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Volumn 31, Issue 6, 2009, Pages 4066-4099

A finite volume method for solving parabolic equations on logically cartesian curved surface meshes

Author keywords

Curved surfaces; Finite volume methods; Laplace Beltrami operator; Parabolic PDEs; Pattern formation

Indexed keywords

ADVECTION; DIFFUSION IN LIQUIDS; LINEAR EQUATIONS; MAPPING; PARTIAL DIFFERENTIAL EQUATIONS; SPHERES;

EID: 73249116154     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/08073322X     Document Type: Article
Times cited : (43)

References (57)
  • 2
    • 0037429531 scopus 로고    scopus 로고
    • Transport and diffusion of material quantities on propagating interfaces via level set methods
    • D. ADALSTEINSSON AND J. A. SETHIAN, Transport and diffusion of material quantities on propagating interfaces via level set methods, J. Comput. Phys., 185 (2003), pp. 271-288.
    • (2003) J. Comput. Phys. , vol.185 , pp. 271-288
    • Adalsteinsson, D.1    Sethian, J.A.2
  • 3
    • 33846309262 scopus 로고    scopus 로고
    • Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes
    • B. ANDREIANOV, F. BOYER, AND F. HUBERT, Discrete duality finite volume schemes for Leray- Lions-type elliptic problems on general 2D meshes, Numer. Methods Partial Differential Equations, 23 (2007), pp. 145-195.
    • (2007) Numer. Methods Partial Differential Equations , vol.23 , pp. 145-195
    • Andreianov, B.1    Boyer, F.2    Hubert, F.3
  • 4
    • 58149339886 scopus 로고    scopus 로고
    • Partial differential equations for selforganization in cellular and developmental biology
    • R. E. BAKER, E. A. GAFFNEY, AND P. K. MAINI, Partial differential equations for selforganization in cellular and developmental biology, Nonlinearity, 21 (2008), pp. 251-290.
    • (2008) Nonlinearity , vol.21 , pp. 251-290
    • Baker, R.E.1    Gaffney, E.A.2    Maini, P.K.3
  • 5
    • 0033038309 scopus 로고    scopus 로고
    • A two-dimensional numerical study of spatial pattern formation in interacting Turing systems
    • R. BARRIO, C. VAREA, J. ARAGÓN, AND P. MAINI, A two-dimensional numerical study of spatial pattern formation in interacting Turing systems, Bull. Math. Biol., 61 (1999), pp. 483-505.
    • (1999) Bull. Math. Biol. , vol.61 , pp. 483-505
    • Barrio, R.1    Varea, C.2    Aragón, J.3    Maini, P.4
  • 7
    • 0035842857 scopus 로고    scopus 로고
    • Variational problems and partial differential equations on implicit surfaces
    • M. BERTALMIO, L. CHENG, S. OSHER, AND G. SAPIRO, Variational problems and partial differential equations on implicit surfaces, J. Comput. Phys., 174 (2001), pp. 759-780.
    • (2001) J. Comput. Phys. , vol.174 , pp. 759-780
    • Bertalmio, M.1    Cheng, L.2    Osher, S.3    Sapiro, G.4
  • 8
    • 55349102655 scopus 로고    scopus 로고
    • Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities
    • F. BOYER AND F. HUBERT, Finite volume method for 2D linear and nonlinear elliptic problems with discontinuities, SIAM J. Numer. Anal., 46 (2008), pp. 3032-3070.
    • (2008) SIAM J. Numer. Anal. , vol.46 , pp. 3032-3070
    • Boyer, F.1    Hubert, F.2
  • 9
    • 0026023209 scopus 로고
    • Complex patterns formed by motile cells in Escherichia coli
    • E. O. BUDRENE AND H. BERG, Complex patterns formed by motile cells in Escherichia coli, Nature, 349 (1991), pp. 630-633.
