.486$	$1.79\times {10}^0$	$2.53\times {10}^{-2}$	13	$23.058\pm 0.593$	$8.94\times {10}^{-7}$	$1.73\times {10}^{-8}$
4	$11.446\pm 0.513$	$3.51\times {10}^{-1}$	$5.67\times {10}^{-3}$	14	$23.872\pm 0.630$	$6.46\times {10}^{-7}$	$6.46\times {10}^{-9}$
5	$12.638\pm 0.434$	$8.23\times {10}^{-2}$	$1.38\times {10}^{-3}$	15	$25.324\pm 0.781$	$8.77\times {10}^{-7}$	$4.27\times {10}^{-9}$
6	$14.070\pm 0.512$	$2.03\times {10}^{-2}$	$3.42\times {10}^{-4}$	16	$26.250\pm 0.733$	$9.34\times {10}^{-7}$	$3.74\times {10}^{-9}$
7	$15.633\pm 0.698$	$5.04\times {10}^{-3}$	$8.54\times {10}^{-5}$	17	$27.248\pm 0.575$	$9.49\times {10}^{-7}$	$3.61\times {10}^{-9}$
8	$16.615\pm 0.518$	$1.26\times {10}^{-3}$	$2.13\times {10}^{-5}$	18	$28.295\pm 0.593$	$9.52\times {10}^{-7}$	$3.59\times {10}^{-9}$
9	$17.862\pm 0.385$	$3.14\times {10}^{-4}$	$5.33\times {10}^{-6}$	19	$29.581\pm 0.598$	$9.53\times {10}^{-7}$	$3.59\times {10}^{-9}$
10	$19.169\pm 0.496$	$7.77\times {10}^{-5}$	$1.33\times {10}^{-6}$	20	$30.271\pm 0.666$	$9.53\times {10}^{-7}$	$3.60\times {10}^{-9}$

TABLE VIII The 297-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-50}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$5.484\pm 0.194$	$3.22\times {10}^0$	$5.31\times {10}^{-2}$	11	$12.470\pm 0.492$	$9.74\times {10}^{-4}$	$4.48\times {10}^{-6}$
2	$6.201\pm 0.192$	$6.70\times {10}^{-1}$	$1.20\times {10}^{-2}$	12	$12.636\pm 0.323$	$9.76\times {10}^{-4}$	$4.48\times {10}^{-6}$
3	$7.005\pm 0.247$	$1.58\times {10}^{-1}$	$2.89\times {10}^{-3}$	13	$13.283\pm 0.418$	$9.77\times {10}^{-4}$	$4.47\times {10}^{-6}$
4	$7.777\pm 0.375$	$3.91\times {10}^{-2}$	$7.14\times {10}^{-4}$	14	$13.839\pm 0.435$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$
5	$8.972\pm 0.420$	$9.73\times {10}^{-3}$	$1.75\times {10}^{-4}$	15	$14.390\pm 0.436$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$
6	$8.941\pm 0.196$	$2.41\times {10}^{-3}$	$4.07\times {10}^{-5}$	16	$14.734\pm 0.246$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$
7	$9.594\pm 0.199$	$5.79\times {10}^{-4}$	$1.01\times {10}^{-5}$	17	$15.396\pm 0.415$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$
8	$10.632\pm 0.443$	$8.27\times {10}^{-4}$	$5.67\times {10}^{-6}$	18	$16.131\pm 0.468$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$
9	$11.033\pm 0.428$	$9.39\times {10}^{-4}$	$4.58\times {10}^{-6}$	19	$16.359\pm 0.452$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$
10	$11.380\pm 0.208$	$9.67\times {10}^{-4}$	$4.49\times {10}^{-6}$	20	$16.888\pm 0.402$	$9.77\times {10}^{-4}$	$4.48\times {10}^{-6}$

