Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems
Abstract
:1. Introduction
2. Fundamental Concepts
- if ;
- if and
- if and repeat the step.
3. Construction of the Proposed Method
4. Algebraic Order and Error Analysis
Analysis of Stability
5. Numerical Experiments
- EEERKN5(3): The new embedded pair constructed in this paper;
- RKN5(3): A 5(3) pair of explicit RKN methods given by Van de Vyver in [14];
- ARKN5(3): A 5(3) pair of explicit ARKN methods derived by Franco in [16];
- RKN6(4)6ER-PFAF: A 6(4) optimized embedded RKN pair obtained by Anastassi and Kosti in [11]; and
- FRKN4: A Runge–Kutta–Nyström pair obtained by Van de Vyver in [17],
6. Discussion
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
RKN | Runge–Kutta–Nyström |
IVP | Initial value problem |
LTE | Local Truncation error |
References
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TOL | METHOD | STEP | FCN | FSTEP | MAXE | TIME(s) |
---|---|---|---|---|---|---|
EEERKN5(3) | 122 | 488 | 0 | 2.570495(−3) | 0.053 | |
RKN5(3) | 122 | 488 | 0 | 1.076884(−2) | 0.094 | |
ARKN5(3) | 242 | 968 | 0 | 9.829659(−1) | 0.271 | |
RKN6(4)6ER-PFAF | 242 | 1452 | 0 | 6.005192(−1) | 0.075 | |
FRKN4 | 484 | 1939 | 1 | 3.086156(−1) | 0.063 | |
EEERKN5(3) | 522 | 2088 | 0 | 4.246848(−7) | 0.050 | |
RKN5(3) | 522 | 2088 | 0 | 7.153723(−6) | 0.055 | |
ARKN5(3) | 1044 | 4179 | 1 | 6.406274(−2) | 0.062 | |
RKN6(4)6ER-PFAF | 1044 | 6269 | 1 | 3.549698(−2) | 0.370 | |
FRKN4 | 4169 | 16,685 | 3 | 4.185123(−3) | 0.102 | |
EEERKN5(3) | 1123 | 4492 | 0 | 4.226820(−9) | 0.047 | |
RKN5(3) | 1123 | 4492 | 0 | 1.541216(−7) | 0.053 | |
ARKN5(3) | 4491 | 17,970 | 2 | 3.460856(−3) | 0.053 | |
RKN6(4)6ER-PFAF | 4491 | 26,956 | 2 | 1.912992(−3) | 0.218 | |
FRKN4 | 35,919 | 143,691 | 5 | 5.635773(−5) | 0.487 | |
EEERKN5(3) | 2420 | 9680 | 0 | 4.243372(−11) | 0.075 | |
RKN5(3) | 2420 | 9680 | 0 | 3.319323(−9) | 0.096 | |
ARKN5(3) | 19,347 | 77,397 | 3 | 1.863540(−4) | 0.130 | |
RKN6(4)6ER-PFAF | 19,347 | 116,097 | 3 | 1.030210(−4) | 0.129 | |
FRKN4 | 309,539 | 1,238,177 | 7 | 7.583321(−7) | 3.248 | |
EEERKN5(3) | 10,422 | 41,694 | 2 | 1.646495(−11) | 0.134 | |
RKN5(3) | 10,421 | 41,687 | 1 | 1.664952(−11) | 0.109 | |
ARKN5(3) | 83,362 | 333,460 | 4 | 1.003239(−5) | 0.338 | |
RKN6(4)6ER-PFAF | 83,362 | 500,192 | 4 | 5.548769(−6) | 0.403 | |
FRKN4 | 2,667,524 | 10,670,123 | 9 | 1.495822(−8) | 26.747 |
TOL | METHOD | STEP | FCN | FSTEP | MAXE | TIME(s) |
---|---|---|---|---|---|---|
EEERKN5(3) | 122 | 488 | 0 | 1.227156(−1) | 0.040 | |
RKN5(3) | 122 | 488 | 0 | 8.478978(−1) | 0.041 | |
