Evaluating a double integral using Euler's method and Richardson extrapolation
We transform a double integral into a second-order initial value problem, which we solve using Euler's method and Richardson extrapolation. For an example we consider, we achieve accuracy close to machine precision (~10-13). We find that the algorithm is capable of determining the error curve...
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| Format: | Article |
| Language: | English |
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Vilnius University Press
2024-12-01
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| Series: | Lietuvos Matematikos Rinkinys |
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| Online Access: | https://ojs.test/index.php/LMR/article/view/38091 |
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| _version_ | 1841561117613096960 |
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| author | Justin Steven Calder Prentice |
| author_facet | Justin Steven Calder Prentice |
| author_sort | Justin Steven Calder Prentice |
| collection | DOAJ |
| description |
We transform a double integral into a second-order initial value problem, which we solve using Euler's method and Richardson extrapolation. For an example we consider, we achieve accuracy close to machine precision (~10-13). We find that the algorithm is capable of determining the error curve for an arbitrary cubature formula, and we use this feature to determine the error curve for a Simpson cubature rule. We also provide a generalization of the method to the case of nonlinear limits in the outer integral.
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| format | Article |
| id | doaj-art-6c095c232a5d4513b7f527e1fce83f45 |
| institution | Kabale University |
| issn | 0132-2818 2335-898X |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Vilnius University Press |
| record_format | Article |
| series | Lietuvos Matematikos Rinkinys |
| spelling | doaj-art-6c095c232a5d4513b7f527e1fce83f452025-01-03T06:33:47ZengVilnius University PressLietuvos Matematikos Rinkinys0132-28182335-898X2024-12-0165A10.15388/LMD.2024.38091Evaluating a double integral using Euler's method and Richardson extrapolationJustin Steven Calder Prentice0Mathsophical, Johannesburg, South Africa We transform a double integral into a second-order initial value problem, which we solve using Euler's method and Richardson extrapolation. For an example we consider, we achieve accuracy close to machine precision (~10-13). We find that the algorithm is capable of determining the error curve for an arbitrary cubature formula, and we use this feature to determine the error curve for a Simpson cubature rule. We also provide a generalization of the method to the case of nonlinear limits in the outer integral. https://ojs.test/index.php/LMR/article/view/38091cubaturedouble integralEulerRichardson extrapolationerror |
| spellingShingle | Justin Steven Calder Prentice Evaluating a double integral using Euler's method and Richardson extrapolation Lietuvos Matematikos Rinkinys cubature double integral Euler Richardson extrapolation error |
| title | Evaluating a double integral using Euler's method and Richardson extrapolation |
| title_full | Evaluating a double integral using Euler's method and Richardson extrapolation |
| title_fullStr | Evaluating a double integral using Euler's method and Richardson extrapolation |
| title_full_unstemmed | Evaluating a double integral using Euler's method and Richardson extrapolation |
| title_short | Evaluating a double integral using Euler's method and Richardson extrapolation |
| title_sort | evaluating a double integral using euler s method and richardson extrapolation |
| topic | cubature double integral Euler Richardson extrapolation error |
| url | https://ojs.test/index.php/LMR/article/view/38091 |
| work_keys_str_mv | AT justinstevencalderprentice evaluatingadoubleintegralusingeulersmethodandrichardsonextrapolation |