TABLE IX The 297-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	MNARD	ANARD	Power	MNARD	ANARD
1	$1.25\times {10}^{-1}$	$2.68\times {10}^{-3}$	11	$1.19\times {10}^{-7}$	$2.56\times {10}^{-9}$
2	$3.13\times {10}^{-2}$	$6.71\times {10}^{-4}$	12	$2.98\times {10}^{-8}$	$6.40\times {10}^{-10}$
3	$7.81\times {10}^{-3}$	$1.68\times {10}^{-4}$	13	$7.45\times {10}^{-9}$	$1.60\times {10}^{-10}$
4	$1.95\times {10}^{-3}$	$4.19\times {10}^{-5}$	14	$1.86\times {10}^{-9}$	$4.06\times {10}^{-11}$
5	$4.88\times {10}^{-4}$	$1.05\times {10}^{-5}$	15	$4.66\times {10}^{-10}$	$1.10\times {10}^{-11}$
6	$1.22\times {10}^{-4}$	$2.62\times {10}^{-6}$	16	$1.19\times {10}^{-10}$	$5.37\times {10}^{-12}$
7	$3.05\times {10}^{-5}$	$6.55\times {10}^{-7}$	17	$3.31\times {10}^{-11}$	$4.68\times {10}^{-12}$
8	$7.63\times {10}^{-6}$	$1.64\times {10}^{-7}$	18	$2.07\times {10}^{-11}$	$8.39\times {10}^{-12}$
9	$1.91\times {10}^{-6}$	$4.09\times {10}^{-8}$	19	$4.91\times {10}^{-11}$	$2.01\times {10}^{-11}$
10	$4.77\times {10}^{-7}$	$1.02\times {10}^{-8}$	20	$9.44\times {10}^{-11}$	$3.45\times {10}^{-11}$

Fig. 3. The 297-nuclide 1.0-s test case relative differences computed for various ADM approximation powers with various cutoffs to the CRAM-48 solution: (a) cutoff of ${10}^{-200}$, (b) cutoff of ${10}^{-150}$, (c) cutoff of ${10}^{-100}$, and (d) cutoff of ${10}^{-50}$.

TABLE X The 297-Nuclide 1.0-s Test Case Run-Time Comparisons for ADM-16 with a Cutoff of ${10}^{-50}$ with Varying $m$ and $v$ Numbers

$m$	$v$	Time (ms)	$m$	$v$	Time (ms)
16	0	$14.734\pm 0.246$	7	9	$15.471\pm 0.300$
15	1	$13.989\pm 0.273$	6	10	$21.639\pm 0.384$
14	2	$13.329\pm 0.297$	5	11	$34.539\pm 0.570$
13	3	$12.677\pm 0.289$	4	12	$60.035\pm 0.894$
12	4	$12.083\pm 0.239$	3	13	$111.901\pm 1.042$
11	5	$11.569\pm 0.195$	2	14	$215.705\pm 1.624$
10	6	$11.405\pm 0.257$	1	15	$419.896\pm 4.308$
9	7	$11.788\pm 0.389$	0	16	$832.604\pm 3.376$
8	8	$12.754\pm 0.255$	—	—	—

TABLE XI The 297-Nuclide 3600- and $3.6\times {10}^6$-s Test Case Comparison of ADM-16 with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Depletion Interval (s)	MNARD	ANARD
3600	$2.32\times {10}^{-9}$	$5.92\times {10}^{-11}$
$3.6\times {10}^6$	$4.00\times {10}^{-8}$	$7.03\times {10}^{-10}$

Fig. 4. The 297-nuclide 3600- and $3.6\times {10}^6$-s test case relative differences computed for ADM-16 with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XII The 1450-Nuclide 1.0-s Test Case Comparison of Various Approximation Orders of CRAM to the CRAM-48 Solution

