Solución bidimensional sin malla de la ecuación no lineal de convección-difusión-reacción mediante el método de Interpolación Local Hermítica
Main Article Content
Keywords
Funciones de Base Radial, Métodos sin malla, Método Simé- trico, Newton-Raphson, Método de Homotopía.
Resumen
Un método sin malla es desarrollado para solucionar una versión genérica de la ecuación no lineal de convección-difusión-reacción en dominios bidimensionales. El método de Interpolación Local Hermítica (LHI) es empleado para la discretización espacial, y diferentes estrategias son implementadas para solucionar el sistema de ecuaciones no lineales resultante, entre estas iteración de Picard, método de Newton-Raphson y el Método de Homotopía truncado (HAM). En el método LHI las Funciones de Base Radial (RBFs) son empleadas para construir una función de interpolación. A diferencia del Método de Kansa, el LHI es aplicado localmente y los operadores diferenciales de las condiciones de frontera y la ecuación gobernante son utilizados para construir la función de interpolación, obteniéndose una matriz de colocación simétrica. El método de Newton-Rapshon se implementa con matriz Jacobiana analítica y numérica, y las derivadas de la ecuación gobernante con respecto al paramétro de homotopía son obtenidas analíticamente. El esquema numérico es verificado mediante la comparación de resultados con las soluciones analíticas de las ecuaciones de Burgers en una dimensión y Richards en dos dimensiones. Similares resultados son obtenidos para todos los solucionadores que se probaron, pero mejores ratas de convergencia son logradas con el método de Newton-Raphson en doble iteración.
MSC: 65H20, 65N35
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[2] E. J. Kansa, “Multiquadrics -a scattered data approximation scheme with applications to computational fluid dynamics-II solution to parabolic, hyperbolic and elliptic partial differential equations,” Comput. Math. Appl., vol. 19, no. 8-9, pp. 127–145, 1990. 23
[3] F. E. Fasshauer, “Solving partial differential equations by collocation with radial basis functions,” Surf. Fit. Multires. Methods, pp. 131–138, 1997. 23, 24, 29
[4] B. Jumarhon, S. Amini, and K. Chen, “The Hermite collocation method using radial basis functions,” Eng. Anal. Boundary Elem., vol. 24, no. 7-8, pp. 607–611, 2000. 23, 31
[5] H. Power and V. Barraco, “A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations,” Comput. Math. Appl., vol. 43, no. 3-5, pp. 551–583, 2002. 23, 31
[6] A. LaRocca, A. Hernandez, and H. Power, “Radial basis functions Hermite collocation approach for the solution of time dependent convenction-diffusion problems,” Eng. Anal. Boundary Elem., vol. 29, no. 4, pp. 359–370, 2005. 24, 30
[7] P. P. Chinchapatnam, K. Djidjeli, and P. B. Nair, “Unsymmetric and symmetric meshless schemes for the unsteady convection?diffusion equation,” Comput. Meth. Appl. Mech. Eng., vol. 195, no. 19–22, pp. 2432–2453, 2006. 24
[8] W. R. Madych and S. A. Nelson, “Multivariate Interpolation and Conditionally Positive Definite Functions,” Math. Comput., vol. 54, no. 1, pp. 211–230, 1990. 24
[9] M. Zerroukat, H. Power, and C. S. Chen, “A numerical method for heat transfer problems using collocation and radial basis functions,” Int. J. Numer. Methods Eng., vol. 42, no. 7, pp. 1263–1278, 1998. 24
[10] J. Li, A. H. D. Cheng, and C. S. Chen, “A comparison of efficiency and error convergence of multiquadric collocation method and finite element method,” Eng. Anal. Boundary Elem., vol. 27, no. 3, pp. 251–257, 2003. 24
[11] A. H.-D. Cheng and J. J. S. P. Cabral, “Direct solution of ill-posed boundary value problems by radial basis function collocation method,” Int. J. Numer. Methods Eng., vol. 64, no. 1, pp. 45–64, 2005. 24
[12] S. Chantasiriwan, “Cartesian grid methods using radial basis functions for solving Poisson, Helmholtz, and diffusion?convection equations,” Eng. Anal. Boundary Elem., vol. 28, no. 12, pp. 1417–1425, 2004. 24
[13] R. Schaback, “Multivariate interpolation and approximation by translates of basis functions,” Approx. Theory, vol. 8, pp. 1–8, 1995. 24
[14] D. Brown, “On approximate cardinal preconditioning methods for solving PDEs with radial basis functions,” Eng. Anal. Boundary Elem., vol. 29, pp. 343–353, 2005. 24
[15] L. Ling and R. Schaback, “On adaptive unsymmetric meshless collocation,” Proceedings of the 2004 international conference on computational and experimental engineering and sciences, 2004. 24
[16] C. Lee, X. Liu, and S. Fan, “Local Multicuadric approximation for solving boundary value problems,” Comput. Mech., vol. 30, pp. 396–409, 2003. 24
[17] B. Sarler and R. Vertnik, “Meshless explicit local radial basis function collocation methods for diffusion problems,” Comput. Math. Appl., vol. 51, pp. 1269–1282, 2006. 24, 25
