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Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems. (English) Zbl 1403.65106
Summary: In this paper, the localized radial basis function collocation method (LRBFCM) is combined with the partial upwind scheme for solving convection-dominated fluid flow problems. The localization technique adopted in LRBFCM has shown to be effective in avoiding the well known ill-conditioning problem of traditional meshless collocation method with globally defined radial basis functions (RBFs). For convection-diffusion problems with dominated convection, stiffness in the form of boundary/interior layers and shock waves emerge as convection overwhelms diffusion. We show in this paper that these kinds of stiff problems can be well tackled by combining the LRBFCM with partial upwind scheme. For verification, several numerical examples are given to demonstrate that this scheme improves the LRBFCM in providing stable, accurate, and oscillation-free solutions to one- and two-dimensional Burger’s equations with shock waves and singular perturbation problems with turning points and boundary layers.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Babuška, I.; Banerjee, U.; Osborn, J. E., Survey of meshless and generalized finite element methods: a unified approach, Acta Numer, 1-125, (2003) · Zbl 1048.65105
[2] Atluri, S. N.; Shen, S., The meshless local Petrov-Galerkin methoda simple & less-costly alternative to the finite element and boundary element methods, CMES: Comput Model Eng Sci, 3, 2, 11-52, (2002) · Zbl 0996.65116
[3] Kansa, E. J., Multiquadrics - a scattered data approximation scheme with application to computational fluid dynamics, part I, Comput Math Appl, 19, 127-145, (1990) · Zbl 0692.76003
[4] Kansa, E. J., Multiquadrics - a scattered data approximation scheme with application to computational fluid dynamics, part II, Comput Math Appl, 19, 147-161, (1990) · Zbl 0850.76048
[5] Fasshauer, G. E., Solving partial differential equations by collocation with radial basis functions, (Mehaute, A. L.; Rabut, C.; Schumaker, L. L., Surface fitting and multiresolution methods, (1997), Vanderbilt University Press Nashville, TN), 131-138 · Zbl 0938.65140
[6] Franke, C.; Schaback, R., Solving partial differential equations by collocation using radial basis functions, Appl Math Comput, 93, 73-82, (1998) · Zbl 0943.65133
[7] Franke, C.; Schaback, R., Convergence order estimates of meshless collocation methods using radial basis functions, Adv Comput Math, 8, 4, 381-399, (1998) · Zbl 0909.65088
[8] Hon, Y. C.; Schaback, R., On unsymmetric collocation by radial basis functions, Appl Math Comput, 119, 177-186, (2001) · Zbl 1026.65107
[9] Wu, Z.; Hon, Y. C., Convergence error estimate in solving free boundary diffusion problem by radial basis functions method, Eng Anal Bound Elem, 27, 73-79, (2003) · Zbl 1040.91058
[10] Wendland, H., Scattered data approximation. Cambridge monographs on applied and computational mathematics, (2005), Cambridge University Press Cambridge
[11] Ling, L.; Kansa, E. J., Preconditioning for radial basis functions with domain decomposition methods, Math Comput Modell, 40, 13, 1413-1427, (2004) · Zbl 1077.41008
[12] Ling, L.; Schaback, R., An improved subspace selection algorithm for meshless collocation methods, Int J Numer Methods Eng, 80, 13, 1623-1639, (2009) · Zbl 1183.65153
[13] Li, J.; Hon, Y. C., Domain decomposition for radial basis meshless methods, Numer Methods Partial Differ Equ, 20, 3, 450-462, (2004) · Zbl 1048.65124
[14] Šarler, B., From global to local radial basis function collocation method for transport phenomena, 257-282, (2007), Springer Berlin · Zbl 1323.65111