Order	Time (ms)	MNARD	ANARD	NNND
4	$7.695\pm 0.323$	$2.56\times {10}^{18}$	$3.97\times {10}^{15}$	210
8	$10.976\pm 0.423$	$2.27\times {10}^{11}$	$1.23\times {10}^9$	149
12	$14.567\pm 0.402$	$4.74\times {10}^8$	$7.84\times {10}^5$	144
16	$17.897\pm 0.453$	$6.53\times {10}^6$	$1.73\times {10}^4$	104
20	$21.283\pm 0.524$	$9.32\times {10}^4$	$2.26\times {10}^2$	50
24	$25.081\pm 0.741$	$6.82\times {10}^2$	$1.54\times {10}^0$	19
28	$28.018\pm 0.576$	$3.00\times {10}^0$	$6.41\times {10}^{-3}$	2
32	$31.366\pm 0.687$	$8.82\times {10}^{-3}$	$1.80\times {10}^{-5}$	0
36	$35.200\pm 0.780$	$1.88\times {10}^{-5}$	$3.70\times {10}^{-8}$	0
40	$38.106\pm 0.668$	$3.08\times {10}^{-8}$	$5.87\times {10}^{-11}$	0
44	$41.362\pm 0.692$	$4.04\times {10}^{-11}$	$7.64\times {10}^{-14}$	0
48	$44.983\pm 0.630$	—	—	0

Fig. 5. The 1450-nuclide 1.0-s test case relative differences computed for various CRAM approximation orders to the CRAM-48 solution.

TABLE XIII The 1450-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$87.527\pm 3.245$	$1.26\times {10}^0$	$1.69\times {10}^{-2}$	11	$171.336\pm 1.837$	$1.06\times {10}^{-6}$	$1.14\times {10}^{-8}$
2	$97.568\pm 1.499$	$2.88\times {10}^{-1}$	$3.37\times {10}^{-3}$	12	$177.757\pm 2.024$	$2.65\times {10}^{-7}$	$2.84\times {10}^{-9}$
3	$106.814\pm 1.122$	$7.02\times {10}^{-2}$	$7.47\times {10}^{-4}$	13	$184.864\pm 1.944$	$6.64\times {10}^{-8}$	$7.11\times {10}^{-10}$
4	$115.902\pm 1.174$	$1.74\times {10}^{-2}$	$1.86\times {10}^{-4}$	14	$191.530\pm 2.048$	$1.66\times {10}^{-8}$	$1.78\times {10}^{-10}$
5	$124.660\pm 1.493$	$4.35\times {10}^{-3}$	$4.66\times {10}^{-5}$	15	$197.297\pm 1.780$	$4.15\times {10}^{-9}$	$4.47\times {10}^{-11}$
6	$133.319\pm 1.476$	$1.09\times {10}^{-3}$	$1.17\times {10}^{-5}$	16	$203.416\pm 1.711$	$1.04\times {10}^{-9}$	$1.28\times {10}^{-11}$
7	$141.832\pm 2.368$	$2.72\times {10}^{-4}$	$2.91\times {10}^{-6}$	17	$209.008\pm 1.493$	$2.59\times {10}^{-10}$	$6.58\times {10}^{-12}$
8	$148.941\pm 1.506$	$6.79\times {10}^{-5}$	$7.29\times {10}^{-7}$	18	$214.349\pm 1.892$	$6.84\times {10}^{-11}$	$9.06\times {10}^{-12}$
9	$156.760\pm 1.196$	$1.70\times {10}^{-5}$	$1.82\times {10}^{-7}$	19	$222.910\pm 6.465$	$6.40\times {10}^{-11}$	$1.64\times {10}^{-11}$
10	$163.923\pm 1.588$	$4.25\times {10}^{-6}$	$4.55\times {10}^{-8}$	20	$225.825\pm 1.937$	$1.30\times {10}^{-10}$	$3.25\times {10}^{-11}$

Fig. 6. The 1450-nuclide 1.0-s test case relative differences computed for various ADM approximation powers with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XIV The 1450-Nuclide 1.0-s Run-Time Comparisons of Various $m$ and $v$ Values for ADM-17 with a Cutoff of ${10}^{-50}$