    • (1991) Nature , vol.349 , pp. 630-633
    • Budrene, E.O.1    Berg, H.2
  • 10
    • 0029023398 scopus 로고
    • Dynamics of formation of symmetrical patterns of chemotactic bacteria
    • E. O. BUDRENE AND H. BERG, Dynamics of formation of symmetrical patterns of chemotactic bacteria, Nature, 376 (1995), pp. 49-53.
    • (1995) Nature , vol.376 , pp. 49-53
    • Budrene, E.O.1    Berg, H.2
  • 11
    • 56349092136 scopus 로고    scopus 로고
    • A finite volume grid for solving hyperbolic problems on the sphere
    • Hyperbolic Problems: Theory, Numerics, Applications, Lyon, Springer, Berlin, 2008
    • D. A. CALHOUN, C. HELZEL, AND R. J. LEVEQUE, A finite volume grid for solving hyperbolic problems on the sphere, in Hyperbolic Problems: Theory, Numerics, Applications, Proceedings of the 11th International Conference on Hyperbolic Problems, Lyon, 2006, Springer, Berlin, 2008, pp. 355-362.
    • (2006) Proceedings of the 11th International Conference on Hyperbolic Problems , pp. 355-362
    • Calhoun, D.A.1    Helzel, C.2    Leveque, R.J.3
  • 12
    • 56349111880 scopus 로고    scopus 로고
    • Logically rectangular grids and finite volume methods for PDEs in circular and spherical domains
    • D. A. CALHOUN, C. HELZEL, AND R. J. LEVEQUE, Logically rectangular grids and finite volume methods for PDEs in circular and spherical domains, SIAM Rev., 50 (2008), pp. 723-752.
    • (2008) SIAM Rev. , vol.50 , pp. 723-752
    • Calhoun, D.A.1    Helzel, C.2    Leveque, R.J.3
  • 13
    • 56549128338 scopus 로고    scopus 로고
    • A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models
    • A. CHERTOCK AND A. KURGANOV, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), pp. 169-205.
    • (2008) Numer. Math. , vol.111 , pp. 169-205
    • Chertock, A.1    Kurganov, A.2
  • 16
    • 0039938830 scopus 로고    scopus 로고
    • Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem
    • Y. COUDIÉre, J. P. VILA, AND P. VILLEDIEU, Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem, ESAIM Math. Model. Numer. Anal., 33 (1999), pp. 493-516.
    • (1999) ESAIM Math. Model. Numer. Anal. , vol.33 , pp. 493-516
    • Coudiére, Y.1    Vila, J.P.2    Villedieu, P.3
  • 17
    • 40049084767 scopus 로고    scopus 로고
    • An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces
    • A. DEMLOW AND G. DZIUK, An adaptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces, SIAM J. Numer. Anal., 45 (2007), pp. 421-442.
    • (2007) SIAM J. Numer. Anal. , vol.45 , pp. 421-442
    • Demlow, A.1    Dziuk, G.2
  • 19
    • 30344434032 scopus 로고    scopus 로고
    • A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids
    • K. DOMELEVO AND P. OMNES, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, ESAIM Math. Model. Numer. Anal., 39 (2005), pp. 1203- 1249.
    • (2005) ESAIM Math. Model. Numer. Anal. , vol.39 , pp. 1203-1249
    • Domelevo, K.1    Omnes, P.2
  • 20
    • 0010974195 scopus 로고
    • Finite elements for the Beltrami operator on arbitrary surfaces, in partial differential equations and calculus of variations
    • S. Hildebrandt and R. Leis, eds., Springer, New York
    • G. DZIUK, Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations, S. Hildebrandt and R. Leis, eds., Lecture Notes in Math. 1357, Springer, New York, 1988, pp. 142-155.
    • (1988) Lecture Notes in Math. 1357 , pp. 142-155
    • Dziuk, G.1
  • 21
    • 34447646892 scopus 로고    scopus 로고
    • Surface finite elements for parabolic equations
    • G. DZIUK AND C. M. ELLIOT, Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), pp. 385-407.
