DOI: 10.1108/09615530410513809. Radial basis function collocation method solution of natural convection in porous media

Radial basis function collocation method solution of natural convection in porous media

10.1108/09615530410513809

Crossref journal-article

Emerald

International Journal of Numerical Methods for Heat & Fluid Flow (140)

Abstract

This paper describes the solution of a steady-state natural convection problem in porous media by the radial basis function collocation method (RBFCM). This mesh-free (polygon-free) numerical method is for a coupled set of mass, momentum, and energy equations in two dimensions structured by the Hardy's multiquadrics with different shape parameter and different order of polynomial augmentation. The solution is formulated in primitive variables and involves iterative treatment of coupled pressure, velocity, pressure correction, velocity correction, and energy equations. Numerical examples include convergence studies with different collocation point density and arrangements for a two-dimensional differentially heated rectangular cavity problem at filtration Rayleigh numbers Ra*=25, 50 and 100, and aspect ratios A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine-mesh finite volume method in terms of mid-plane velocity components, mid-plane and insulated surface temperatures, streamfunction minimum, and Nusselt number.

Authors 3

Božidar Šarler (first)
Janez Perko (additional)
Ching-Shyang Chen (additional)

References 40 Referenced 41

Bulgakov, V. , Šarler, B. and Kuhn, G. (1998), “Iterative solution of systems of equations in the dual reciprocity boundary element method for diffusion equation”, International Journal of Numerical Methods in Engineering, Vol. 43, pp. 713-32. (10.1002/(SICI)1097-0207(19981030)43:4<713::AID-NME445>3.0.CO;2-8)
Carlson, R.E. and Foley, T.A. (1991), “The parameter R2 in multiquadrics interpolation”, Computers & Mathematics with Applications, Vol. 21, pp. 29-42. (10.1016/0898-1221(91)90123-L)
Chan, B.K.C. , Ivey, C.M. and Barry, J.M. (1970), “Natural convection in enclosed porous media with rectangular boundaries”, Wärme und Stoffübertragung, Vol. 7, pp. 22-30. (10.1115/1.3449641)
Chen, W. and Tanaka, M. (2000), “New insights into boundary-only and domain-type RBF methods”, Journal of Nonlinear Science and Numerical Simulation, Vol. 1, pp. 145-51. (10.1515/IJNSNS.2000.1.3.145)
Chen, W. and Tanaka, M. (2002), “A meshless, exponential convergence, intergation-free, and boundary-only RBF technique”, Computers & Mathematics with Application, Vol. 43, pp. 379-91. (10.1016/S0898-1221(01)00293-0)
Chen, C.S. , Brebbia, C.A. and Power, H. (1999), “Dual reciprocity method using compactly supported radial basis functions”, Communications in Numerical Methods in Engineering, Vol. 15, pp. 137-50. (10.1002/(SICI)1099-0887(199902)15:2<137::AID-CNM233>3.0.CO;2-9)
Darcy, H. Les Fontaines Publique de la Ville de Dijon, Victor Delmont, Paris, France.
Fasshauer, G.E. (1999), “Solving differential equations with radial basis functions: multilevel methods and smoothing”, Advances in Computational Mathematics, Vol. 11, pp. 139-59. (10.1023/A:1018919824891)
Franke, J. (1982), “Scattered data interpolation: tests of some methods”, Mathematics of Computation, Vol. 48, pp. 181-200. (10.1090/S0025-5718-1982-0637296-4)