ARKN5(3) | 270 | 1083 | 1 | 1.804551(+0) | 0.044 | |
RKN6(4)6ER-PFAF | 363 | 2188 | 2 | 1.815228(+0) | 0.047 | |
FRKN4 | 484 | 1939 | 1 | 1.942861(+0) | 0.043 | |
EEERKN5(3) | 522 | 2088 | 0 | 3.621045(−5) | 0.041 | |
RKN5(3) | 522 | 2088 | 0 | 6.990118(−4) | 0.047 | |
ARKN5(3) | 1044 | 4179 | 1 | 1.480069(−1) | 0.045 | |
RKN6(4)6ER-PFAF | 1044 | 6269 | 1 | 2.961366(−1) | 0.041 | |
FRKN4 | 4169 | 16,685 | 3 | 1.130567(−2) | 0.078 | |
EEERKN5(3) | 1123 | 4492 | 0 | 3.722093(−7) | 0.054 | |
RKN5(3) | 1123 | 4492 | 0 | 1.520229(−5) | 0.063 | |
ARKN5(3) | 4491 | 17,970 | 2 | 5.473843(−3) | 0.051 | |
RKN6(4)6ER-PFAF | 4491 | 26,956 | 2 | 6.609722(−3) | 0.060 | |
FRKN4 | 35,919 | 143,691 | 5 | 1.165965(−4) | 0.361 | |
EEERKN5(3) | 2420 | 9680 | 0 | 3.718493(−9) | 0.058 | |
RKN5(3) | 2420 | 9680 | 0 | 3.282692(−7) | 0.139 | |
ARKN5(3) | 19,347 | 77,397 | 3 | 4.588825(−4) | 0.122 | |
RKN6(4)6ER-PFAF | 19,347 | 116,097 | 3 | 2.397332(−4) | 0.090 | |
FRKN4 | 309,539 | 1,238,177 | 7 | 1.515111(−6) | 2.676 | |
EEERKN5(3) | 10,422 | 41,694 | 2 | 1.717850(−11) | 0.120 | |
RKN5(3) | 10,421 | 41,687 | 1 | 2.058225(−10) | 0.054 | |
ARKN5(3) | 83,362 | 333,460 | 4 | 2.680717(−5) | 0.254 | |
RKN6(4)6ER-PFAF | 83,362 | 500,192 | 4 | 1.145927(−5) | 0.247 | |
FRKN4 | 2,667,524 | 10,670,123 | 9 | 1.557560(−8) | 22.647 |
TOL | METHOD | STEP | FCN | FSTEP | MAXE | TIME(s) |
---|---|---|---|---|---|---|
EEERKN5(3) | 122 | 488 | 0 | 2.591319(−3) | 0.062 | |
RKN5(3) | 122 | 488 | 0 | 1.078825(−2) | 0.062 | |
ARKN5(3) | 242 | 968 | 0 | 9.806283(−1) | 0.100 | |
RKN6(4)6ER-PFAF | 242 | 1452 | 0 | 5.976002(−1) | 0.300 | |
FRKN4 | 484 | 1939 | 1 | 3.076264(−1) | 0.092 | |
EEERKN5(3) | 522 | 2088 | 0 | 4.299671(−7) | 0.065 | |
RKN5(3) | 522 | 2088 | 0 | 7.172465(−6) | 0.074 | |
ARKN5(3) | 1044 | 4179 | 1 | 6.403390(−2) | 0.191 | |
RKN6(4)6ER-PFAF | 1044 | 6269 | 1 | 3.548404(−2) | 0.165 | |
FRKN4 | 4169 | 16,685 | 3 | 4.181655(−3) | 0.126 | |
EEERKN5(3) | 1123 | 4492 | 0 | 4.355510(−9) | 0.064 | |
RKN5(3) | 1123 | 4491 | 0 | 1.542823(−7) | 0.066 | |
ARKN5(3) | 4491 | 17,970 | 2 | 3.458063(−3) | 0.152 | |
RKN6(4)6ER-PFAF | 4491 | 26,956 | 2 | 1.911456(−3) | 0.143 | |
FRKN4 | 35,919 | 143,691 | 5 | 5.631169(−5) | 0.566 | |
EEERKN5(3) | 2420 | 9680 | 0 | 4.516099(−11) | 0.081 | |
RKN5(3) | 2420 | 9680 | 0 | 3.324929(−9) | 0.095 | |
ARKN5(3) | 19,347 | 77,397 | 3 | 1.862032(−4) | 0.243 | |
RKN6(4)6ER-PFAF | 19,347 | 116,097 | 3 | 1.029369(−4) | 0.276 | |
FRKN4 | 309,539 | 1,238,177 | 7 | 7.577131(−7) | 3.588 | |
EEERKN5(3) | 10,422 | 41,694 | 2 | 1.643935(−11) | 0.166 | |