    • (2007) J. Comput. Math. , vol.25 , pp. 385-407
    • Dziuk, G.1    Elliot, C.M.2
  • 22
    • 70350322945 scopus 로고    scopus 로고
    • The finite volume method
    • P. Ciarlet and J. L. Lions, eds. North-Holland, Amsterdam
    • R. EYMARD, T. GALLOUËT, AND R. HERBIN, The finite volume method, in Handbook of Numerical Analysis, Vol. VII, P. Ciarlet and J. L. Lions, eds., North-Holland, Amsterdam, 2000, pp. 713-1020.
    • (2000) Handbook of Numerical Analysis , vol.7 , pp. 713-1020
    • Eymard, R.1    Gallouët, T.2    Herbin, R.3
  • 23
    • 0037779152 scopus 로고    scopus 로고
    • A generic geometric transformation that unifies a wide range of natural and abstract shapes
    • J. GIELIS, A generic geometric transformation that unifies a wide range of natural and abstract shapes, Amer. J. Botany, 90 (2003), pp. 333-338.
    • (2003) Amer. J. Botany , vol.90 , pp. 333-338
    • Gielis, J.1
  • 24
    • 33646157612 scopus 로고    scopus 로고
    • An improvement of a recent Eulerian method for solving PDEs on general geometries
    • J. GREER, An improvement of a recent Eulerian method for solving PDEs on general geometries, J. Sci. Comput., 29 (2006), pp. 321-352.
    • (2006) J. Sci. Comput. , vol.29 , pp. 321-352
    • Greer, J.1
  • 25
    • 67349146905 scopus 로고    scopus 로고
    • Benchmark on discretization schemes for anisotropic diffusion problems on general grids
    • R. Eymard and J.-M. Hèrard, eds. ISTE/Wiley, New York
    • R. HERBIN AND F. HUBERT, Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids, in Finite Volumes for Complex Applications V: Problems and Perspectives, R. Eymard and J.-M. Hèrard, eds., ISTE/Wiley, New York, 2008, pp. 659- 692.
    • (2008) Finite Volumes for Complex Applications V: Problems and Perspectives , pp. 659-692
    • Herbin, R.1    Hubert, F.2
  • 26
    • 73249144199 scopus 로고    scopus 로고
    • F. HERMELINE, private discussions, CEA/DIF, DSSI/SNEC, Commissariat à l'Energie Atomique (C.E.A.), Bruyères le Châtel, France, 2007
    • F. HERMELINE, private discussions, CEA/DIF, DSSI/SNEC, Commissariat à l'Energie Atomique (C.E.A.), Bruyères le Châtel, France, 2007.
  • 27
    • 0001072763 scopus 로고    scopus 로고
    • A finite volume method for the approximation of diffusion operators on distorted meshes
    • F. HERMELINE, A finite volume method for the approximation of diffusion operators on distorted meshes, J. Comput. Phys., 160 (2000), pp. 481-499.
    • (2000) J. Comput. Phys. , vol.160 , pp. 481-499
    • Hermeline, F.1
  • 28
    • 0038702483 scopus 로고    scopus 로고
    • Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes
    • F. HERMELINE, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 1939-1959.
    • (2003) Comput. Methods Appl. Mech. Engrg. , vol.192 , pp. 1939-1959
    • Hermeline, F.1
  • 29
    • 0035842844 scopus 로고    scopus 로고
    • Variational mesh adaptation: Isotropy and equidistribution
    • W. HUANG, Variational mesh adaptation: Isotropy and equidistribution, J. Comput. Phys., 174 (2001), pp. 903-924.
    • (2001) J. Comput. Phys. , vol.174 , pp. 903-924
    • Huang, W.1
  • 30
    • 0008599281 scopus 로고    scopus 로고
    • Analysis of the cell-centered finite volume method for the diffusion equation
    • W. P. JONES AND K. R. MENZIES, Analysis of the cell-centered finite volume method for the diffusion equation, J. Comput. Phys., 165 (2000), pp. 45-68.