Gobin, D. and Bennacer, R. (1996a), “Cooperating thermosolutal convection in enclosures, 1. Scale analysis and mass transfer”, International Journal of Heat and Mass Transfer, Vol. 39, pp. 2671-81. (10.1016/0017-9310(95)00350-9)
Gobin, D. and Bennacer, R. (1996b), “Cooperating thermosolutal convection in enclosures, 2. Heat transfer and flow structure”, International Journal of Heat and Mass Transfer, Vol. 39, pp. 2683-97. (10.1016/0017-9310(95)00351-7)
Golberg, M.A. and Chen, C.S. (1997), Discrete Projection Methods for Integral Equations, CMP, Southampton, UK.
Golberg, M.A. , Chen, C.S. and Karur, S.R. (1996), “Improved multiquadric approximation for partial differential equations”, Engineering Analysis with Boundary Elements, Vol. 18, pp. 9-17. (10.1016/S0955-7997(96)00033-1)
Hardy, R.L. (1990), “Theory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968-1988)”, Computers & Mathematics with Applications, Vol. 19, pp. 163-208. (10.1016/0898-1221(90)90272-L)
Hickox, G.E. and Gartling, D.K. (1981), “A numerical study of natural convection in a horizontal porous layer subjected to an end-to-end temperature difference”, Journal of Heat Transfer, Vol. 103, pp. 797-802. (10.1115/1.3244544)
Jecl, R. , Škerget, L. and Petrišin, E. (2001), “Boundary domain integral method for transport phenomena in porous media”, International Journal for Numerical Methods in Fluids, Vol. 35, pp. 39-54. (10.1002/1097-0363(20010115)35:1<39::AID-FLD81>3.0.CO;2-3)
Kansa, E.J. (1990a), “Muliquadrics – a scattered data approximation scheme with application to computational fluid dynamics – I, Surface approximations and partial derivative estimates”, Computers & Mathematics with Applications, Vol. 19, pp. 127-45. (10.1016/0898-1221(90)90270-T)
Kansa, E.J. (1990b), “Muliquadrics – a scattered data approximation scheme with application to computational fluid dynamics – II, Solutions to parabolic, hyperbolic and elliptic partial differential equations”, Computers & Mathematics with Applications, Vol. 19, pp. 147-61. (10.1016/0898-1221(90)90271-K)
Kaviany, M. (1995), Principles of Heat Transfer in Porous Media, Springer-Verlag, Berlin, Germany. (10.1007/978-1-4612-4254-3)
Li, J.C. , Hon, Y.C. and Chen, C.S. (2002), “Numerical comparisons of two meshless methods using radial basis functions”, Engineering Analysis with Boundary Elements, Vol. 26, pp. 205-25. (10.1016/S0955-7997(01)00101-1)
Mai-Dui, N. and Tran-Cong, T. (2001a), “Numerical solution of Navier-Stokes equations using multiquadric radial basis function networks”, International Journal for Numerical Methods in Fluids, Vol. 37, pp. 65-86. (10.1002/fld.165)
Mai-Dui, N. and Tran-Cong, T. (2001b), “Numerical solution of differential equations using multiquadric radial basis function networks”, Neural Networks, Vol. 14, pp. 185-99. (10.1016/S0893-6080(00)00095-2)
Mai-Dui, N. and Tran-Cong, T. (2002), “Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations”, Engineering Analysis with Boundary Elements, Vol. 26, pp. 133-56. (10.1016/S0955-7997(01)00092-3)
Nield, D.A. and Bejan, A. (1992), Convection in Porous Media, Springer-Verlag, Berlin. (10.1007/978-1-4757-2175-1)
Perko, J. , Chen, C.S. and Šarler, B. (2001), “A polygon-free numerical solution of steady natural convection in solid-liquid systems”, in Šarler, B. and Brebbia, C.A. (Eds), Computational Modelling of Moving and Free Boundary Problems, Wit Press, Southampton, UK, pp. 111-22.
Power, H. and Barraco, V. (2002), “A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations”, Computers & Mathematics with Applications, Vol. 43, pp. 551-83. (10.1016/S0898-1221(01)00305-4)
Prasad, V. and Kulacki, F.A. (1984), “Convective heat transfer in a rectangular porous cavity – effect of aspect ratio on flow structure and heat transfer”, Journal of Heat Transfer, Vol. 106, pp. 158-65. (10.1115/1.3246629)
Raghavan, R. and Ozkan, E. (1992), A Method for Computing Unsteady Flows in Porous Media, Pitman Research Notes in Mathematics Series, Vol. 318, Longman Scientific & Technical, Harlow, UK.
Sadat, H. and Couturier, S. (2000), “Performance and accuracy of a meshless method for laminar natural convection”, Numerical Heat Transfer, Vol. 37B, pp. 455-67. (10.1080/10407790050051146)
Sahimi, M. (1995), Flow and Transport in Porous Media and Fractured Rock, VCH Verlagsgesellschaft mbH, Weinheim, Germany.
Šarler, B. , Perko, J. and Chen, C.S. (2002), “Natural convection in porous media – radial basis function collocation method solution of the Darcy model”, in Mang, H.A., Rammerstorfer, F.G. and Eberhardsteiner, J. (Eds), Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V).
Šarler, B. , Perko, J., Chen, C.S. and Kuhn, G. (2001), “A meshless approach to natural convection”, in Atluri, S.N. (Ed.), International Conference on Computational Engineering and Sciences – 2001, CD-rom Proceedings, pp. 3-9.
Šarler, B. , Gobin, D., Goyeau, B., Perko, J. and Power, H. (2000), “Natural convection in porous media – dual reciprocity boundary element method solution of the Darcy model”, International Journal for Numerical Methods in Fluids, Vol. 33, pp. 279-312. (10.1002/(SICI)1097-0363(20000530)33:2<279::AID-FLD18>3.3.CO;2-E)
Wang, J.G. and Liu, G.R. (2002), “On the optimal shape parameters of radial basis functions used for 2-D meshless methods”, Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 2611-30. (10.1016/S0045-7825(01)00419-4)
Zerroukat, M. , Djidjeli, K. and Charafi, A. (2000), “Explicit and implicit meshless methods for linear advection-diffusion-type partial differential equations”, International Journal for Numerical Methods in Engineering, Vol. 48, pp. 19-35. (10.1002/(SICI)1097-0207(20000510)48:1<19::AID-NME862>3.0.CO;2-3)
Zerroukat, M. , Power, H. and Chen, C.S. (1998), “A numerical method for heat transfer problems using collocation and radial basis functions”, International Journal for Numerical Methods in Engineering, Vol. 42, pp. 1263-78. (10.1002/(SICI)1097-0207(19980815)42:7<1263::AID-NME431>3.0.CO;2-I)
10.1007/s004660000181 / Comput. Mech. / Meshless methods based on collocation with radial basis functions by Zhang (2000)
Chen, C.S. , Ganesh, M., Golberg, M.A. and Cheng, A.H.D. (2002), “Multilevel compact radial functions based computational schemes for some elliptic problems”, Computers & Mathematics with Applications, Vol. 43, pp. 359-78. (10.1016/S0898-1221(01)00292-9)
Golberg, M.A. , Chen, C.S. and Ganesh, M. (2000), “Particular solutions of 3D Helmoltz-type equations using compactly supported radial basis functions”, Engineering Analysis with Boundary Elements, Vol. 24, pp. 539-47. (10.1016/S0955-7997(00)00034-5)
Šarler, B. (2002), “Towards a mesh-free computation of transport phenomena”, Engineering Analysis with Boundary Elements, Vol. 26, pp. 731-8. (10.1016/S0955-7997(02)00044-9)

