The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis.
Citation: Jin Li. Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation[J]. AIMS Mathematics, 2023, 8(7): 16494-16510. doi: 10.3934/math.2023843
The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis.
| [1] |
L. Jones Tarcius Doss, A. P. Nandini, A fourth-order $H^{1}$-Galerkin mixed finite element method for Kuramoto-Sivashinsky equation, Numer. Methods Partial Differ. Equ., 35 (2019), 445–477. https://doi.org/10.1002/num.22306 doi: 10.1002/num.22306
|
| [2] |
H. Otomo, B. M. Boghosian, F. Dubois, Efficient lattice Boltzmann models for the Kuramoto-Sivashinsky equation, Comput. Fluids, 172 (2018), 683–688. https://doi.org/10.1016/j.compfluid.2018.01.036 doi: 10.1016/j.compfluid.2018.01.036
|
| [3] |
G. Akrivis, Y. S. Smyrlis, Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation, Appl. Numer. Math., 51 (2004), 151–169. https://doi.org/10.1016/j.apnum.2004.03.002 doi: 10.1016/j.apnum.2004.03.002
|
| [4] |
A. Mouloud, H. Felloua, B. A. Wade, M. Kessal, Time discretization and stability regions for dissipative-dispersive Kuramoto-Sivashinsky equation arising in turbulent gas flow over laminar liquid, J. Comput. Appl. Math., 330 (2018), 605–617. http://dx.doi.org/10.1016/j.cam.2017.09.014 doi: 10.1016/j.cam.2017.09.014
|
| [5] |
B. Chentouf, A. Guesmia, Well-posedness and stability results for the Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations with infinite memory: a history approach, Nonlinear Anal., 65 (2022), 103508. https://doi.org/10.1016/j.nonrwa.2022.103508 doi: 10.1016/j.nonrwa.2022.103508
|
| [6] |
Shallu, V. K. Kukreja, An improvised extrapolated collocation algorithm for solving Kuramoto-Sivashinsky equation, Math. Methods Appl. Sci., 45 (2021), 1451–1467. https://doi.org/10.1002/mma.7865 doi: 10.1002/mma.7865
|
| [7] |
A. Kumari, V. K. Kukreja, Study of $4^{th}$ order Kuramoto-Sivashinsky equation by septic Hermite collocation method, Appl. Numer. Math., 188 (2023), 88–105. https://doi.org/10.1016/j.apnum.2023.03.001 doi: 10.1016/j.apnum.2023.03.001
|
| [8] |
M. Hosseininia, M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, Z. Avazzadeh, Numerical method based on the Chebyshev cardinal functions for the variable-order fractional version of the fourth-order 2D Kuramoto-Sivashinsky equation, Math. Methods Appl. Sci., 44 (2021), 1831–1842. https://doi.org/10.1002/mma.6881 doi: 10.1002/mma.6881
|
| [9] |
M. H. Heydari, Z. Avazzadeh, C. Cattani, Numerical solution of variable-order space-time fractional KdV-Burgers-Kuramoto equation by using discrete Legendre polynomials, Eng. Comput., 38 (2022), 859–869. https://doi.org/10.1007/s00366-020-01181-x doi: 10.1007/s00366-020-01181-x
|
| [10] |
J. P. Berrut, S. A. Hosseini, G. Klein, The linear barycentric rational quadrature method for Volterra integral equations, SIAM J. Sci. Comput., 36 (2014), 105–123. https://doi.org/10.1137/120904020 doi: 10.1137/120904020
|
| [11] |
J. P. Berrut, G. Klein, Recent advances in linear barycentric rational interpolation, J. Comput. Appl. Math., 259 (2014), 95–107. https://doi.org/10.1016/j.cam.2013.03.044 doi: 10.1016/j.cam.2013.03.044
|
| [12] |
E. Cirillo, H. Kai, On the Lebesgue constant of barycentric rational Hermite interpolants at uniform partition, J. Comput. Appl. Math., 349 (2019), 292–301. https://doi.org/10.1016/j.cam.2018.06.011 doi: 10.1016/j.cam.2018.06.011
|
| [13] |
M. S. Floater, K. Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107 (2007), 315–331. https://doi.org/10.1007/s00211-007-0093-y doi: 10.1007/s00211-007-0093-y