    • (2000) J. Comput. Phys. , vol.165 , pp. 45-68
    • Jones, W.P.1    Menzies, K.R.2
  • 31
    • 29744434340 scopus 로고    scopus 로고
    • Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles nonstructurés
    • C. LEPOTIER, Schéma volumes finis monotone pour des opérateurs de diffusion fortement anisotropes sur des maillages de triangles nonstructurés, C. R. Acad. Sci. Paris, 341 (2005), pp. 787-792.
    • (2005) C. R. Acad. Sci. Paris , vol.341 , pp. 787-792
    • LePotier, C.1
  • 34
    • 73249127433 scopus 로고    scopus 로고
    • Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems
    • Philadelphia
    • R. J. LEVEQUE, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, Philadelphia, 2007.
    • (2007) SIAM
    • Leveque, R.J.1
  • 35
    • 0035474955 scopus 로고    scopus 로고
    • Turing model for the patterns of lady beetles
    • paper 041909
    • S. S. Liaw, C. C. Yang, R. T. Liu, and J. T. Hong, Turing model for the patterns of lady beetles, Phys. Rev. E, 64 (2001), paper 041909.
    • (2001) Phys. Rev. E , vol.64
    • Liaw, S.S.1    Yang, C.C.2    Liu, R.T.3    Hong, J.T.4
  • 36
    • 35348844627 scopus 로고    scopus 로고
    • Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes
    • K. LIPNIKOV, M. SHASHKOV, D. SVYATSKIY, AND Y. VASSILEVSKI, Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes, J. Comput. Phys., 227 (2007), pp. 492-512.
    • (2007) J. Comput. Phys. , vol.227 , pp. 492-512
    • Lipnikov, K.1    Shashkov, M.2    Svyatskiy, D.3    Vassilevski, Y.4
  • 37
    • 38849083276 scopus 로고    scopus 로고
    • A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes
    • D. LIU, G. XU, AND Q. ZHANG, A discrete scheme of Laplace-Beltrami operator and its convergence over quadrilateral meshes, Comput. Math. Appl., 55 (2008), pp. 1081-1093.
    • (2008) Comput. Math. Appl. , vol.55 , pp. 1081-1093
    • Liu, D.1    Xu, G.2    Zhang, Q.3
  • 38
    • 0013020004 scopus 로고    scopus 로고
    • Mathematical models for biological pattern formation
    • P. K. MAINI AND H. G. OTHMER, EDS., Springer, New York
    • P. K. MAINI AND H. G. OTHMER, EDS., Mathematical Models for Biological Pattern Formation, IMA Vol. Math. Appl. 121, Springer, New York, 2000.
    • (2000) IMA Vol., Math. Appl. , vol.121
  • 42
    • 34548697081 scopus 로고    scopus 로고
    • Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology
    • I. NOVAK, F. GAO, Y. CHOI, D. RESASCO, J. SCHAFF, AND B. SLEPCHENKO, Diffusion on a curved surface coupled to diffusion in the volume: Application to cell biology, J. Comput. Phys., 226 (2007), pp. 1271-1290.
    • (2007) J. Comput. Phys. , vol.226 , pp. 1271-1290
    • Novak, I.1    Gao, F.2    Choi, Y.3    Resasco, D.4    Schaff, J.5    Slepchenko, B.6
  • 43
    • 84950826016 scopus 로고
    • Computing discrete minimal surfaces and their conjugates
    • U. PINKALL AND K. POLTHIER, Computing discrete minimal surfaces and their conjugates, Exper. Math., 2 (1993), pp. 15-36.
    • (1993) Exper. Math. , vol.2 , pp. 15-36
    • Pinkall, U.1    Polthier, K.2
  • 45
    • 35348927504 scopus 로고    scopus 로고
    • Finite-volume transport on various cubed-sphere grids
    • W. M. PUTMAN AND S. J. LIN, Finite-volume transport on various cubed-sphere grids, J. Comput. Phys., 227 (2007), pp. 55-78.
