Numerical Modelling of Nonlinear Diffusion Phenomena on a Sphere

  • Yuri N. SkibaEmail author
  • Denis M. Filatov
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 197)


A new method for the numerical modelling of physical phenomena described by nonlinear diffusion equations on a sphere is developed. The key point of the method is the splitting of the differential equation by coordinates that reduces the original 2D problem to a pair of 1D problems. Due to the splitting, while solving the 1D problems separately one from another we involve the procedure of map swap — the same sphere is covered by either one or another of two different coordinate grids, which allows employing periodic boundary conditions for both 1D problems, despite the sphere is, actually, not a doubly periodic domain. Hence, we avoid the necessity of dealing with cumbersome mathematical procedures, such as the construction of artificial boundary conditions at the poles, etc. As a result, second-order finite difference schemes for the one-dimentional problems implemented as systems of linear algebraic equations with tridiagonal matrices are constructed. It is essential that each split one-dimentional finite difference scheme keeps all the substantial properties of the corresponding differential problem: the spatial finite difference operator is negative definite, whereas the scheme itself is balanced and dissipative. The results of several numerical simulations are presented and thoroughly analysed. Increase of the accuracy of the finite difference schemes to the fourth approximation order in space is discussed.


Finite Difference Scheme Nonlinear Diffusion North Pole Periodic Domain Nonlinear Diffusion Equation 
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  1. 1.
    Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1988)zbMATHGoogle Scholar
  2. 2.
    Catté, F., Lions, P.-L., Morel, J.-M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29, 182–193 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Glicksman, M.E.: Diffusion in Solids: Field Theory, Solid-State Principles and Applications. John Wiley & Sons, New York (2000)Google Scholar
  4. 4.
    Marchuk, G.I.: Methods of Computational Mathematics. Springer, Berlin (1982)Google Scholar
  5. 5.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge (2007)zbMATHGoogle Scholar
  6. 6.
    Skiba, Y.N., Filatov, D.M.: On an efficient splitting-based method for solving the diffusion equation on a sphere. Numer. Meth. Part. Diff. Eq. (2011), doi: 10.1002/num.20622Google Scholar
  7. 7.
    Tsynkov, S.V.: Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27, 465–532 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Vorob’yov, A.K.: Diffusion Problems in Chemical Kinetics. Moscow University Press, Moscow (2003)Google Scholar
  9. 9.
    Wu, Z., Zhao, J., Yin, J., Li, H.: Nonlinear Diffusion Equations. World Scientific Publishing, Singapore (2001)zbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Centro de Ciencias de la Atmósfera (CCA)Universidad Nacional Autónoma de México (UNAM)México D.F.México
  2. 2.Instituto Politécnico Nacional (IPN)Centro de Investigación en Computación (CIC)México D.F.México

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