Dates

Type	When
Created	21 years, 6 months ago (Feb. 17, 2004, 1:44 p.m.)
Deposited	3 weeks, 2 days ago (July 28, 2025, 5:20 p.m.)
Indexed	2 weeks, 5 days ago (Aug. 2, 2025, 3:03 p.m.)
Issued	21 years, 5 months ago (March 1, 2004)
Published	21 years, 5 months ago (March 1, 2004)
Published Print	21 years, 5 months ago (March 1, 2004)

Funders 0

None

BibTeX

@article{_arler_2004, title={Radial basis function collocation method solution of natural convection in porous media}, volume={14}, ISSN={1758-6585}, url={http://dx.doi.org/10.1108/09615530410513809}, DOI={10.1108/09615530410513809}, number={2}, journal={International Journal of Numerical Methods for Heat & Fluid Flow}, publisher={Emerald}, author={Šarler, Božidar and Perko, Janez and Chen, Ching-Shyang}, year={2004}, month=mar, pages={187–212} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        2
      ]
    ],
    "date-time": "2025-08-02T19:03:36Z",
    "timestamp": 1754161416797,
    "version": "3.41.2"
  },
  "reference-count": 40,
  "publisher": "Emerald",
  "issue": "2",
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        2004,
        3,
        1
      ]
    ]
  },
  "abstract": "<jats:p>This paper describes the solution of a steady-state natural convection problem in porous media by the radial basis function collocation method (RBFCM). This mesh-free (polygon-free) numerical method is for a coupled set of mass, momentum, and energy equations in two dimensions structured by the Hardy's multiquadrics with different shape parameter and different order of polynomial augmentation. The solution is formulated in primitive variables and involves iterative treatment of coupled pressure, velocity, pressure correction, velocity correction, and energy equations. Numerical examples include convergence studies with different collocation point density and arrangements for a two-dimensional differentially heated rectangular cavity problem at filtration Rayleigh numbers Ra*=25, 50 and 100, and aspect ratios A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine-mesh finite volume method in terms of mid-plane velocity components, mid-plane and insulated surface temperatures, streamfunction minimum, and Nusselt number.</jats:p>",
  "DOI": "10.1108/09615530410513809",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2004,
        2,
        17
      ]
    ],
    "date-time": "2004-02-17T18:44:33Z",
    "timestamp": 1077043473000
  },
  "page": "187-212",
  "source": "Crossref",
  "is-referenced-by-count": 41,
  "title": "Radial basis function collocation method solution of natural convection in porous media",
  "prefix": "10.1108",
  "volume": "14",
  "author": [
    {
      "given": "Bo\u017eidar",
      "family": "\u0160arler",
      "sequence": "first",
      "affiliation": [
        {
          "name": "Nova Gorica Polytechnic, Nova Gorica, Slovenia"
        }
      ]
    },
    {
      "given": "Janez",
      "family": "Perko",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "Nova Gorica Polytechnic, Nova Gorica, Slovenia"
        }
      ]
    },
    {
      "given": "Ching-Shyang",
      "family": "Chen",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "University of Nevada, Las Vegas, Nevada, USA"
        }
      ]
    }
  ],
  "member": "140",
  "reference": [
    {
      "key": "2025072817203234200_b1",
      "doi-asserted-by": "crossref",
      "unstructured": "Bulgakov, V.\n          , \u0160arler, B. and Kuhn, G. (1998), \u201cIterative solution of systems of equations in the dual reciprocity boundary element method for diffusion equation\u201d, International Journal of Numerical Methods in Engineering, Vol. 43, pp. 713-32.",
      "DOI": "10.1002/(SICI)1097-0207(19981030)43:4<713::AID-NME445>3.0.CO;2-8"
    },
    {
      "key": "2025072817203234200_b2",
      "doi-asserted-by": "crossref",
