×
Sun, Zhengjie; Gao, Yuyan

High order multiquadric trigonometric quasi-interpolation method for solving time-dependent partial differential equations. (English) Zbl 07720463

Numer. Algorithms 93, No. 4, 1719-1739 (2023).
Summary: In this paper, we propose a high order multiquadric trigonometric quasi-interpolation method for function approximation and derivative approximation based on periodic sampling data. Moreover, we apply it to solve time-dependent nonlinear partial differential equations (PDEs) with periodic solutions. Learning from the construction of the trigonometric B-spline quasi-interpolation, we derive the new quasi-interpolation scheme by replacing the truncated trigonometric polynomial with a high degree, infinitely smooth multiquadric trigonometric kernel. By appropriately choosing the shape parameter in the kernel, we prove that the proposed method achieves higher order of convergence than the existing multiquadric trigonometric quasi-interpolation method. Finally, we conduct extensive numerical experiments to demonstrate the accuracy and efficiency of the proposed method for approximating unknown function and solving different types of PDEs including the one-dimensional KdV equation and two-dimensional Allen-Cahn equation.

Cited in 3 Documents

MSC:

65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems

Keywords:

quasi-interpolation; multiquadric trigonometric kernel; PDE; KdV equation; Allen-Cahn equation

Software:

Matlab
Cite Review PDF
Full Text: DOI

References:

[1] Beatson, RK; Powell, MJD, Univariate multiquadric approximation: quasi-interpolation to scattered data, Constr. Approx., 8, 3, 275-288 (1992) · Zbl 0763.41012 · doi:10.1007/BF01279020
[2] Bernstein, S., Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités (proof of the theorem of weierstrass based on the calculus of probabilities), Comm. Kharkov Math. Soc., 13, 1-2 (1912) · JFM 43.0301.03
[3] Buhmann, MD, Convergence of univariate quasi-interpolation using multiquadrics, IMA J. Numer. Anal., 8, 3, 365-383 (1988) · Zbl 0659.41003 · doi:10.1093/imanum/8.3.365
[4] Buhmann, M.D.: Radial basis functions: theory and implementations, vol. 12, Cambridge University Press (2003) · Zbl 1038.41001
[5] Chen, RH; Wu, ZM, Applying multiquadric quasi-interpolation to solve Burgers’ equation, Appl. Math. Comput., 172, 1, 472-484 (2006) · Zbl 1088.65086
[6] De Boor, C.; Höllig, K.; Riemenschneider, S., Box Splines (1993), Berlin: Springer, Berlin · Zbl 0814.41012 · doi:10.1007/978-1-4757-2244-4
[7] Duan, Y.; Rong, F., A numerical scheme for nonlinear Schrödinger equation by MQ quasi-interpolation, Eng. Anal. Bound Elem., 37, 1, 89-94 (2013) · Zbl 1352.65391 · doi:10.1016/j.enganabound.2012.08.006
[8] Fabien, MS, Numerical error analysis for an energy-stable HDG method for the Allen-Cahn equation, J. Comput. Appl. Comput., 402, 113800 (2022) · Zbl 1524.65794 · doi:10.1016/j.cam.2021.113800
[9] Fasshauer, G.E.: Meshfree approximation methods with MATLAB, vol. 6, World Scientific (2007) · Zbl 1123.65001
[10] Gao, QJ; Wu, ZM; Zhang, SG, Applying multiquadric quasi-interpolation for boundary detection, Comput. Math Appl., 62, 12, 4356-4361 (2011) · Zbl 1236.94020 · doi:10.1016/j.camwa.2011.09.069
[11] Gao, WW; Sun, XP; Wu, ZM; Zhou, X., Multivariate Monte Carlo approximation based on scattered data, SIAM J. Sci Comput., 42, 4, A2262-A2280 (2020) · Zbl 1489.65020 · doi:10.1137/19M1249138
[12] Gao, WW; Sun, ZJ, High-order numerical solution of time-dependent differential equations with quasi-interpolation, Appl. Numer Math., 146, 276-290 (2019) · Zbl 1447.65059 · doi:10.1016/j.apnum.2019.07.011
[13] Gao, WW; Wu, ZM, A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines, J. Comput. Appl Math., 271, 20-30 (2014) · Zbl 1326.65021 · doi:10.1016/j.cam.2014.03.012
[14] Gao, WW; Wu, ZM, Approximation orders and shape preserving properties of the multiquadric trigonometric B-spline quasi-interpolant, Comput. Math Appl., 69, 7, 696-707 (2015) · Zbl 1443.41010 · doi:10.1016/j.camwa.2015.02.008
[15] Hong, Q.; Gong, YZ; Zhao, J.; Wang, Q., Arbitrarily high order structure-preserving algorithms for the Allen-Cahn model with a nonlocal constraint, Appl. Numer Math., 170, 321-339 (2021) · Zbl 1501.65080 · doi:10.1016/j.apnum.2021.08.002
[16] Jeong, B.; Kersey, SN; Yoon, J., Approximation of multivariate functions on sparse grids by kernel-based quasi-interpolation, SIAM J. Sci Comput., 43, 2, A953-A979 (2021) · doi:10.1137/20M1318055
[17] Jerri, AJ, The Shannon sampling theorem—its various extensions and applications: a tutorial review, Proc. IEEE, 65, 11, 1565-1596 (1977) · Zbl 0442.94002 · doi:10.1109/PROC.1977.10771
[18] Jiang, ZW; Wang, RH, Numerical solution of one-dimensional Sine-Gordon equation using high accuracy multiquadric quasi-interpolation, Appl. Math Comput., 218, 15, 7711-7716 (2012) · Zbl 1242.65163
[19] Lamnii, A.; Nour, MY; Sbibih, D.; Zidna, A., Generalized spline quasi-interpolants and applications to numerical analysis, J. Comput. Appl. Math., 114100, 408 (2022) · Zbl 1484.41008
[20] Li, YB; Lee, HG; Jeong, D.; Kim, J., An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation, Comput. Math Appl., 60, 1591-1606 (2010) · Zbl 1202.65112 · doi:10.1016/j.camwa.2010.06.041
[21] Lyche, T.; Schumaker, LL; Stanley, S., Quasi-interpolants based on trigonometric splines, J. Approx. Theory, 95, 2, 280-309 (1998) · Zbl 0912.41008 · doi:10.1006/jath.1998.3196
[22] Lyche, T.; Winther, R., A stable recurrence relation for trigonometric B-splines, J Approx. Theory, 25, 3, 266-279 (1979) · Zbl 0414.41005 · doi:10.1016/0021-9045(79)90017-0
[23] Ma, LM; Wu, ZM, Approximation to the k-th derivatives by multiquadric quasi-interpolation method, J. Comput. Appl Math., 231, 2, 925-932 (2009) · Zbl 1236.65020 · doi:10.1016/j.cam.2009.05.017
[24] Ma, LM; Wu, ZM, Stability of multiquadric quasi-interpolation to approximate high order derivatives, Sci China Math., 53, 4, 985-992 (2010) · Zbl 1191.65018 · doi:10.1007/s11425-010-0068-9
[25] Sablonnière, P., Univariate spline quasi-interpolants and applications to numerical analysis., Rend. Sem. Mat. Univ. Pol. Torino, 63, 211-222 (2005) · Zbl 1115.41009
[26] Schumaker, LL; Traas, C., Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines, Numer. Math., 60, 1, 133-144 (1991) · Zbl 0744.65008 · doi:10.1007/BF01385718
[27] Sun, ZJ, A conservative scheme for two-dimensional Schrödinger equation based on multiquadric trigonometric quasi-interpolation approach, Appl. Math. Comput., 126996, 423 (2022) · Zbl 1510.65268
[28] Sun, ZJ; Gao, WW, A meshless scheme for Hamiltonian partial differential equations with conservation properties, Appl. Numer Math., 119, 115-125 (2017) · Zbl 1368.65257 · doi:10.1016/j.apnum.2017.04.005
[29] Sun, ZJ; Wu, ZM; Gao, WW, An iterated quasi-interpolation approach for derivative approximation, Numer Algorithms, 85, 1, 255-276 (2020) · Zbl 1446.41002 · doi:10.1007/s11075-019-00812-9
[30] Wendland, H.: Scattered Data Approximation, vol. 17, Cambridge University Press (2004) · Zbl 1185.65022
[31] Wu, RF; Wu, TR; Li, HL, A family of multivariate multiquadric quasi-interpolation operators with higher degree polynomial reproduction, J. Comput. Appl Math., 274, 88-108 (2015) · Zbl 1297.65016 · doi:10.1016/j.cam.2014.07.008
[32] Wu, ZM; Schaback, R., Shape preserving properties and convergence of univariate multiquadric quasi-interpolation, Acta Math. Appl. Sin., 10, 441-446 (1994) · Zbl 0822.41025 · doi:10.1007/BF02016334
[33] Wu, ZM; Zhang, R., Data-driven modeling for the motion of a sphere falling through a non-Newtonian fluid, Commun. Math Sci., 16, 2, 425-439 (2018) · Zbl 1390.34160 · doi:10.4310/CMS.2018.v16.n2.a6
[34] Wu, ZM; Zhang, R., Learning physics by data for the motion of a sphere falling in a non-Newtonian fluid, Commun. Non. Sci. Numer Simul., 67, 577-593 (2019) · Zbl 1508.76008 · doi:10.1016/j.cnsns.2018.05.007
[35] Wu, ZM; Zhang, SL, Conservative multiquadric quasi-interpolation method for Hamiltonian wave equations, Eng. Anal. Bound Elem., 37, 7-8, 1052-1058 (2013) · Zbl 1287.65139 · doi:10.1016/j.enganabound.2013.04.011
[36] Zhang, W.X., Wu, Z.M.: Shape-preserving MQ-B-splines quasi-interpolation. In: Geometric modeling and processing, 2004. Proceedings, pp. 85-92. IEEE (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.