Adaptive moving knots meshless method for simulating time dependent partial differential equations. (English) Zbl 1403.65065
Summary: This paper presents an improvement of Huang’s moving knots method [W. Huang et al., SIAM J. Numer. Anal. 31, No. 3, 709–730 (1994; Zbl 0806.65092)] for solving the PDE on the contours of Equidistribution Principle (EP-contours). Firstly, a new moving knots strategy is generated by discovering a significant factor ignored (dropped) by Huang [loc. cit.]. The proposed strategy ensures that arbitrary initial knots could asymptotically converge to the EP-contours. Then the moving knots strategy and PDE are simulated by using multi-quadric (MQ) quasi-interpolation step by Step (3.3, 3.4). Error estimates of the algorithm are given. At last, numerical experiments are provided to illustrate the validity of the algorithm. Both theoretical analysis and numerical results show that the proposed algorithm benefits the methods in [Huang et al., loc. cit.] and [Z. Wu, Eng. Anal. Bound. Elem. 29, No. 4, 354–358 (2005; Zbl 1182.76933)].
MSC:
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Lefloch, P. G., Hyperbolic systems of conservation laws, (2000), Oxford University Press · Zbl 0997.35002 |
| [2] | Ames, W. F., Numerical methods for partial differential equations, (2014), Academic Press · Zbl 0219.35007 |
| [3] | Hundsdorfer, W.; Verwer, J. G., Numerical solution of time-dependent advection-diffusion-reaction equations, (2013), Springer Science & Business Media |
| [4] | Liseikin, V. D., Grid generation methods, (2009), Springer New York · Zbl 0949.65098 |
| [5] | Huang, W.; Russell, R. D., Adaptive moving mesh methods, (2011), Springer New York · Zbl 1227.65090 |
| [6] | De Boor, C., Good approximation by splines with variable knots. II Conference on the numerical solution of differential equations, 12-20, (1974), Springer Berlin Heidelberg · Zbl 0343.65005 |
| [7] | Huang, W.; Ren, Y.; Russell, R. D., Moving mesh partial differential equations (MMPDES) based on the equidistribution principle, SIAM J Numer Anal, 31, 3, 709-730, (1994) · Zbl 0806.65092 |
| [8] | Li, R.; Tang, T.; Zhang, P., Moving mesh methods in multiple dimensions based on harmonic maps, J Comput Phys, 170, 2, 562-588, (2001) · Zbl 0986.65090 |
| [9] | Zhang Q, Zhao Y, Levesley J. Adaptive radial basis function interpolation using an error indicator. numer algori. 2017. 1-31.; Zhang Q, Zhao Y, Levesley J. Adaptive radial basis function interpolation using an error indicator. numer algori. 2017. 1-31. |
| [10] | Wu, Z. M., Dynamical knot and shape parameter setting for simulating shock wave by using multi-quadric quasi-interpolation, Eng Anal Bound Elem, 29, 4, 354-358, (2005) · Zbl 1182.76933 |
| [11] | Ngo, C.; Huang, W. Z., A study on moving mesh finite element solution of the porous medium equation, J Comput Phys, 331, 357-380, (2017) · Zbl 1378.76110 |
| [12] | Lee, T. E.; Baines, M. J.; Langdon, S., A finite difference moving mesh method based on conservation for moving boundary problems, J Comput Appl Math, 288, 1-17, (2015) · Zbl 1320.65118 |
| [13] | Esmaeilbeigi, M.; Hosseini, M. M., Dynamic node adaptive strategy for nearly singular problems on large domains, Eng Anal Bound Elem, 36, 9, 1311-1321, (2012) · Zbl 1352.65564 |
| [14] | Huang, W. Z., Practical aspects of formulation and solution of moving mesh partial differential equations, J Comput Phys, 171, 2, 753-775, (2001) · Zbl 0990.65107 |
| [15] | Budd, C. J., Adaptivity with moving grids, Acta Numer, 18, 18, 111-241, (2009) · Zbl 1181.65122 |
| [16] | Hon, Y. C.; Mao, X. Z., An efficient numerical scheme for burgers’ equation, Appl Math Comput, 95, 1, 37-50, (1998) · Zbl 0943.65101 |
| [17] | Duan, Y.; Rong, F., A numerical scheme for nonlinear schrodinger equation by MQ quasi-interpolation, Eng Anal Bound Elem, 37, 1, 89-94, (2013) · Zbl 1352.65391 |
| [18] | Gao, W. W.; Wu, Z. M., Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation, Appl Math Comput, 253, 377-386, (2015) · Zbl 1338.65205 |
| [19] | Gao, Q. J.; Zhang, S. G., Moving mesh strategies of adaptive methods for solving nonlinear partial differential equations, Algorithms, 9, 4, 86, (2016) · Zbl 1461.65235 |
| [20] | Hardy, R., Multiquadric equations of topography and other irregular surfaces, J Geoph Res, 76, 1905-1915, (1971) |
| [21] | Beatson, R.; Powell, M., Univariate multiquadric approximation: quasi-interpolation to scattered data, Constr Approx, 8, 275-288, (1992) · Zbl 0763.41012 |
| [22] | Buhmann, M., Convergence of univariate quasi-interpolation using multiquadrics, IMA J Numer Anal, 8, 365-383, (1988) · Zbl 0659.41003 |
| [23] | Wu, Z. M.; Liu, J. P., Generalized strang-fix condition for scattered data quasi-interpolation, Adv Comput Math, 23, 201-214, (2005) · Zbl 1068.65028 |
| [24] | Ma, L. M.; Wu, Z. M., Approximation to the kth derivatives by multiquadric quasi-interpolation method, J Comput Appl Math N, 2, 925-932, (2009) · Zbl 1236.65020 |
| [25] | Sun, Z. J.; Wu, Z. M.; Gao, W. W., Order preserving derivative approximation with quasi-interpolation, preprint, (2018) |
| [26] | Ma, L. M.; Wu, Z. M., Stability of multiquadric quasi-interpolation to approximate high order derivatives, Sci China Math, 53, 4, 985-992, (2010) · Zbl 1191.65018 |
| [27] | Kansa, E., Multiquadrics- a scattered data approximation scheme with applications to computational fluid-dynamics-i: surface approximation and partial derivative estimates, Comput Math Appl, 19, 127-145, (1990) · Zbl 0692.76003 |
| [28] | Chen, R. H.; Wu, Z. M., Applying multiquadric quasi-interpolation to solve Burgers equation, Appl Math Comput, 172, 472-484, (2006) · Zbl 1088.65086 |
| [29] | Gao, Q. J.; Wu, Z. M.; Zhang, S. G., Applying multiquadric quasi-interpolation for boundary detection, Comput & Math Appl, 62, 12, 4356-4361, (2011) · Zbl 1236.94020 |
| [30] | Gao, W. W.; Zhang, R., Multiquatric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective, Numer Algorit, (2017) |
| [31] | Cole, J. D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart Appl Math, 9, 225-236, (1951) · Zbl 0043.09902 |
| [32] | Christie, I.; Mitchell, A. R., Upwinding of high order Galerkin methods in conduction- convection problems, Int J Numer Methods Eng, 12, 1764-1771, (1978) · Zbl 0391.65034 |
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.