|
| [14] |
G. Klein, J. P. Berrut, Linear rational finite differences from derivatives of barycentric rational interpolants, SIAM J. Numer. Anal., 50 (2012), 643–656. https://doi.org/10.1137/110827156 doi: 10.1137/110827156
|
| [15] |
G. Klein, J. P. Berrut, Linear barycentric rational quadrature, BIT Numer. Math., 52 (2012), 407–424. https://doi.org/10.1007/s10543-011-0357-x doi: 10.1007/s10543-011-0357-x
|
| [16] |
J. Li, Y. Cheng, Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation, Comput. Appl. Math., 39 (2020), 92. https://doi.org/10.1007/s40314-020-1114-z doi: 10.1007/s40314-020-1114-z
|
| [17] |
J. Li, Y. Cheng, Linear barycentric rational collocation method for solving heat conduction equation, Numer. Methods Partial Differ. Equ., 37 (2021), 533–545. https://doi.org/10.1002/num.22539 doi: 10.1002/num.22539
|
| [18] |
J. Li, Y. Cheng, Barycentric rational method for solving biharmonic equation by depression of order, Numer. Methods Partial Differ. Equ., 37 (2021), 1993–2007. https://doi.org/10.1002/num.22638 doi: 10.1002/num.22638
|
| [19] |
J. Li, Y. Cheng, Barycentric rational interpolation method for solving KPP equation, Electron. Res. Arch., 31 (2023), 3014–3029. https://doi.org/10.3934/era.2023152 doi: 10.3934/era.2023152
|
| [20] |
J. Li, X. Su, K. Zhao, Barycentric interpolation collocation algorithm to solve fractional differential equations, Math. Comput. Simul., 205 (2023), 340–367. https://doi.org/10.1016/j.matcom.2022.10.005 doi: 10.1016/j.matcom.2022.10.005
|
| [21] |
J. Li, Y. Cheng, Z. Li, Z. Tian, Linear barycentric rational collocation method for solving generalized Poisson equations, Math. Biosci. Eng., 20 (2023), 4782–4797. https://doi.org/10.3934/mbe.2023221 doi: 10.3934/mbe.2023221
|
| [22] | S. Li, Z. Wang, Meshless barycentric interpolation collocation method–algorithmics, programs & applications in engineering, Beijing: Science Publishing, 2012. |
| [23] | Z. Wang, S. Li, Barycentric interpolation collocation method for nonlinear problems, Beijing: National Defense Industry Press, 2015. |
| [24] | Z. Wang, Z. Xu, J. Li, Mixed barycentric interpolation collocation method of displacement-pressure for incompressible plane elastic problems, Chin. J. Appl. Mech., 35 (2018), 195–201. |
| [25] | Z. Wang, L. Zhang, Z. Xu, J. Li, Barycentric interpolation collocation method based on mixed displacement-stress formulation for solving plane elastic problems, Chin. J. Appl. Mech., 35 (2018), 304–309. |
| [26] |
J. Li, Linear barycentric rational collocation method for solving biharmonic equation, Demonstr. Math., 55 (2022), 587–603. https://doi.org/10.1515/dema-2022-0151 doi: 10.1515/dema-2022-0151
|
| [27] |
J. Li, Barycentric rational collocation method for semi-infinite domain problems, AIMS Math., 8 (2023), 8756–8771. https://doi.org/10.3934/math.2023439 doi: 10.3934/math.2023439
|
| [28] |
J. Li, Barycentric rational collocation method for fractional reaction-diffusion equation, AIMS Math., 8 (2023), 9009–9026. https://doi.org/10.3934/math.2023451 doi: 10.3934/math.2023451
|
| [29] |
J. Li, Linear barycentric rational collocation method to solve plane elastic problems, Math. Biosci. Eng., 20 (2023), 8337–8357. https://doi.org/10.3934/mbe.2023365 doi: 10.3934/mbe.2023365
|
Figures(9) / Tables(11)
Jin Li. Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation[J]. AIMS Mathematics, 2023, 8(7): 16494-16510. doi: 10.3934/math.2023843
DownLoad: