Skip to main content
SpringerLink
Log in

A Hyperbolic Theory for Advection-Diffusion Problems: Mathematical Foundations and Numerical Modeling

  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Abstract

Linear parabolic diffusion theories based on Fourier’s or Fick’s laws predict that disturbances can propagate at infinite speed. Although in some applications, the infinite speed paradox may be ignored, there are many other applications in which a theory that predicts propagation at finite speed is mandatory. As a consequence, several alternatives to the linear parabolic diffusion theory, that aim at avoiding the infinite speed paradox, have been proposed over the years. This paper is devoted to the mathematical, physical and numerical analysis of a hyperbolic convection-diffusion theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ali YM, Zhang LC (2005) Relativistic heat conduction. Int J Heat Mass Transfer 48:2397–2406

    Article  Google Scholar 

  2. Antontsev S, Díaz JI, Shmarev S (2002) Energy methods for free boundary problems. Applications to nonlinear PDEs and Fluid Mechanics, Series Progress in Nonlinear Differential Equations and Their Applications. Birkäuser, Basel

    Google Scholar 

  3. Arora M (1996) Explicit characteristic-based high resolution algorithms for hyperbolic conservation laws with stiff source terms. PhD dissertation, University of Michigan

  4. Arnold DN, Brezzi F, Cockburn B, Marini D (2001) Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J Numer Anal 39:1749–1779

    Article  MathSciNet  Google Scholar 

  5. Barenblatt GI (1952) On some unsteady motions of fluids and gases in a porous medium. Prikl Mat Mekh 16:67–78

    MATH  MathSciNet  Google Scholar 

  6. Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  7. Barenblatt GI, Vishik MI (1956) On the finite speed of propagation in the problems of unsteady filtration of fluid and gas in a porous medium. Appl Math Mech (PMM) 34:655–662

    Article  Google Scholar 

  8. Bassi F, Rebay S (1997) A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J Comput Phys 131:267–279

    Article  MATH  MathSciNet  Google Scholar 

  9. Baumann CE, Oden JT (1999) A discontinuous hp finite element method for convection-diffusion problems. Comput Methods Appl Mech Eng 175:311–341

    Article  MATH  MathSciNet  Google Scholar 

  10. Baumann CE, Oden JT (1999) A discontinuous hp finite element method for the Euler and the Navier-Stokes equations. Int J Numer Methods Eng 31:79–95

    Article  MATH  MathSciNet  Google Scholar 

  11. Brooks A, Hughes TJR (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32:199–259

    Article  MATH  MathSciNet  Google Scholar 

  12. Carey GF, Tsai M (1982) Hyperbolic heat transfer with reflection. Numer Heat Transfer 5:309–327

    Article  Google Scholar 

  13. Castillo P (2002) Performance of discontinuous Galerkin methods for elliptic PDEs. SIAM J Sci Comput 24:524–547

    Article  MATH  MathSciNet  Google Scholar 

  14. Castillo P, Cockburn B, Perugia I, Schötzau D (2000) An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J Numer Anal 38:1676–1706

    Article  MATH  MathSciNet  Google Scholar 

  15. Cattaneo MC (1948) Sulla conduzione de calore. Atti Semin Mat Fis Univ Modena 3:3

    Google Scholar 

  16. Cattaneo MC (1958) Sur une forme de l’equation de la chaleur éliminant le paradoxe d’une propagation instantaneé. C R Acad Sci, Ser I Math 247:431–433

    MathSciNet  Google Scholar 

  17. Christov CI, M Jordan P (2005) Heat conduction paradox involving second-sound propagation in moving media. Phys Rev Lett 94:4301–4304

    Article  Google Scholar 

  18. Cockburn B, Karniadakis GE, Shu C-W (eds) (1999) Discontinuous Galerkin methods. Theory, computation and applications. Springer, Berlin

    Google Scholar 

  19. Cockburn B, Lin S-Y, Shu C-W (1989) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J Comput Phys 84:90–113

    Article  MATH  MathSciNet  Google Scholar 

  20. Cockburn B, Shu C-W (1989) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Math Comput 52:411–435

    Article  MATH  MathSciNet  Google Scholar 

  21. Cockburn B, Shu C-W (1998) The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: multidimensional systems. J Comput Phys 141:199–224

    Article  MATH  MathSciNet  Google Scholar 

  22. Cockburn B, Shu C-W (2001) Runge-Kutta discontinuous Galerkin methods for convection dominated flows. J Sci Comput 16:173–261