RKN5(3) | 10,421 | 41,687 | 1 | 1.658362(−11) | 0.153 | |
ARKN5(3) | 83,362 | 333,460 | 4 | 1.002430(−5) | 0.470 | |
RKN6(4)6ER-PFAF | 83,362 | 500,192 | 4 | 5.544237(−6) | 0.470 | |
FRKN4 | 2,667,524 | 10,670,123 | 9 | 1.494338(−8) | 26.650 |
TOL | METHOD | STEP | FCN | FSTEP | MAXE | TIME(s) |
---|---|---|---|---|---|---|
EEERKN5(3) | 122 | 515 | 9 | 1.170545(−3) | 0.055 | |
RKN5(3) | 122 | 536 | 16 | 2.777502(−3) | 0.066 | |
ARKN5(3) | 123 | 504 | 4 | 2.849535(−1) | 0.141 | |
RKN6(4)6ER-PFAF | 124 | 744 | 0 | 2.653779(−1) | 0.062 | |
FRKN4 | 439 | 1825 | 23 | 5.355312(−2) | 0.062 | |
EEERKN5(3) | 262 | 1072 | 8 | 1.356514(−5) | 0.047 | |
RKN5(3) | 262 | 1075 | 9 | 7.208088(−5) | 0.062 | |
ARKN5(3) | 510 | 2076 | 12 | 4.321049(−2) | 0.078 | |
RKN6(4)6ER-PFAF | 513 | 3128 | 10 | 2.781421(−2) | 0.094 | |
FRKN4 | 1815 | 7362 | 34 | 2.600772(−3) | 0.088 | |
EEERKN5(3) | 573 | 2337 | 15 | 9.010356(−8) | 0.062 | |
RKN5(3) | 562 | 2260 | 4 | 1.659456(−6) | 0.078 | |
ARKN5(3) | 2085 | 8439 | 33 | 1.815951(−3) | 0.141 | |
RKN6(4)6ER-PFAF | 2140 | 13,005 | 33 | 1.236425(−3) | 0.071 | |
FRKN4 | 14,666 | 58,868 | 68 | 4.713910(−5) | 0.266 | |
EEERKN5(3) | 1959 | 7932 | 32 | 9.751267(−10) | 0.078 | |
RKN5(3) | 2324 | 9392 | 32 | 7.596427(−9) | 0.141 | |
ARKN5(3) | 9091 | 36,487 | 41 | 1.005629(−4) | 0.148 | |
RKN6(4)6ER-PFAF | 9258 | 55,773 | 45 | 6.549156(−5) | 0.187 | |
FRKN4 | 134,843 | 539,678 | 102 | 4.210467(−7) | 2.129 | |
EEERKN5(3) | 5213 | 21,140 | 96 | 4.741679(−12) | 0.109 | |
RKN5(3) | 5183 | 20,816 | 28 | 4.524522(−11) | 0.125 | |
ARKN5(3) | 39,556 | 158,425 | 67 | 5.609382(−6) | 0.281 | |
RKN6(4)6ER-PFAF | 40,226 | 241,631 | 55 | 3.583863(−6) | 0.299 | |
FRKN4 | 1,209,331 | 4,837,732 | 136 | 5.483721(−9) | 14.829 |
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Demba, M.A.; Kumam, P.; Watthayu, W.; Phairatchatniyom, P. Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems. Computation 2020, 8, 32. https://doi.org/10.3390/computation8020032
Demba MA, Kumam P, Watthayu W, Phairatchatniyom P. Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems. Computation. 2020; 8(2):32. https://doi.org/10.3390/computation8020032
Chicago/Turabian StyleDemba, Musa Ahmed, Poom Kumam, Wiboonsak Watthayu, and Pawicha Phairatchatniyom. 2020. "Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems" Computation 8, no. 2: 32. https://doi.org/10.3390/computation8020032
APA StyleDemba, M. A., Kumam, P., Watthayu, W., & Phairatchatniyom, P. (2020). Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems. Computation, 8(2), 32. https://doi.org/10.3390/computation8020032