Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation
Four explicit numerical schemes are collected, which are stable and efficient for the diffusion equation. Using these diffusion solvers, several new methods are constructed for the nonlinear Huxley’s equation. Then, based on many successive numerical case studies in one and two space dimensions, the...
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MDPI AG
2025-01-01
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| author | Husniddin Khayrullaev Issa Omle Endre Kovács |
| author_facet | Husniddin Khayrullaev Issa Omle Endre Kovács |
| author_sort | Husniddin Khayrullaev |
| collection | DOAJ |
| description | Four explicit numerical schemes are collected, which are stable and efficient for the diffusion equation. Using these diffusion solvers, several new methods are constructed for the nonlinear Huxley’s equation. Then, based on many successive numerical case studies in one and two space dimensions, the least performing methods are gradually dropped out to keep only the best ones. During the tests, not only one but all the relevant time step sizes are considered, and for them, running-time measurements are performed. A major aspect is computational efficiency, which means that an acceptable solution is produced in the shortest possible time. Parameter sweeps are executed for the coefficient of the nonlinear term, the stiffness ratio, and the length of the examined time interval as well. We obtained that usually, the leapfrog–hopscotch method with Strang-type operator-splitting is the most efficient and reliable, but the method based on the Dufort–Frankel scheme can also be very efficient. |
| format | Article |
| id | doaj-art-693e067febcd4e8881676540da43b1c8 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-693e067febcd4e8881676540da43b1c82025-01-24T13:39:45ZengMDPI AGMathematics2227-73902025-01-0113220710.3390/math13020207Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s EquationHusniddin Khayrullaev0Issa Omle1Endre Kovács2Institute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, HungaryInstitute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, HungaryInstitute of Physics and Electrical Engineering, University of Miskolc, 3515 Miskolc, HungaryFour explicit numerical schemes are collected, which are stable and efficient for the diffusion equation. Using these diffusion solvers, several new methods are constructed for the nonlinear Huxley’s equation. Then, based on many successive numerical case studies in one and two space dimensions, the least performing methods are gradually dropped out to keep only the best ones. During the tests, not only one but all the relevant time step sizes are considered, and for them, running-time measurements are performed. A major aspect is computational efficiency, which means that an acceptable solution is produced in the shortest possible time. Parameter sweeps are executed for the coefficient of the nonlinear term, the stiffness ratio, and the length of the examined time interval as well. We obtained that usually, the leapfrog–hopscotch method with Strang-type operator-splitting is the most efficient and reliable, but the method based on the Dufort–Frankel scheme can also be very efficient.https://www.mdpi.com/2227-7390/13/2/207nonlinear PDEsdiffusion–reaction equationsHuxley’s equationexplicit numerical methodsstiff equations |
| spellingShingle | Husniddin Khayrullaev Issa Omle Endre Kovács Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation Mathematics nonlinear PDEs diffusion–reaction equations Huxley’s equation explicit numerical methods stiff equations |
| title | Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation |
| title_full | Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation |
| title_fullStr | Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation |
| title_full_unstemmed | Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation |
| title_short | Exploring the Performance of Some Efficient Explicit Numerical Methods with Good Stability Properties for Huxley’s Equation |
| title_sort | exploring the performance of some efficient explicit numerical methods with good stability properties for huxley s equation |
| topic | nonlinear PDEs diffusion–reaction equations Huxley’s equation explicit numerical methods stiff equations |
| url | https://www.mdpi.com/2227-7390/13/2/207 |
| work_keys_str_mv | AT husniddinkhayrullaev exploringtheperformanceofsomeefficientexplicitnumericalmethodswithgoodstabilitypropertiesforhuxleysequation AT issaomle exploringtheperformanceofsomeefficientexplicitnumericalmethodswithgoodstabilitypropertiesforhuxleysequation AT endrekovacs exploringtheperformanceofsomeefficientexplicitnumericalmethodswithgoodstabilitypropertiesforhuxleysequation |