    Article  MATH  MathSciNet  Google Scholar 

  23. Cockburn B, Shu C-W (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal 35:2440–2463

    Article  MATH  MathSciNet  Google Scholar 

  24. Codina R (1993) Stability analysis of the forward Euler scheme for the convection-diffusion equation using the SUPG formulation in space. Int J Numer Methods Eng 36:1445–1464

    Article  MATH  Google Scholar 

  25. Compte A (1997) The generalized Cattaneo equation for the description of anomalous transport processes. J Phys A, Math Gen 30:7277–7289

    Article  MATH  MathSciNet  Google Scholar 

  26. Courant R, Hilbert D (1989) Methods of mathematical physics, vol II. Wiley, New York

    Google Scholar 

  27. Courant R, Friedrichs KO (1999) Supersonic flow and shock waves. Springer, Berlin

    Google Scholar 

  28. Donea J (1984) A Taylor-Galerkin method for convective transport problems. Int J Numer Methods Eng 20:101–120

    Article  MATH  Google Scholar 

  29. Donea J, Huerta A (2003) Finite element methods for flow problems. Wiley, New York

    Book  Google Scholar 

  30. Donea J, Quartapelle L, Selmin V (1987) An analysis of time discretization in the finite element solution of hyperbolic problems. J Comput Phys 70:463–499

    Article  MATH  MathSciNet  Google Scholar 

  31. Donea J, Quartapelle L (1992) An introduction to finite element methods for transient advection problems. Comput Methods Appl Mech Eng 95:169–203

    Article  MATH  MathSciNet  Google Scholar 

  32. Duhamel P (2004) Application of a new finite integral transform method to the wave model of conduction. Int J Heat Mass Transfer 47:573–588

    Article  MATH  Google Scholar 

  33. Fick A (1855) Uber diffusion. Poggendorff’s Ann Phys Chem 94:59–86

    Article  Google Scholar 

  34. Figueroa CA, Colominas I, Mosqueira G, Navarrina F, Casteleiro M (1999) A stabilized finite element approach for advective-diffusive transport problems. In: Pimenta, PM, Brasil, R, Almeida, E (eds) Proceedings of the XX Iberian Latin-American congress on computational methods in engineering (CDROM), São Paulo, Brasil

  35. Fourier JB (1822) Théorie analytique de la chaleur, Jacques Gabay

  36. Godlewski E, Raviart P-A (1991) Numerical approximation of hyperbolic systems of conservation laws. Springer, Berlin

    Google Scholar 

  37. Goldstein S (1951) On diffusion by discontinuous movements and on the telegraph equation. Q J Mech Appl Math IV 2:2

    Google Scholar 

  38. Gomez H (2003) A new formulation for the advective-diffusive transport problem. Technical report, University of A Coruña (in Spanish)

  39. Gomez H, Colominas I, Navarrina F, Casteleiro M (2004) An alternative formulation for the advective-diffusive transport problem. In: Mota Soares, CA, Batista, AL, Bugeda, G, Casteleiro, M, Goicolea, JM, Martins, JAC, Pina, CAB, Rodrigues, HC (eds) 7th congress on computational methods in engineering, Lisbon, Portugal

  40. Gomez H, Colominas I, Navarrina F, Casteleiro M (2004) On the intrinsic instability of the advection–diffusion equation. In: Neittaanmäki, P, Rossi, T, Korotov, S, Oñate, E, Périaux, J, Knörzer, D (eds) Proc of the 4th European congress on computational methods in applied sciences and engineering (CDROM), Jyväskylä, Finland

  41. Gomez H (2006) A hyperbolic formulation for convective-diffusive problems in CFD. PhD dissertation, University of A Coruña (in Spanish)

  42. Gomez H, Colominas I, Navarrina F, Casteleiro M (2006) A finite element formulation for a convection-diffusion equation based on Cattaneo’s law. Comput Methods Appl Mech Eng 196:1757–1766

    Article  Google Scholar 

  43. Gomez H, Colominas I, Navarrina F, Casteleiro M (2007) A discontinuous Galerkin method for a hyperbolic model for convection-diffusion problems in CFD. Int J Numer Methods Eng 71:1342–1364

    Article  MathSciNet  Google Scholar 

  44. Gomez H, Colominas I, Navarrina F, Casteleiro M (2008) A mathematical model and a numerical model for hyperbolic mass transport in compressible flows. Heat Mass Transfer 45:219–226