[15] Lee, C.; Liu, X.; Fan, S., Local multiquadric approximation for solving boundary value problems, Comput Mech, 30, 396-409, (2003) · Zbl 1035.65136
[16] Li, M.; Chen, W.; Chen, C. S., The localized RBFs collocation methods for solving high dimensional pdes, Eng Anal Bound Elem, 37, 10, 1300-1304, (2013) · Zbl 1287.65115
[17] Šarler, B.; Vertnik, R., Meshfree explicit local radial basis function collocation method for diffusion problems, Comput Math Appl, 21, 1269-1282, (2006) · Zbl 1168.41003
[18] Hon, Y. C.; Šarler, B.; Yun, D. F., Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface, Eng Anal Bound Elem, 57, 2-8, (2015) · Zbl 1403.76140
[19] Sarra, S. A., A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains, Appl Math Comput, 218, 9853-9865, (2012) · Zbl 1245.65144
[20] Godunov, S. K., A finite difference method for the numerical computation of discontinuous solutions of the fluid dynamics, Mat Sb, 47, 271-290, (1959)
[21] Osher, S., Nonlinear singular perturbation problems and one sided difference schemes, SIAM J Numer Anal, 18, 129-144, (1981) · Zbl 0471.65069
[22] Kopteva, N., Error expansion for an upwind scheme applied to a two-dimensional convection-diffusion problem, SIAM J Numer Anal, 41, 5, 1851-1869, (2004) · Zbl 1055.65119
[23] Pandian, M. C., A partial upwind difference scheme for nonlinear parabolic equations, J Comput Appl Math, 26, 219-233, (1989) · Zbl 0676.65094
[24] Knobloch P. Numerical solution of convection-diffusion equations using upwinding techniques satisfying the discrete maximum principle. In: Proceedings of the Czech-Japanese seminar in applied mathematics 2005, Kuju Training Center, Oita, Japan, September 15-18, 2005. p. 69-76. · Zbl 1143.65394
[25] Heinrich, J. C.; Huyakorn, P. S.; Zienkiewicz, O. C.; Mitchell, A. R., An upwind finite element scheme for two-dimensional convective transport equation, Int J Numer Methods Eng, 11, 1, 131-143, (1977) · Zbl 0353.65065
[26] Tabata, M., A finite element approximation corresponding to the upwind finite differencing, Mem Numer Anal, 4, 47-63, (1977) · Zbl 0358.65102
[27] Shu, C. W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Adv Numer Approx Nonlinear Hyperbolic Equ, 1697, 325-432, (2006) · Zbl 0927.65111
[28] Shu, C. W., High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev, 51, 82-126, (2009) · Zbl 1160.65330
[29] Shu CW. On high-order accurate weighted essentially non-oscillatory and discontinuous Galerkin schemes for compressible turbulence simulations. Philos Trans R Soc A 2013;371:20120172. · Zbl 1353.76039
[30] Gu, Y. T.; Liu, G. R., Meshless techniques for convection dominated problems, Comput Mech, 38, 171-182, (2006) · Zbl 1138.76402
[31] Chandhini, G.; Sanyasiraju, Y. V.S. S., Local RBF solutions for steady convective diffusion problems, Int J Numer Methods Eng, 72, 352-378, (2007) · Zbl 1194.76174
[32] Sanyasiraju, Y. V.S. S.; Chandhini, G., A note on two upwind strategies for RBF-based grid-free schemes to solve steady convection-diffusion equations, Short Commun Int J Numer Methods Fluids, 61, 1053-1062, (2009) · Zbl 1252.65195
[33] Siraj-ul-Islam; Šarler, B.; Vertnik, R.; Kosec, G., Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled burgers׳ equations, Appl Math Modell, 36, 3, 1148-1160, (2012) · Zbl 1243.76076
[34] El Zahab, Z.; Divo, E.; Kassab, A. J., A localized collocation meshless method (LCMM) for incompressible flows CFD modeling with applications to transient hemodynamics, Eng Anal Bound Elem, 33, 1045-1061, (2009) · Zbl 1244.76073
[35] Shu, C.; Ding, H.; Chen, H. Q.; Wang, T. G., An upwind local RBF-DQ method for simulation of inviscid compressible flows, Comput Methods Appl Mech Eng, 194, 2001-2017, (2005) · Zbl 1093.76052
