Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits
Abstract
:1. Preliminary Discussion
2. Theory Numerov-Type Methods Using Off-Step Nodes
3. Performance of the Schemes in a Set of Keplerian-like Problems
4. Training the Free Parameters in a Wide Set of Keplerian-like Problems
5. Numerical Tests
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
References
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e | Steps | ACM17 | PL18 | MIN |
---|---|---|---|---|
60 | ||||
120 | ||||
180 | ||||
240 | ||||
300 | ||||
360 | ||||
420 | ||||
80 | ||||
160 | ||||
240 | ||||
320 | ||||
400 | ||||
480 | ||||
560 | ||||
150 | ||||
300 | ||||
450 | ||||
600 | ||||
750 | ||||
900 | ||||
1050 | ||||
200 | ||||
400 | ||||
600 | ||||
800 | ||||
1000 | ||||
1200 | ||||
1400 | ||||
500 | ||||
1000 | ||||
1500 | ||||
2000 | ||||
2500 | ||||
3000 | ||||
3500 |
Steps | ACM17 | PL18 | MIN | |
---|---|---|---|---|
50 | ||||
100 | ||||
150 | ||||
200 | ||||
250 | ||||
300 | ||||
350 | ||||
50 | ||||
100 | ||||
150 | ||||
200 | ||||
250 | ||||
300 | ||||
350 | ||||
50 | ||||
100 | ||||
150 | ||||
200 | ||||
250 | ||||
300 | ||||
350 | ||||
60 | 2.5 | 2.5 | 2.2 | |
120 | 5.2 | 4.8 | 4.9 | |
180 | 6.7 | 6.2 | 6.5 | |
240 | 7.8 | 7.2 | 7.6 | |
300 | 8.6 | 8.0 | 8.5 | |
360 | 9.3 | 8.7 | 9.3 | |
420 | 9.9 | 9.2 | 9.9 | |
60 | 2.4 | 2.4 | 2.2 | |
120 | 5.1 | 4.7 | 4.8 | |
180 | 6.6 | 6.2 | 6.4 | |
240 | 7.7 | 7.2 | 7.6 | |
300 | 8.6 | 8.0 | 8.4 | |
360 | 9.2 | 8.6 | 9.2 | |
420 | 9.8 | 9.2 | 9.8 |
Steps | ACM17 | PL18 | MIN | |
---|---|---|---|---|
10,000 | ||||
15,000 | ||||
20,000 | ||||
25,000 | ||||
30,000 | ||||
35,000 | ||||
40,000 | ||||
10,000 | ||||
20,000 | ||||
30,000 | ||||
40,000 | ||||
50,000 | ||||
60,000 | ||||
70,000 |
Steps | ACM17 | PL18 | MIN | |
---|---|---|---|---|
3 | 3000 | |||
3000 | ||||
4500 | ||||
6000 | ||||
7500 | ||||
9000 | ||||
10,500 | ||||
12,000 | ||||
4 | 4000 | |||
6000 | ||||
8000 | ||||
10,000 | ||||
12,000 | ||||
14,000 | ||||
16,000 |
e | Steps | NEW8 |
---|---|---|
60 | ||
120 | ||
180 | ||
240 | ||
300 | ||
360 | ||
420 | ||
80 | ||
160 | ||
240 | ||
320 | ||
400 | ||
480 | ||
560 | ||
150 | ||
300 | ||
450 | ||
600 | ||
750 | ||
900 | ||
1050 | ||
200 | ||
400 | ||
600 | ||
800 | ||
1000 | ||
1200 | ||
1400 | ||
500 | ||
1000 | ||
1500 | ||
2000 | ||
2500 | ||
3000 | ||
3500 |
Steps | NEW8 | |
---|---|---|
50 | ||
100 | ||
150 | ||
200 | ||
250 | ||
300 | ||
350 | ||
50 | ||
100 | ||
150 | ||
200 | ||
250 | ||
300 | ||
350 | ||
50 | ||
100 | ||
150 | ||
200 | ||
250 | ||
300 | ||
350 | ||
60 | ||
120 | ||
180 | ||
240 | ||
300 | ||
360 | ||
420 | ||
60 | ||
120 | ||
180 | ||
240 | ||
300 | ||
360 | ||
420 |
Steps | NEW8 | Steps | NEW8 |
---|---|---|---|
10,000 | 3.8 | 10,000 | 1.1 |
15,000 | 5.4 | 20,000 | 1.6 |
20,000 | 6.7 | 30,000 | 3.2 |
25,000 | 7.6 | 40,000 | 4.5 |
30,000 | 8.4 | 50,000 | 5.8 |
35,000 | 9.1 | 60,000 | 7.1 |
40,000 | 9.7 | 70,000 | 8.8 |
Steps | NEW8 |
---|---|
3000 | 3.1 |
4500 | 4.3 |
6000 | 5.3 |
7500 | 6.1 |
9000 | 6.8 |
10,500 | 7.3 |
12,000 | 7.8 |
Steps | NEW8 |
4000 | 2.6 |
6000 | 3.8 |
8000 | 4.9 |
10,000 | 5.7 |
12,000 | 6.3 |
14,000 | 6.9 |
16,000 | 7.4 |
e | Steps | NEW8 | MIN |
---|---|---|---|
0.1 | 2 · 60 | 3.1 | 1.5 |
2 · 120 | 5.4 | 4.0 | |
2 · 180 | 6.7 | 5.6 | |
2 · 240 | 7.6 | 6.7 | |
2 · 300 | 8.4 | 7.6 | |
2 · 360 | 9.0 | 8.3 | |
2 · 420 | 9.5 | 8.9 | |
0.3 | 2 · 80 | 1.6 | 1.0 |
2 · 160 | 4.2 | 3.3 | |
2 · 240 | 5.8 | 4.8 | |
2 · 320 | 7.0 | 5.9 | |
2 · 400 | 7.8 | 6.8 | |
2 · 480 | 8.4 | 7.5 | |
2 · 560 | 8.8 | 8.1 | |
0.5 | 2 · 150 | 1.3 | 1.0 |
2 · 300 | 3.8 | 3.2 | |
2 · 450 | 5.5 | 4.7 | |
2 · 600 | 6.8 | 5.8 | |
2 · 750 | 7.8 | 6.7 | |
2 · 900 | 8.4 | 7.4 | |
2100 | 8.8 | 8.0 | |
0.7 | 2 · 200 | 0.2 | -0.3 |
2 · 400 | 1.4 | 1.1 | |
2 · 600 | 2.8 | 2.4 | |
2 · 800 | 3.9 | 3.4 | |
2000 | 4.8 | 4.2 | |
2400 | 5.6 | 4.9 | |
2800 | 6.3 | 5.5 | |
0.9 | 2000 | 2.2 | 1.8 |
3000 | 3.2 | 2.7 | |
4000 | 4.0 | 3.5 | |
5000 | 4.7 | 4.1 | |
6000 | 5.3 | 4.6 | |
7000 | 5.8 | 5.1 | |
8000 | 6.3 | 5.5 |
Steps | NEW8 | MIN | |
---|---|---|---|
Steps | NEW8 | MIN |
---|---|---|
5000 | 2.6 | 3.2 |
10,000 | 5.1 | 5.2 |
15,000 | 6.9 | 6.5 |
20,000 | 7.9 | 7.5 |
25,000 | 8.7 | 8.4 |
30,000 | 9.3 | 9.0 |
35,000 | 9.8 | 9.6 |
Steps | NEW8 | MIN |
10,000 | 1.5 | 2.1 |
20,000 | 4.3 | 4.1 |
30,000 | 6.4 | 5.5 |
40,000 | 8.0 | 6.6 |
50,000 | 8.2 | 7.4 |
60,000 | 8.7 | 8.1 |
70,000 | 9.4 | 8.7 |
Steps | NEW8 | MIN |
---|---|---|
5000 | 2.7 | 2.7 |
7500 | 4.1 | 4.0 |
10,000 | 5.1 | 5.0 |
12,500 | 5.9 | 5.8 |
15,000 | 6.6 | 6.5 |
17,500 | 7.2 | 7.1 |
20,000 | 7.6 | 7.6 |
Steps | NEW8 | MIN |
7000 | 3.4 | 3.3 |
10,500 | 4.8 | 4.7 |
14,000 | 5.9 | 5.7 |
17,500 | 6.7 | 6.6 |
21,000 | 7.3 | 7.3 |
24,500 | 7.9 | 7.8 |
28,000 | 8.4 | 8.4 |
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Kovalnogov, V.N.; Fedorov, R.V.; Chukalin, A.V.; Simos, T.E.; Tsitouras, C. Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits. Mathematics 2021, 9, 3071. https://doi.org/10.3390/math9233071
Kovalnogov VN, Fedorov RV, Chukalin AV, Simos TE, Tsitouras C. Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits. Mathematics. 2021; 9(23):3071. https://doi.org/10.3390/math9233071
Chicago/Turabian StyleKovalnogov, Vladislav N., Ruslan V. Fedorov, Andrey V. Chukalin, Theodore E. Simos, and Charalampos Tsitouras. 2021. "Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits" Mathematics 9, no. 23: 3071. https://doi.org/10.3390/math9233071
APA StyleKovalnogov, V. N., Fedorov, R. V., Chukalin, A. V., Simos, T. E., & Tsitouras, C. (2021). Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits. Mathematics, 9(23), 3071. https://doi.org/10.3390/math9233071