    Article  Google Scholar 

  45. Gurtin ME, Pipkin AC (1968) A general theory of heat conduction with finite wave speeds. Arch Ration Mech Anal 31:113

    Article  MATH  MathSciNet  Google Scholar 

  46. Grad H (1951) Principles of the kinetic theory of gases. In: Handbuch der Physik 12: Thermodynamics of gases. Springer, Berlin, p 205

    Google Scholar 

  47. Haji-Sheikh A, Minkowycz WJ, Sparrow EM (2002) Certain anomalies in the analysis of hyperbolic heat conduction. J Heat Transfer 124:307–319

    Article  Google Scholar 

  48. Han-Taw C, Jae-Yuh L (1993) Numerical analysis for hyperbolic heat conduction. Int J Heat Mass Transfer 36:2891–2898

    Article  MATH  Google Scholar 

  49. Hänel D, Schwane R, Seider G (1987) On the accuracy of upwind schemes for the resolution of the Navier-Stokes equations. AIAA Paper, 87-1005

  50. Hoashi E, Yokomine T, Shimizu A, Kunugi T (2003) Numerical analysis of wave-type heat transfer propagating in a thin foil irradiated by short-pulsed laser. Int J Heat Mass Transfer 46:4083–4095

    Article  MATH  Google Scholar 

  51. Hirsch C (2002) Numerical computation of internal and external flows, vol. II. Wiley, New York

    Google Scholar 

  52. Holley ER (1996) Diffusion and dispersion. In: Singh VP, Hager WH (eds) Environmental hydraulics. Kluwer Academic, Dordrecht, pp 111–151

    Google Scholar 

  53. Hughes TJR (2000) The finite element method. Linear static and dynamic finite element analysis. Dover, New York

    Google Scholar 

  54. Isaacson E, Keller HB (1966) Analysis of numerical methods. Wiley, New York

    MATH  Google Scholar 

  55. Jiang F, Sousa ACM (2005) Analytical solution for hyperbolic heat conduction in a hollow sphere. AIAA J Thermophys Heat Transfer 19:595–598

    Article  Google Scholar 

  56. Jiang F (2006) Solution and analysis of hyperbolic heat propagation in hollow spherical objects. Heat Mass Transfer 42:1083–1091

    Article  Google Scholar 

  57. Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–73

    Article  MATH  MathSciNet  Google Scholar 

  58. Joseph DD, Preziosi L (1990) Addendum to the paper “Heat waves”. Rev Mod Phys 62:375–391

    Article  MathSciNet  Google Scholar 

  59. Jou D, Casas-Vázquez J, Lebon G (2001) Extended irreversible thermodynamics. Springer, Berlin

    MATH  Google Scholar 

  60. Kalashnikov AS (1987) Some problems of qualitative theory of non-linear second-order parabolic equations. Russ Math Surv 42:169–222

    Article  MATH  MathSciNet  Google Scholar 

  61. Kalatnikov IM (1965) Introduction to the theory of superfluidity. Benjamin, Elmsford

    Google Scholar 

  62. Kaliski S (1965) Wave equation for heat conduction. Bull Acad Pol Sci 4:211–219

    Google Scholar 

  63. Klages R, Radons G, Sokolov IM (2008) Anomalous transport: foundations and applications. Wiley, New York

    Book  Google Scholar 

  64. Landau LD, Lifshitz EM (1959) Fluid mechanics. Pergamon, Elmsford

    Google Scholar 

  65. Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  66. Manzari MT, Manzari MT (1999) On numerical solution of hyperbolic heat equation. Commun Numer Methods Eng 15:853–866

    Article  MATH  MathSciNet  Google Scholar 

  67. Maxwell JC (1867) On the dynamical theory of gases. Philos Trans R Soc Lond 157:49–88

    Article  Google Scholar 

  68. Oleynik OA, Kalashnikov AS, Yui-lin C (1958) The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv USSR Acad Sci Ser Mat 22:667–704

    Google Scholar 

  69. Özisik MN, Tzou DY (1994) On the wave theory in heat conduction. ASME J Heat Transfer 116:526–535

    Article  Google Scholar 

  70. Pattle RE (1959) Diffusion from an instantaneous point source with a concentration-dependent coefficient. Q J Mech Appl Math 12:407–409