    • (2007) J. Comput. Phys. , vol.227 , pp. 55-78
    • Putman, W.M.1    Lin, S.J.2
  • 46
    • 37249003717 scopus 로고    scopus 로고
    • A simple embedding method for solving partial differential equations on surfaces
    • S. RUUTH AND B. MERRIMAN, A simple embedding method for solving partial differential equations on surfaces, J. Comput. Phys., 227 (2008), pp. 1943-1961.
    • (2008) J. Comput. Phys. , vol.227 , pp. 1943-1961
    • Ruuth, S.1    Merriman, B.2
  • 47
    • 0000644762 scopus 로고
    • Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids
    • R. SADOURNY, Conservative finite-difference approximations of the primitive equations on quasi-uniform spherical grids, Monthly Weather Rev., 100 (1972), pp. 211-224.
    • (1972) Monthly Weather Rev. , vol.100 , pp. 211-224
    • Sadourny, R.1
  • 48
    • 33646179555 scopus 로고    scopus 로고
    • Simulations of (an)isotropic diffusion on curved biological surfaces
    • I. V. SBALZARINI, A. HAYER, A. HELENIUS, AND P. KOUMOUTSAKOS, Simulations of (an)isotropic diffusion on curved biological surfaces, Biophys. J., 90 (2006), pp. 878-885.
    • (2006) Biophys. J. , vol.90 , pp. 878-885
    • Sbalzarini, I.V.1    Hayer, A.2    Helenius, A.3    Koumoutsakos, P.4
  • 50
    • 43049109588 scopus 로고    scopus 로고
    • A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes
    • Z. SHENG AND G. YUAN, A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes, SIAM J. Sci. Comput., 30 (2008), pp. 1341-1361.
    • (2008) SIAM J. Sci. Comput. , vol.30 , pp. 1341-1361
    • Sheng, Z.1    Yuan, G.2
  • 52
    • 0026238610 scopus 로고
    • Convergence of finite volume schemes for Poisson's equation on nonuniform meshes
    • E. SÜLI, Convergence of finite volume schemes for Poisson's equation on nonuniform meshes, SIAM J. Numer. Anal., 28 (1991), pp. 1419-1430.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 1419-1430
    • Süli, E.1
  • 53
    • 85028881989 scopus 로고
    • Generating textures on arbitrary surfaces using reaction-diffusion
    • G. TURK, Generating textures on arbitrary surfaces using reaction-diffusion, Computer Graphics, 25 (1991), pp. 289-298.
    • (1991) Computer Graphics , vol.25 , pp. 289-298
    • Turk, G.1
  • 54
    • 0033112265 scopus 로고    scopus 로고
    • Model and analysis of chemotactic bacterial patterns in a liquid medium
    • R. TYSON, S. LUBKIN, AND J. D. MURRAY, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), pp. 357-375.
    • (1999) J. Math. Biol. , vol.38 , pp. 357-375
    • Tyson, R.1    Lubkin, S.2    Murray, J.D.3
  • 55
    • 0034331541 scopus 로고    scopus 로고
    • Fractional step methods applied to a chemotaxis model
    • R. TYSON, L. G. STERN, AND R. J. LEVEQUE, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), pp. 455-475.
    • (2000) J. Math. Biol. , vol.41 , pp. 455-475
    • Tyson, R.1    Stern, L.G.2    Leveque, R.J.3
  • 56
    • 0033215734 scopus 로고    scopus 로고
    • Turing patterns on a sphere
    • C. VAREA, J. ARAGON, AND R. BARRIO, Turing patterns on a sphere, Phys. Rev. E, 60 (1999), pp. 4588-4592.
    • (1999) Phys. Rev. E , vol.60 , pp. 4588-4592
    • Varea, C.1    Aragon, J.2    Barrio, R.3
  • 57
    • 4544385874 scopus 로고    scopus 로고
    • Discrete Laplace-Beltrami operators and their convergence
    • G. XU, Discrete Laplace-Beltrami operators and their convergence, Comput. Aided Geom. Design, 21 (2004), pp. 767-784.
    • (2004) Comput. Aided Geom. Design , vol.21 , pp. 767-784
    • Xu, G.1


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