      "unstructured": "Carlson, R.E.\n           and Foley, T.A. (1991), \u201cThe parameter R2 in multiquadrics interpolation\u201d, Computers & Mathematics with Applications, Vol. 21, pp. 29-42.",
      "DOI": "10.1016/0898-1221(91)90123-L"
    },
    {
      "key": "2025072817203234200_b3",
      "doi-asserted-by": "crossref",
      "unstructured": "Chan, B.K.C.\n          , Ivey, C.M. and Barry, J.M. (1970), \u201cNatural convection in enclosed porous media with rectangular boundaries\u201d, W\u00e4rme und Stoff\u00fcbertragung, Vol. 7, pp. 22-30.",
      "DOI": "10.1115/1.3449641"
    },
    {
      "key": "2025072817203234200_b4",
      "doi-asserted-by": "crossref",
      "unstructured": "Chen, W.\n           and Tanaka, M. (2000), \u201cNew insights into boundary-only and domain-type RBF methods\u201d, Journal of Nonlinear Science and Numerical Simulation, Vol. 1, pp. 145-51.",
      "DOI": "10.1515/IJNSNS.2000.1.3.145"
    },
    {
      "key": "2025072817203234200_b5",
      "doi-asserted-by": "crossref",
      "unstructured": "Chen, W.\n           and Tanaka, M. (2002), \u201cA meshless, exponential convergence, intergation-free, and boundary-only RBF technique\u201d, Computers & Mathematics with Application, Vol. 43, pp. 379-91.",
      "DOI": "10.1016/S0898-1221(01)00293-0"
    },
    {
      "key": "2025072817203234200_b6",
      "doi-asserted-by": "crossref",
      "unstructured": "Chen, C.S.\n          , Brebbia, C.A. and Power, H. (1999), \u201cDual reciprocity method using compactly supported radial basis functions\u201d, Communications in Numerical Methods in Engineering, Vol. 15, pp. 137-50.",
      "DOI": "10.1002/(SICI)1099-0887(199902)15:2<137::AID-CNM233>3.0.CO;2-9"
    },
    {
      "key": "2025072817203234200_b7",
      "unstructured": "Darcy, H.\n          \u2008Les Fontaines Publique de la Ville de Dijon, Victor Delmont, Paris, France."
    },
    {
      "key": "2025072817203234200_b8",
      "doi-asserted-by": "crossref",
      "unstructured": "Fasshauer, G.E.\n           (1999), \u201cSolving differential equations with radial basis functions: multilevel methods and smoothing\u201d, Advances in Computational Mathematics, Vol. 11, pp. 139-59.",
      "DOI": "10.1023/A:1018919824891"
    },
    {
      "key": "2025072817203234200_b9",
      "doi-asserted-by": "crossref",
      "unstructured": "Franke, J.\n           (1982), \u201cScattered data interpolation: tests of some methods\u201d, Mathematics of Computation, Vol. 48, pp. 181-200.",
      "DOI": "10.1090/S0025-5718-1982-0637296-4"
    },
    {
      "key": "2025072817203234200_b10",
      "doi-asserted-by": "crossref",
      "unstructured": "Gobin, D.\n           and Bennacer, R. (1996a), \u201cCooperating thermosolutal convection in enclosures, 1. Scale analysis and mass transfer\u201d, International Journal of Heat and Mass Transfer, Vol. 39, pp. 2671-81.",
      "DOI": "10.1016/0017-9310(95)00350-9"
    },
    {
      "key": "2025072817203234200_b11",
      "doi-asserted-by": "crossref",
      "unstructured": "Gobin, D.\n           and Bennacer, R. (1996b), \u201cCooperating thermosolutal convection in enclosures, 2. Heat transfer and flow structure\u201d, International Journal of Heat and Mass Transfer, Vol. 39, pp. 2683-97.",
      "DOI": "10.1016/0017-9310(95)00351-7"
    },
    {
      "key": "2025072817203234200_b12",
      "unstructured": "Golberg, M.A.\n           and Chen, C.S. (1997), Discrete Projection Methods for Integral Equations, CMP, Southampton, UK."
    },
    {
      "key": "2025072817203234200_b13",
      "doi-asserted-by": "crossref",
      "unstructured": "Golberg, M.A.\n          , Chen, C.S. and Karur, S.R. (1996), \u201cImproved multiquadric approximation for partial differential equations\u201d, Engineering Analysis with Boundary Elements, Vol. 18, pp. 9-17.",