    Article  MATH  MathSciNet  Google Scholar 

  71. Peshkov V (1944) Second sound in helium II. J Phys 8:381

    Google Scholar 

  72. Reed WH, Hill TR (1973) Triangular mesh methods for the neutron transport equation. Technical report LA-UR-73-479, Los Alamos Scientific Laboratory

  73. Reverberi AP, Bagnerini P, Maga L, Bruzzone AG (2008) On the non-linear Maxwell-Cattaneo equation with non-constant diffusivity: Shock and discontinuity waves. Int J Heat Mass Transfer 51:5327–5332

    Article  MATH  Google Scholar 

  74. Ruggeri T, Muracchini A, Seccia L (1990) Shock waves and second sound in a rigid heat conductor: a critical temperature for NaF and Bi. Phys Rev Lett 64:2640–2643

    Article  Google Scholar 

  75. Sarrate J, Huerta A (2000) Efficient unstructured quadrilateral mesh generation. Int J Numer Methods Eng 49:1327–1350

    Article  MATH  Google Scholar 

  76. Sharma KR (2008) On the solution of damped wave conduction and relaxation equation in a semi-infinite medium subject to constant wall flux. Int J Heat Mass Transfer 51:6024–6031

    Article  MATH  Google Scholar 

  77. Shu C-W, Osher S (1988) Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 77:439–471

    Article  MATH  MathSciNet  Google Scholar 

  78. Tavernier J (1962) Sue l’equation de conduction de la chaleur. C R Acad Sci 254:69

    MathSciNet  Google Scholar 

  79. Toro EF (1999) Riemann solvers and numerical methods for fluid dynamics: A practical introduction. Springer, Berlin

    MATH  Google Scholar 

  80. Van Leer B (1982) Flux vector splitting for the Euler equations. Lect Notes Phys 170:507–512

    Article  Google Scholar 

  81. Vázquez JL (2006) The porous medium equation. Mathematical theory. Oxford University Press, Oxford

    Book  Google Scholar 

  82. Vick B, Özisik MN (1983) Growth and decay of a thermal pulse predicted by the hyperbolic heat conduction equation. ASME J Heat Transfer 105:902–907

    Article  Google Scholar 

  83. Wesseling P (2001) Principles of computational fluid dynamics. Springer, Berlin

    Google Scholar 

  84. Whitham GB (1999) Linear and nonlinear waves. Wiley, New York

    MATH  Google Scholar 

  85. Wilhelm HE, Choi SH (1975) Nonlinear hyperbolic theory of thermal waves in metals. J Chem Phys 63:2119

    Article  MathSciNet  Google Scholar 

  86. Wu W, Li X (2006) Application of the time discontinuous Galerkin finite element method to heat wave simulation. Int J Heat Mass Transfer 49:1679–1684

    Article  Google Scholar 

  87. Yabe T, Ishikawa T, Wang PY, Aoki T, Kadota Y, Ikeda F (1991) A universal solver for hyperbolic equations by cubic-polynomial interpolation II. Two- and three-dimensional solvers. Comput Phys Commun 66:223–242

    MathSciNet  Google Scholar 

  88. Yang HQ (1992) Solution of two-dimensional hyperbolic heat conduction by high resolution numerical methods. AIAA-922937

  89. Zakari M, Jou D (1993) Equations of state and transport equations in viscous cosmological models. Phys Rev D 48:1597–1601

    Article  Google Scholar 

  90. Zel’dovich YB, Kompaneets AS (1950) On the theory of propagation of heat with thermal conductivity depending on temperature. In: Collection of papers dedicated to 70th birthday of AF Ioffe 61–71

  91. Zienkiewicz OC, Taylor RL (2000) The finite element method, vol 3. Fluid dynamics. Butterworth Heinemann, Stoneham

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Methods, Civil Engineering School, A Coruña, Spain

    Hector Gomez, Ignasi Colominas, Fermín Navarrina, José París & Manuel Casteleiro

Authors
  1. Hector Gomez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ignasi Colominas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fermín Navarrina
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. José París
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Manuel Casteleiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ignasi Colominas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gomez, H., Colominas, I., Navarrina, F. et al. A Hyperbolic Theory for Advection-Diffusion Problems: Mathematical Foundations and Numerical Modeling. Arch Computat Methods Eng 17, 191–211 (2010). https://doi.org/10.1007/s11831-010-9042-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11831-010-9042-5

Keywords

Search

Navigation