$m$	$v$	Time (ms)	$m$	$v$	Time (ms)
17	0	$209.008\pm 1.493$	8	9	$187.608\pm 1.666$
16	1	$195.543\pm 1.462$	7	10	$239.931\pm 1.727$
15	2	$185.906\pm 2.944$	6	11	$351.685\pm 1.954$
14	3	$176.691\pm 1.550$	5	12	$577.568\pm 2.041$
13	4	$168.687\pm 2.010$	4	13	$1028.398\pm 2.842$
12	5	$161.333\pm 1.298$	3	14	$1925.427\pm 4.519$
11	6	$156.377\pm 1.274$	2	15	$3688.520\pm 33.674$
10	7	$156.375\pm 1.264$	1	16	$7122.559\pm 142.493$
9	8	$167.378\pm 3.972$	0	17	$14333.752\pm 290.456$

TABLE XV The 1450-Nuclide 3600- and $3.6\times {10}^6$-s Test Case Comparison of ADM-17 with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Depletion Interval (s)	MNARD	ANARD
3600	$3.21\times {10}^{-9}$	$1.91\times {10}^{-11}$
$3.6\times {10}^6$	$3.38\times {10}^{-6}$	$2.61\times {10}^{-9}$

Fig. 7. The 1450-nuclide 3600- and $3.6\times {10}^6$-s test case relative differences computed for ADM-17 with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XVI The 1599-Nuclide 1.0-s Test Case Comparison of Various Approximation Orders of CRAM to the CRAM-48 Solution

Order	Time (ms)	MNARD	ANARD	NNND
4	$8.482\pm 0.382$	$2.56\times {10}^{18}$	$3.60\times {10}^{15}$	193
8	$11.862\pm 0.359$	$2.27\times {10}^{11}$	$1.11\times {10}^9$	137
12	$15.419\pm 0.441$	$4.74\times {10}^8$	$7.07\times {10}^5$	119
16	$18.678\pm 0.468$	$6.53\times {10}^6$	$1.56\times {10}^4$	88
20	$22.072\pm 0.498$	$9.32\times {10}^4$	$2.04\times {10}^2$	43
24	$25.574\pm 0.593$	$6.82\times {10}^2$	$1.39\times {10}^0$	18
28	$29.179\pm 0.511$	$3.00\times {10}^0$	$5.80\times {10}^{-3}$	2
32	$32.891\pm 0.603$	$8.82\times {10}^{-3}$	$1.63\times {10}^{-5}$	0
36	$36.080\pm 0.751$	$1.88\times {10}^{-5}$	$3.35\times {10}^{-8}$	0
40	$39.386\pm 0.714$	$3.08\times {10}^{-8}$	$5.32\times {10}^{-11}$	0
44	$42.699\pm 0.778$	$4.04\times {10}^{-11}$	$6.92\times {10}^{-14}$	0
48	$46.326\pm 0.759$	—	—	0

Fig. 8. The 1599-nuclide 1.0-s test case relative differences computed for various CRAM approximation orders to the CRAM-48 solution.

TABLE XVII The 1599-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$106.663\pm 7.069$	$1.26\times {10}^0$	$8.39\times {10}^{-2}$	11	$200.935\pm 6.343$	$1.28\times {10}^{-2}$	$1.07\times {10}^{-5}$
2	$117.649\pm 7.568$	$1.00\times {10}^0$	$5.75\times {10}^{-2}$	12	$209.311\pm 6.492$	$2.65\times {10}^{-7}$	$2.69\times {10}^{-9}$
3	$127.183\pm 5.396$	$1.00\times {10}^0$	$3.78\times {10}^{-2}$	13	$215.783\pm 6.615$	$6.64\times {10}^{-8}$	$6.66\times {10}^{-10}$
4	$138.153\pm 6.418$	$1.00\times {10}^0$	$1.53\times {10}^{-2}$	14	$224.172\pm 6.700$	$1.66\times {10}^{-8}$	$1.67\times {10}^{-10}$
5	$149.537\pm 7.497$	$9.99\times {10}^{-1}$	$1.63\times {10}^{-3}$	15	$231.779\pm 7.775$	$4.15\times {10}^{-9}$	$4.20\times {10}^{-11}$
6	$158.449\pm 7.637$	$9.96\times {10}^{-1}$	$8.40\times {10}^{-4}$	16	$238.346\pm 6.112$	$1.04\times {10}^{-9}$	$1.21\times {10}^{-11}$
7	$168.013\pm 6.623$	$9.83\times {10}^{-1}$	$8.21\times {10}^{-4}$	17	$244.053\pm 6.819$	$2.59\times {10}^{-10}$	$6.39\times {10}^{-12}$
8	$177.483\pm 7.421$	$9.34\times {10}^{-1}$	$7.79\times {10}^{-4}$	18	$249.185\pm 5.953$	$6.84\times {10}^{-11}$	$9.03\times {10}^{-12}$
9	$184.055\pm 6.217$	$7.62\times {10}^{-1}$	$6.34\times {10}^{-4}$	19	$256.358\pm 6.934$	$6.40\times {10}^{-11}$	$1.68\times {10}^{-11}$
10	$192.735\pm 6.577$	$3.36\times {10}^{-1}$	$2.80\times {10}^{-4}$	20	$262.023\pm 5.549$	$1.30\times {10}^{-10}$	$3.27\times {10}^{-11}$