      "DOI": "10.1016/S0955-7997(96)00033-1"
    },
    {
      "key": "2025072817203234200_b14",
      "doi-asserted-by": "crossref",
      "unstructured": "Hardy, R.L.\n           (1990), \u201cTheory and applications of the multiquadrics-biharmonic method (20 years of discovery 1968-1988)\u201d, Computers & Mathematics with Applications, Vol. 19, pp. 163-208.",
      "DOI": "10.1016/0898-1221(90)90272-L"
    },
    {
      "key": "2025072817203234200_b15",
      "doi-asserted-by": "crossref",
      "unstructured": "Hickox, G.E.\n           and Gartling, D.K. (1981), \u201cA numerical study of natural convection in a horizontal porous layer subjected to an end-to-end temperature difference\u201d, Journal of Heat Transfer, Vol. 103, pp. 797-802.",
      "DOI": "10.1115/1.3244544"
    },
    {
      "key": "2025072817203234200_b16",
      "doi-asserted-by": "crossref",
      "unstructured": "Jecl, R.\n          , \u0160kerget, L. and Petri\u0161in, E. (2001), \u201cBoundary domain integral method for transport phenomena in porous media\u201d, International Journal for Numerical Methods in Fluids, Vol. 35, pp. 39-54.",
      "DOI": "10.1002/1097-0363(20010115)35:1<39::AID-FLD81>3.0.CO;2-3"
    },
    {
      "key": "2025072817203234200_b17",
      "doi-asserted-by": "crossref",
      "unstructured": "Kansa, E.J.\n           (1990a), \u201cMuliquadrics \u2013 a scattered data approximation scheme with application to computational fluid dynamics \u2013 I, Surface approximations and partial derivative estimates\u201d, Computers & Mathematics with Applications, Vol. 19, pp. 127-45.",
      "DOI": "10.1016/0898-1221(90)90270-T"
    },
    {
      "key": "2025072817203234200_b18",
      "doi-asserted-by": "crossref",
      "unstructured": "Kansa, E.J.\n           (1990b), \u201cMuliquadrics \u2013 a scattered data approximation scheme with application to computational fluid dynamics \u2013 II, Solutions to parabolic, hyperbolic and elliptic partial differential equations\u201d, Computers & Mathematics with Applications, Vol. 19, pp. 147-61.",
      "DOI": "10.1016/0898-1221(90)90271-K"
    },
    {
      "key": "2025072817203234200_b19",
      "doi-asserted-by": "crossref",
      "unstructured": "Kaviany, M.\n           (1995), Principles of Heat Transfer in Porous Media, Springer-Verlag, Berlin, Germany.",
      "DOI": "10.1007/978-1-4612-4254-3"
    },
    {
      "key": "2025072817203234200_b20",
      "doi-asserted-by": "crossref",
      "unstructured": "Li, J.C.\n          , Hon, Y.C. and Chen, C.S. (2002), \u201cNumerical comparisons of two meshless methods using radial basis functions\u201d, Engineering Analysis with Boundary Elements, Vol. 26, pp. 205-25.",
      "DOI": "10.1016/S0955-7997(01)00101-1"
    },
    {
      "key": "2025072817203234200_b21",
      "doi-asserted-by": "crossref",
      "unstructured": "Mai-Dui, N.\n           and Tran-Cong, T. (2001a), \u201cNumerical solution of Navier-Stokes equations using multiquadric radial basis function networks\u201d, International Journal for Numerical Methods in Fluids, Vol. 37, pp. 65-86.",
      "DOI": "10.1002/fld.165"
    },
    {
      "key": "2025072817203234200_b22",
      "doi-asserted-by": "crossref",
      "unstructured": "Mai-Dui, N.\n           and Tran-Cong, T. (2001b), \u201cNumerical solution of differential equations using multiquadric radial basis function networks\u201d, Neural Networks, Vol. 14, pp. 185-99.",
      "DOI": "10.1016/S0893-6080(00)00095-2"
    },
    {
      "key": "2025072817203234200_b23",
      "doi-asserted-by": "crossref",
      "unstructured": "Mai-Dui, N.\n           and Tran-Cong, T. (2002), \u201cMesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations\u201d, Engineering Analysis with Boundary Elements, Vol. 26, pp. 133-56.",
      "DOI": "10.1016/S0955-7997(01)00092-3"
    },
    {