[18] E. Divo and K. Kassab, “An efficient localised radial basis function collocation method for fluid flow and conjugate heat transfer,” J. Heat Transfer, vol. 212, pp. 99–123, 2006. 24, 25
[19] G. Wright and B. Fornberg, “Scattered node compact finite difference-type formulas generated from radial basis functions,” J. Comput. Phys., vol. 212, no. 1, pp. 99–123, 2006. 24
[20] C. K. Lee, X. Liu, and S. C. Fan, “Local multiquadric approximation for solving boundary value problems,” Comput. Mech., vol. 30, no. 5-6, pp. 396–409, 2003. 24
[21] T. J. Moroney and I. W. Turner, “A finite volume method based on radial basis functions for two-dimensional nonlinear diffusion equations,” Appl. Math. Modell., vol. 30, no. 10, pp. 1118–1133, 2006. 25
[22] ——, “A three dimensional finite volume method based on radial basis functions for the accurate computacional modelling of nonlinear diffusion equations,” J. Comput. Phys., vol. 225, no. 2, pp. 1409–1426, 2007. 25
[23] P. Orsini, H. Power, and H. Morovan, “Improving Volume Element Methods by Meshless Radial Basis Function Techniques,” Comput. model. Eng. Scien., vol. 769, no. 1, pp. 1–21, 2008. 25
[24] D. Stevens, H. Power, and H. Morvan, “An order-N complexity meshless algorithm for transport-type PDEs,based on local Hermitian interpolation,” Eng. Anal. Boundary Elem., vol. 33, no. 4, pp. 425–441, 2009. 25, 28, 31, 32
[25] D. Stevens, H. Power, M. Lees, and H. Morvan, “The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems,” J. Comput. Phys., vol. 228, no. 12, pp. 4606–4624, 2009. 25, 28, 29, 30, 31
[26] B.-C. Shin, M. T. Darvishi, and C.-H. Kim, “A Comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear
systems,” Appl. Math. Comput., vol. 217, no. 7, pp. 3190–3198, 2010. 25, 32, 43
[27] Y. I. Kim and Y. H. Geum, “A cubic-order variant of Newton’s method for finding multiple roots of nonlinear equations,” Comput. Math. Appl., vol. 62, pp. 1634–1640, 2011. 25
[28] X. Cui and J.-Y. Yue, “A nonlinear iteration method for solving a twodimensional nonlinear coupled system of parabolic and hyperbolic equations,” J. Comput. Appl. Math., vol. 234, no. 2, pp. 343–364, 2010. 25
[29] P. S. Mohan, P. B. Nair, and A. J. Keane, “Inexact Picard iterative scheme for steady-state nonlinear diffusion in random heterogeneous media,” Phys. Rev. E, vol. 79, no. 4, pp. 1–9, 2009. 25
[30] D. Stevens, H. Power, M. Lees, and H. Morvan, “A Meshless Solution Technique for the Solution of 3D Unsaturated Zone Problems, Based on Local Hermitian Interpolation with Radial Basis Functions,” Transp. Porous Media, vol. 79, no. 2, pp. 149–169, 2009. 25
[31] S. J. Liao, “On the general boundary element method,” Eng. Anal. Boundary Elem., vol. 21, no. 1, pp. 39–51, 1998. 26
[32] ——, “Boundary element method for general nonlinear differential operators,” Eng. Anal. Boundary Elem., vol. 20, no. 2, pp. 91–99, 1997. 26, 35
[33] ——, “A direct boundary element approach for unsteady non-linear heat transfer problems,” Eng. Anal. Boundary Elem., vol. 26, pp. 55–59, 2002. 26, 36
[34] Z. Lin and S. Liao, “The scaled boundary FEM for nonlinear problems,” Commun. Nonlinear Sci. Numer. Simul., vol. 16, no. 1, pp. 63–75, 2011. 26
[35] S. S. Motsa, P. Sibanda, and S. Shateyi, “A new spectral-homotopy analysis method for solving a nonlinear second order BVP,” Commun. Nonlinear Sci. Numer. Simul., vol. 15, no. 9, pp. 2293–2302, 2010. 26
[36] H. Zhu, H. Shu, and M. Ding, “Numerical solutions of partial differential equations by discrete homotopy analysis method,” Appl. Math. Comput., vol. 216, no. 12, pp. 3592–3615, 2010. 26
[37] M. Noskov and M. D. Smooke, “An implicit compact scheme solver with application to chemically reacting flows,” J. Comput. Phys., vol. 203, pp. 700–730, 2005. 38
[38] F. T. Tracy, “Clean two- and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers,” Water Resour. Res., vol. 42, no. 8, pp. 1–11, 2006. 43
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Publicado
mar 20, 2013
Article Details
Cómo citar
Bustamante Chaverra, C. A., Power, H., Florez Escobar, W. F., & Hill Betancourt, A. F. (2013). Solución bidimensional sin malla de la ecuación no lineal de convección-difusión-reacción mediante el método de Interpolación Local Hermítica. Ingeniería Y Ciencia, 9(17), 21–51. https://doi.org/10.17230/ingciecia.9.17.2
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COLCIENCIAS - Empresas Públicas de Medellín (EPM) - Energy Research and Innovation Centre (CIIEN)
Biografía del autor/a
Carlos A Bustamante Chaverra, Universidad Pontificia Bolivariana
PhD in Computational Mechanics
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