Application of a Local Discontinuous Galerkin Method to the 1D Compressible Reynolds Equation

  • Iñigo ArreguiEmail author
  • J. Jesús Cendán
  • María González
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 18)


In this work we present a numerical method to approximate the solution of the steady-state compressible Reynolds equation with additional first-order slip flow terms. This equation models the hydrodynamic features of read/write processes in magnetic storage devices such as hard disks. The numerical scheme is based on the local discontinuous Galerkin method proposed by Cockburn and Shu (SIAM J Numer Anal 35:2440–2463, 1998), which shows good properties in the presence of internal layers appearing in convection-diffusion problems. Several test examples illustrate the good performance of the method.


Compressible flows Reynolds equation Local discontinuous Galerkin method 



This research was partially supported by the Spanish Government (Ministerio de Economía y Competitividad) under projects MTM2013-47800-C2-1-P and MTM2016-76497-R.


  1. 1.
    Arregui, I., Cendán, J.J., Vázquez, C.: A duality method for the compressible Reynolds equation. Application to simulation of read/write processes in magnetic storage devices. J. Comput. Appl. Math. 175(1), 31–40 (2005)zbMATHGoogle Scholar
  2. 2.
    Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayada, G., Chambat, M.: The transition between the Stokes equations and the Reynolds equation: a mathematical proof. Appl. Math. Optim. 14(1), 73–93 (1986)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bhushan, B.: Tribology and Mechanics of Magnetic Storage Devices. Springer, New York (1990)CrossRefGoogle Scholar
  5. 5.
    Buscaglia, G., Ciuperca, S., Jai, M.: Existence and uniqueness for several nonlinear elliptic problems arising in lubrication theory. J. Differ. Equ. 218, 187–215 (2005)CrossRefGoogle Scholar
  6. 6.
    Chipot, M., Luskin, M.: The compressible Reynolds lubrication equation. In: Metastability and Incompletely Posed Problems. IMA Volumes in Mathematics and Its Applications, vol. 3, pp. 61–75. Springer, New York (1987)CrossRefGoogle Scholar
  7. 7.
    Ciuperca, S., Hafidi, I., Jai, M.: Analysis of a parabolic compressible first-order slip Reynolds equation with discontinuous coefficients. Nonlinear Anal. 69, 1219–1234 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jai, M.: Homogenization and two-scale convergence of the compressible Reynolds lubrification equation modelling the flying characteristics of a rough magnetic head over a rough rigid-disk surface. Math. Model. Numer. Anal. 29, 199–233 (1995)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jai, M.: Existence and uniqueness of solutions of the parabolic nonlinear compressible Reynolds lubrication equation. Nonlinear Anal. 43, 655–682 (2001)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Oden, J.T., Wu, S.R.: Existence of solutions to the Reynolds’ equation of elastohydrodynamic lubrication. Int. J. Eng. Sci. 23(2), 207–215 (1985)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Reynolds, O.: On the theory of lubrication and its application to Mr. Beauchamps tower’s experiments, including an experimental determination of the viscosity of olive oil. Philos. Trans. 177, 157–234 (1886)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Iñigo Arregui
    • 1
    Email author
  • J. Jesús Cendán
    • 1
  • María González
    • 1
  1. 1.Departamento de MatemáticasUniversidade da CoruñaCoruñaSpain

Personalised recommendations