Fig. 9. The 1599-nuclide 1.0-s test case relative differences computed for various ADM approximation powers with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XVIII The 1599-Nuclide 1.0-s Test Case Run-Time Comparisons for Various $m$ and $v$ Values for ADM-17 with a Cutoff of ${10}^{-50}$

$m$	$v$	Time (ms)	$m$	$v$	Time (ms)
17	0	$244.053\pm 6.819$	8	9	$204.485\pm 1.116$
16	1	$234.120\pm 12.414$	7	10	$258.295\pm 2.988$
15	2	$212.482\pm 6.629$	6	11	$367.385\pm 0.949$
14	3	$198.908\pm 1.089$	5	12	$593.511\pm 3.287$
13	4	$188.696\pm 1.460$	4	13	$1044.161\pm 2.669$
12	5	$181.396\pm 2.622$	3	14	$1956.950\pm 58.433$
11	6	$176.521\pm 2.888$	2	15	$3704.073\pm 14.029$
10	7	$175.256\pm 2.989$	1	16	$7180.476\pm 138.903$
9	8	$182.397\pm 3.520$	0	17	$13952.969\pm 137.753$

TABLE XIX The 1599-Nuclide 3600- and $3.6\times {10}^6$-s Test Case Comparison of ADM-17 with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Depletion Interval (s)	MNARD	ANARD	NNND
3600	$0.00704$	$4.92\times {10}^{-6}$	0
$3.6\times {10}^6$	$3.53$	$0.00228$	1

Fig. 10. The 1599-nuclide 3600- and $3.6\times {10}^6$-s test case relative differences computed for ADM-17 with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XX The 1599-Nuclide 3600- and $3.6\times {10}^6$-s Comparison of ADM-25 with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Depletion Interval (s)	MNARD	ANARD
3600	$3.63\times {10}^{-9}$	$1.31\times {10}^{-9}$
$3.6\times {10}^6$	$4.55\times {10}^{-9}$	$5.14\times {10}^{-10}$

Fig. 11. The 1599-nuclide 3600- and $3.6\times {10}^6$-s test case relative differences computed for ADM-25 with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XXI The 35-Nuclide 1.0-s Test Case Comparison of Various CRAM Approximation Orders to the CRAM-48 Solution

Order	Time (ms)	MNARD	ANARD	NNND
4	$0.051\pm 0.006$	$5.19\times {10}^2$	$2.49\times {10}^1$	9
8	$0.076\pm 0.007$	$1.68\times {10}^1$	$6.36\times {10}^{-1}$	2
12	$0.106\pm 0.020$	$5.75\times {10}^{-2}$	$1.99\times {10}^{-3}$	0
16	$0.126\pm 0.012$	$8.56\times {10}^{-5}$	$2.82\times {10}^{-6}$	0
20	$0.154\pm 0.013$	$7.76\times {10}^{-8}$	$2.48\times {10}^{-9}$	0
24	$0.174\pm 0.012$	$6.20\times {10}^{-5}$	$3.93\times {10}^{-5}$	0
28	$0.193\pm 0.012$	$2.55\times {10}^{-14}$	$2.14\times {10}^{-15}$	0
32	$0.229\pm 0.014$	$9.48\times {10}^{-15}$	$4.20\times {10}^{-15}$	0
36	$0.248\pm 0.015$	$5.54\times {10}^{-15}$	$1.11\times {10}^{-15}$	0
40	$0.268\pm 0.022$	$4.36\times {10}^{-15}$	$6.65\times {10}^{-16}$	0
44	$0.301\pm 0.020$	$9.48\times {10}^{-15}$	$2.01\times {10}^{-15}$	0
48	$0.317\pm 0.018$	—	—	0