      "key": "2025072817203234200_b24",
      "doi-asserted-by": "crossref",
      "unstructured": "Nield, D.A.\n           and Bejan, A. (1992), Convection in Porous Media, Springer-Verlag, Berlin.",
      "DOI": "10.1007/978-1-4757-2175-1"
    },
    {
      "key": "2025072817203234200_b25",
      "unstructured": "Perko, J.\n          , Chen, C.S. and \u0160arler, B. (2001), \u201cA polygon-free numerical solution of steady natural convection in solid-liquid systems\u201d, in \u0160arler, B. and Brebbia, C.A. (Eds), Computational Modelling of Moving and Free Boundary Problems, Wit Press, Southampton, UK, pp. 111-22."
    },
    {
      "key": "2025072817203234200_b26",
      "doi-asserted-by": "crossref",
      "unstructured": "Power, H.\n           and Barraco, V. (2002), \u201cA comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations\u201d, Computers & Mathematics with Applications, Vol. 43, pp. 551-83.",
      "DOI": "10.1016/S0898-1221(01)00305-4"
    },
    {
      "key": "2025072817203234200_b27",
      "doi-asserted-by": "crossref",
      "unstructured": "Prasad, V.\n           and Kulacki, F.A. (1984), \u201cConvective heat transfer in a rectangular porous cavity \u2013 effect of aspect ratio on flow structure and heat transfer\u201d, Journal of Heat Transfer, Vol. 106, pp. 158-65.",
      "DOI": "10.1115/1.3246629"
    },
    {
      "key": "2025072817203234200_b28",
      "unstructured": "Raghavan, R.\n           and Ozkan, E. (1992), A Method for Computing Unsteady Flows in Porous Media, Pitman Research Notes in Mathematics Series, Vol. 318, Longman Scientific & Technical, Harlow, UK."
    },
    {
      "key": "2025072817203234200_b29",
      "doi-asserted-by": "crossref",
      "unstructured": "Sadat, H.\n           and Couturier, S. (2000), \u201cPerformance and accuracy of a meshless method for laminar natural convection\u201d, Numerical Heat Transfer, Vol. 37B, pp. 455-67.",
      "DOI": "10.1080/10407790050051146"
    },
    {
      "key": "2025072817203234200_b30",
      "unstructured": "Sahimi, M.\n           (1995), Flow and Transport in Porous Media and Fractured Rock, VCH Verlagsgesellschaft mbH, Weinheim, Germany."
    },
    {
      "key": "2025072817203234200_b31",
      "unstructured": "\u0160arler, B.\n          , Perko, J. and Chen, C.S. (2002), \u201cNatural convection in porous media \u2013 radial basis function collocation method solution of the Darcy model\u201d, in Mang, H.A., Rammerstorfer, F.G. and Eberhardsteiner, J. (Eds), Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V)."
    },
    {
      "key": "2025072817203234200_b32",
      "unstructured": "\u0160arler, B.\n          , Perko, J., Chen, C.S. and Kuhn, G. (2001), \u201cA meshless approach to natural convection\u201d, in Atluri, S.N. (Ed.), International Conference on Computational Engineering and Sciences \u2013 2001, CD-rom Proceedings, pp. 3-9."
    },
    {
      "key": "2025072817203234200_b33",
      "doi-asserted-by": "crossref",
      "unstructured": "\u0160arler, B.\n          , Gobin, D., Goyeau, B., Perko, J. and Power, H. (2000), \u201cNatural convection in porous media \u2013 dual reciprocity boundary element method solution of the Darcy model\u201d, International Journal for Numerical Methods in Fluids, Vol. 33, pp. 279-312.",
      "DOI": "10.1002/(SICI)1097-0363(20000530)33:2<279::AID-FLD18>3.3.CO;2-E"
    },
    {
      "key": "2025072817203234200_b34",
      "doi-asserted-by": "crossref",
      "unstructured": "Wang, J.G.\n           and Liu, G.R. (2002), \u201cOn the optimal shape parameters of radial basis functions used for 2-D meshless methods\u201d, Computer Methods in Applied Mechanics and Engineering, Vol. 191, pp. 2611-30.",
      "DOI": "10.1016/S0045-7825(01)00419-4"
    },
    {
      "key": "2025072817203234200_b35",
      "doi-asserted-by": "crossref",