[36] Shen, Q., Local RBF-based differential quadrature collocation method for the boundary layer problems, Eng Anal Bound Elem, 34, 213-228, (2010) · Zbl 1244.65118
[37] Chan, Y. L.; Shen, L. H.; Wu, C. T.; Young, D. L., A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows, Comput Fluids, 89, 157-166, (2014) · Zbl 1391.76529
[38] Sanyasiraju, Y. V.S. S.; Satyanarayana, C., Upwind strategies for local RBF scheme to solve convection dominated problems, Eng Anal Bound Elem, 48, 1-13, (2014) · Zbl 1403.65175
[39] Siraj-ul-Islam; Vertnik, R.; Šarler, B., Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations, Appl Numer Math, 67, 136-151, (2013) · Zbl 1263.65099
[40] Fornberg, B.; Lehto, E., Stabilization of RBF-generated finite difference methods for convective pdes, J Comput Phys, 230, 2270-2285, (2011) · Zbl 1210.65154
[41] Flyer, N.; Lehto, E., Rotational transport on a spherelocal node refinement with radial basis functions, J Comput Phys, 229, 1954-1969, (2010) · Zbl 1303.76128
[42] Micchelli, C. A., Interpolation of scattered datadistance matrices and conditionally positive definite functions, Constr Approx, 2, 11-22, (1986) · Zbl 0625.41005
[43] Cole, J. D., On a quasi-linear parabolic equations occurring in aerodynamics, Q Appl Math, 9, 225-236, (1951) · Zbl 0043.09902
[44] Caldwell, J.; Smith, P., Solution of Burgers equation with a large Reynolds number, Appl Math Modell, 6, 381-385, (1982) · Zbl 0496.76029
[45] Hon, Y. C.; Mao, X. Z., An efficient numerical scheme for burgers׳ equation, Appl Math Comput, 95, 37-50, (1998) · Zbl 0943.65101
[46] Christie, I.; Griffiths, D. F.; Mitchell, A. R., Product approximation for nonlinear problems in the finite element method, IMA J Numer Anal, 1, 3, 253-266, (1981) · Zbl 0469.65072
[47] Hashemian, A.; Shodja, H. M., A meshless approach for solution of burgers׳ equation, J Comput Appl Math, 220, 226-239, (2008) · Zbl 1149.65079
[48] Caldwell, J.; Wanless, P.; Cook, A. E., Solution of burgers׳ equation for large Reynolds number using finite elements with moving nodes, J Appl Math Modell, 11, 211-214, (1987) · Zbl 0622.76063
[49] Li, J.; Hon, Y. C.; Chen, C. S., Numerical comparisons of two meshless methods using radial basis functions, Eng Anal Bound Elem, 26, 205-225, (2002) · Zbl 1003.65132
[50] Sun, P. T.; Chen, L.; Xu, J. C., Numerical studies of adaptive finite element methods for two dimensional convection-dominated problems, J Sci Comput, 43, 24-43, (2010) · Zbl 1203.65261
[51] Xie, S. S.; Heob, S.; Kimc, S.; Wooc, G.; Yi, S., Numerical solution of one-dimensional burgers׳ equation using reproducing kernel function, J Comput Appl Math, 214, 417-434, (2008)
[52] Holland CJ. Comments on singularly perturbed elliptic partial equations. Personal communication; 1991.
[53] Adjerid, S.; Aiffa, M.; Flaherty, J. E., High-order finite element methods for singularly perturbed elliptic and parabolic problems, SIAM J Appl Math, 55, 2, 520-543, (1995) · Zbl 0827.65097
[54] Tang WP. Numerical solution of a turning point problem. In: Chan T, Keyes D, Meurant G, Scroggs S, Voigt R, editors. Fifth international conference on domain decomposition methods for partial differential equations, Norfolk, 1991.
[55] Adjerid, S.; Flaherty, J. E.; Moore, P. K.; Wang, Y. J., High-order adaptive methods for parabolic systems, Physica D, 60, 94-111, (1992) · Zbl 0790.65088
[56] Grassman, J.; Matkowsky, B. J., A variational approach to singularly perturbed boundary value problems for ordinary and partial differential equations with turning points, SIAM J Appl Math, 32, 588-596, (1977) · Zbl 0392.34038
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