Fig. 12. The 35-nuclide 1.0-s test case relative differences computed for various CRAM approximation orders relative to the CRAM-48 solution.

TABLE XXII The 35-Nuclide 1.0-s Test Case Comparisons for Various Approximation Powers of ADM with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	$0.061\pm 0.011$	$1.25\times {10}^{-1}$	$1.00\times {10}^{-2}$	11	$0.197\pm 0.015$	$1.19\times {10}^{-7}$	$9.54\times {10}^{-9}$
2	$0.078\pm 0.008$	$3.13\times {10}^{-2}$	$2.50\times {10}^{-3}$	12	$0.209\pm 0.013$	$2.98\times {10}^{-8}$	$2.38\times {10}^{-9}$
3	$0.087\pm 0.008$	$7.81\times {10}^{-3}$	$6.25\times {10}^{-4}$	13	$0.222\pm 0.017$	$7.45\times {10}^{-9}$	$5.96\times {10}^{-10}$
4	$0.106\pm 0.013$	$1.95\times {10}^{-3}$	$1.56\times {10}^{-4}$	14	$0.241\pm 0.020$	$1.86\times {10}^{-9}$	$1.50\times {10}^{-10}$
5	$0.118\pm 0.013$	$4.88\times {10}^{-4}$	$3.91\times {10}^{-5}$	15	$0.250\pm 0.015$	$4.65\times {10}^{-10}$	$3.82\times {10}^{-11}$
6	$0.129\pm 0.012$	$1.22\times {10}^{-4}$	$9.77\times {10}^{-6}$	16	$0.263\pm 0.018$	$1.19\times {10}^{-10}$	$1.22\times {10}^{-11}$
7	$0.140\pm 0.011$	$3.05\times {10}^{-5}$	$2.44\times {10}^{-6}$	17	$0.275\pm 0.014$	$3.27\times {10}^{-11}$	$7.22\times {10}^{-12}$
8	$0.155\pm 0.013$	$7.63\times {10}^{-6}$	$6.10\times {10}^{-7}$	18	$0.290\pm 0.021$	$2.00\times {10}^{-11}$	$9.76\times {10}^{-12}$
9	$0.167\pm 0.012$	$1.91\times {10}^{-6}$	$1.53\times {10}^{-7}$	19	$0.303\pm 0.019$	$4.86\times {10}^{-11}$	$1.90\times {10}^{-11}$
10	$0.187\pm 0.012$	$4.77\times {10}^{-7}$	$3.81\times {10}^{-8}$	20	$0.320\pm 0.023$	$8.70\times {10}^{-11}$	$3.72\times {10}^{-11}$

Fig. 13. The 35-nuclide 1.0-s test case relative differences computed for various ADM approximation powers with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XXIII The 35-Nuclide 1.0-s Test Case Run-Time Comparisons of Various m and $v$ Values for ADM-16 with a Cutoff of 10⁻⁵⁰ with Varying m and v Numbers

$m$	$v$	Time (ms)	$m$	$v$	Time (ms)
16	0	$0.263\pm 0.018$	7	9	$0.434\pm 0.032$
15	1	$0.272\pm 0.022$	6	10	$0.699\pm 0.034$
14	2	$0.245\pm 0.021$	5	11	$1.247\pm 0.058$
13	3	$0.234\pm 0.019$	4	12	$2.343\pm 0.076$
12	4	$0.223\pm 0.021$	3	13	$4.581\pm 0.150$
11	5	$0.211\pm 0.012$	2	14	$8.994\pm 0.360$
10	6	$0.218\pm 0.015$	1	15	$17.400\pm 0.312$
9	7	$0.240\pm 0.016$	0	16	$32.036\pm 0.489$
8	8	$0.298\pm 0.019$	—	—	—