      "unstructured": "Zerroukat, M.\n          , Djidjeli, K. and Charafi, A. (2000), \u201cExplicit and implicit meshless methods for linear advection-diffusion-type partial differential equations\u201d, International Journal for Numerical Methods in Engineering, Vol. 48, pp. 19-35.",
      "DOI": "10.1002/(SICI)1097-0207(20000510)48:1<19::AID-NME862>3.0.CO;2-3"
    },
    {
      "key": "2025072817203234200_b36",
      "doi-asserted-by": "crossref",
      "unstructured": "Zerroukat, M.\n          , Power, H. and Chen, C.S. (1998), \u201cA numerical method for heat transfer problems using collocation and radial basis functions\u201d, International Journal for Numerical Methods in Engineering, Vol. 42, pp. 1263-78.",
      "DOI": "10.1002/(SICI)1097-0207(19980815)42:7<1263::AID-NME431>3.0.CO;2-I"
    },
    {
      "issue": "4",
      "key": "2025072817203234200_b37",
      "doi-asserted-by": "crossref",
      "first-page": "333",
      "DOI": "10.1007/s004660000181",
      "article-title": "Meshless methods based on collocation with radial basis functions",
      "volume": "26",
      "author": "Zhang",
      "year": "2000",
      "journal-title": "Comput. Mech."
    },
    {
      "key": "2025072817203234200_frd1",
      "doi-asserted-by": "crossref",
      "unstructured": "Chen, C.S.\n          , Ganesh, M., Golberg, M.A. and Cheng, A.H.D. (2002), \u201cMultilevel compact radial functions based computational schemes for some elliptic problems\u201d, Computers & Mathematics with Applications, Vol. 43, pp. 359-78.",
      "DOI": "10.1016/S0898-1221(01)00292-9"
    },
    {
      "key": "2025072817203234200_frd3",
      "doi-asserted-by": "crossref",
      "unstructured": "Golberg, M.A.\n          , Chen, C.S. and Ganesh, M. (2000), \u201cParticular solutions of 3D Helmoltz-type equations using compactly supported radial basis functions\u201d, Engineering Analysis with Boundary Elements, Vol. 24, pp. 539-47.",
      "DOI": "10.1016/S0955-7997(00)00034-5"
    },
    {
      "key": "2025072817203234200_frd2",
      "doi-asserted-by": "crossref",
      "unstructured": "\u0160arler, B.\n           (2002), \u201cTowards a mesh-free computation of transport phenomena\u201d, Engineering Analysis with Boundary Elements, Vol. 26, pp. 731-8.",
      "DOI": "10.1016/S0955-7997(02)00044-9"
    }
  ],
  "container-title": "International Journal of Numerical Methods for Heat &amp; Fluid Flow",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "http://www.emeraldinsight.com/doi/full-xml/10.1108/09615530410513809",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "https://www.emerald.com/insight/content/doi/10.1108/09615530410513809/full/xml",
      "content-type": "application/xml",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "https://www.emerald.com/hff/article-pdf/14/2/187/484307/09615530410513809.pdf",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "syndication"
    },
    {
      "URL": "https://www.emerald.com/hff/article-pdf/14/2/187/484307/09615530410513809.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2025,
        7,
        28
      ]
    ],
    "date-time": "2025-07-28T21:20:44Z",
    "timestamp": 1753737644000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.emerald.com/hff/article/14/2/187/78149/Radial-basis-function-collocation-method-solution"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        2004,
        3,
        1
      ]
    ]
  },
  "references-count": 40,
  "journal-issue": {
    "issue": "2",
    "published-print": {
      "date-parts": [
        [
          2004,
          3,
          1
        ]
      ]
    }
  },
  "URL": "http://dx.doi.org/10.1108/09615530410513809",
  "relation": {},
  "ISSN": [
    "0961-5539",
    "1758-6585"
  ],
  "subject": [],
  "published": {
    "date-parts": [
      [
        2004,
        3,
        1
      ]
    ]
  }
}