TABLE XXIV The 35-Nuclide 3600- and $3.6\times {10}^6$-s Test Case Comparison of ADM-16 with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Depletion Interval [s]	MNARD	ANARD
3,600	$9.64\times {10}^{-9}$	$9.27\times {10}^{-10}$
$3.6\times {10}^6$	$8.18\times {10}^{-9}$	$6.62\times {10}^{-10}$

Fig. 14. The 35-nuclide 3600- and $3.6\times {10}^6$-s test case relative differences computed for ADM-16 with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XXV The 693-Nuclide 1.0-s Test Case Comparison of Various CRAM Approximation Orders to the CRAM-48 Solution

Order	Time (ms)	MNARD	ANARD	NNND
4	3.203 ± 0.154	1.14 × 10^15	3.27 × 10^12	122
8	5.048 ± 0.196	4.60 × 10^10	9.88 × 10^7	106
12	6.795 ± 0.280	8.06 × 10^6	2.94 × 10^4	75
16	8.702 ± 0.323	3.92 × 10^5	9.80 × 10^2	29
20	10.414 ± 0.324	4.10 × 10^3	9.47 × 10^0	13
24	12.360 ± 0.279	2.27 × 10^1	4.92 × 10^–2	3
28	13.956 ± 0.376	7.71 × 10^–2	1.60 × 10^–4	0
32	15.745 ± 0.310	1.80 × 10^–4	3.59 × 10^–7	0
36	17.429 ± 0.348	3.11 × 10^–7	6.00 × 10^–10	0
40	19.490 ± 0.444	4.19 × 10^–10	7.89 × 10^–13	0
44	20.636 ± 0.329	4.66 × 10^–13	2.06 × 10^–15	0
48	21.151 ± 0.269	—	—	0

Fig. 15. The 693-nuclide 1.0-s test case relative differences computed for various CRAM approximation orders to the CRAM-48 solution.

TABLE XXVI The 693-Nuclide 1.0-s Test Case Comparison of Various Approximation Powers of ADM with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}$ $\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Power	Time (ms)	MNARD	ANARD	Power	Time (ms)	MNARD	ANARD
1	24.981 ± 1.129	$1.25\times {10}^{-1}$	1.94 × 10^–3	11	$50.984\pm 1.917$	1.19 × 10^–7	$1.85\times {10}^{-9}$
2	27.879 ± 1.182	$3.13\times {10}^{-2}$	4.86 × 10^–4	12	$52.586\pm 1.563$	2.98 × 10^–8	$4.63\times {10}^{-10}$
3	30.591 ± 1.325	$7.81\times {10}^{-3}$	1.21 × 10^–4	13	$56.029\pm 2.324$	7.45 × 10^–9	$1.16\times {10}^{-10}$
4	33.356 ± 1.729	$1.95\times {10}^{-3}$	3.03 × 10^–5	14	$56.882\pm 1.786$	1.86 × 10^–9	$2.94\times {10}^{-11}$
5	37.009 ± 2.186	$4.88\times {10}^{-4}$	7.59 × 10^–6	15	$58.338\pm 1.297$	4.66 × 10^–10	$8.15\times {10}^{-12}$
6	39.350 ± 2.037	$1.22\times {10}^{-4}$	1.90 × 10^–6	16	$60.555\pm 1.836$	1.17 × 10^–10	$3.82\times {10}^{-12}$
7	41.469 ± 1.609	$3.05\times {10}^{-5}$	4.74 × 10^–7	17	$62.926\pm 2.174$	3.08 × 10^–11	$4.06\times {10}^{-12}$
8	44.338 ± 2.290	$7.63\times {10}^{-6}$	1.19 × 10^–7	18	$64.077\pm 1.540$	2.42 × 10^–11	$8.01\times {10}^{-12}$
9	45.645 ± 1.003	$1.91\times {10}^{-6}$	2.96 × 10^–8	19	$66.941\pm 2.232$	3.83 × 10^–11	$1.51\times {10}^{-11}$
10	49.492 ± 2.171	$4.77\times {10}^{-7}$	7.41 × 10^–9	20	$67.366\pm 1.362$	9.61 × 10^–11	$3.32\times {10}^{-11}$

TABLE XXVII The 693-Nuclide 1.0-s Test Case Run-Time Comparisons for Various $m$ and $v$ Values for ADM-16 with a Cutoff of 10⁻⁵⁰

$m$	$v$	Time (ms)	$m$	$v$	Time (ms)
16	0	$60.555\pm 1.836$	7	9	57.013 ± 0.923
15	1	$56.369\pm 1.110$	6	10	76.021 ± 0.762
14	2	$53.203\pm 0.732$	5	11	115.709 ± 1.302
13	3	$50.530\pm 0.831$	4	12	196.419 ± 1.969
12	4	$48.084\pm 0.691$	3	13	355.936 ± 3.382
11	5	$46.024\pm 0.650$	2	14	674.263 ± 2.653
10	6	$45.036\pm 0.585$	1	15	1312.431 ± 3.271
9	7	$45.171\pm 0.808$	0	16	2521.742 ± 6.839
8	8	$48.311\pm 0.734$	—	—	—

Fig. 16. The 693-nuclide 1.0-s test case relative differences computed for various ADM approximation powers with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XXVIII The 693-Nuclide 3600- and $3.6\times {10}^6$-s Test Case Comparison of ADM-16 with a Cutoff of ${10}^{-50}$ to the CRAM-48 Solution for Nuclides with Number Densities Greater Than ${10}^{-30}\frac{\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}}{\mathrm{b}\cdot \mathrm{c}\mathrm{m}}$

Depletion Interval [s]	MNARD	ANARD
3,600	$6.45\times {10}^{-9}$	$9.71\times {10}^{-11}$
$3.6\times {10}^6$	$5.55\times {10}^{-8}$	$1.15\times {10}^{-9}$

Fig. 17. The 693-nuclide 3600- and $3.6\times {10}^6$-s test case relative differences computed for ADM-16 with a cutoff of ${10}^{-50}$ to the CRAM-48 solution.

TABLE XXIX Run Times of ADM at Various Approximation Powers with a Cutoff of ${10}^{-50}$ and CRAM-48 for All Nuclide Test Cases Assuming Constant and Cached At

Nuclides	ADM Power	ADM Time (ms)	CRAM-48 Time (ms)	ADM:CRAM-48 Ratio
35	16	$7.24\times {10}^{-4}\pm 6.3\times {10}^{-5}$	0.292200 ± 0.019927	0.0025
297	16	$0.017187\pm 0.003966$	7.613530 ± 0.285358	0.0023
693	16	$0.056262\pm 0.006114$	20.326711 ± 0.498023	0.0028
1450	17	$0.149685\pm 0.012129$	39.862447 ± 0.661700	0.0038
1599	17	$0.154266\pm 0.010152$	40.856141 ± 0.659615	0.0038

TABLE XXX Comparison of the Run Times of Best-Performing Versions of ADM with a Cutoff of ${10}^{-50}$ to CRAM-48

Nuclides	Power	$m$	$v$	ADM Time (ms)	CRAM-48 Time (ms)	ADM:CRAM-48
35	16	11	5	0.211 ± 0.012	$0.317\pm 0.018$	0.67
297	16	10	6	11.405 ± 0.389	$8.499\pm 0.162$	1.34
693	16	10	6	45.036 ± 0.585	$21.151\pm 0.269$	2.13
1450	17	10	7	156.375 ± 1.264	$44.983\pm 0.630$	3.48
1599	17	10	7	175.256 ± 2.989	$46.326\pm 0.759$	3.78