Advertisement

Numerical investigation on the transport equation in spherical coordinates via generalized moving least squares and moving kriging least squares approximations

  • Vahid Mohammadi
  • Mehdi DehghanEmail author
  • Amirreza Khodadadian
  • Thomas Wick
Original Article
  • 61 Downloads

Abstract

The main aim of this paper is to present new and simple numerical methods for solving the time-dependent transport equation on the sphere in spherical coordinates. We use two techniques, namely generalized moving least squares and moving kriging least squares to find the new formulations for approximating the advection operator in spherical coordinates such that any singularities have been omitted. Another feature of the methods considered here is that they do not depend on the background mesh or triangulation for approximation, which they could be applied on the transport equation in spherical coordinates easily with different distribution points. Furthermore, due to the eigenvalue stability of the dicretized advection operator via two proposed approximations, an implicit-explicit linear multistep method has been applied to discretize the time variable. The fully discrete scheme obtained here yields a linear system of algebraic equations at each time step, which is solved via the biconjugate gradient stabilized algorithm with zero-fill incomplete lower-upper (ILU) preconditioner. Three well-known test problems, namely “solid body rotation”, “vortex roll-up” and “deformational flow” are solved to demonstrate our developments. Also for the first test problem, we apply a simple positivity-preserving filter at the end of each time step, which keeps the transported variable positive.

Keywords

Transport equation on the sphere Meshless methods Generalized moving least squares approximation Moving kriging least squares interpolation An implicit-explicit linear multistep method Biconjugate gradient-stabilized method 

Mathematics Subject Classification

35R01 74G15 

Notes

Acknowledgements

The authors are grateful to the reviewers for carefully reading this paper and their comments and suggestions, which have improved the paper.

References

  1. 1.
    Abbaszadeh M, Khodadadian A, Parvizi M, Dehghan M, Heitzinger C (2019) A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations. Eng Anal Bound Elem 98:253–264MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Akram S, Aly EH, Afzal F, Nadeem S (2019) Effect of the variable viscosity on the peristaltic flow of Newtonian fluid coated with magnetic field: application of adomian decomposition method for endoscope. Coatings 9(8):524CrossRefGoogle Scholar
  3. 3.
    Afenyo M, Khan F, Veitch B, Yang M (2016) Modeling oil weathering and transport in sea ice. Mar Pollut Bull 107(1):206–215CrossRefGoogle Scholar
  4. 4.
    Atkinson K, Han W (2012) Spherical harmonics and approximations on the unit sphere: an introduction, vol 2044. Springer Science & Business Media, BerlinzbMATHCrossRefGoogle Scholar
  5. 5.
    Bender LC (1996) Modification of the physics and numerics in a third-generation ocean wave model. J Atomos Ocean Technol 13(3):726–750CrossRefGoogle Scholar
  6. 6.
    Chen C, Xiao F (2008) Shallow water model on cubed-sphere by multi-moment finite volume method. J Comput Phys 227(10):5019–5044MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cheruvu V, Nair RD, Tufo HM (2007) A spectral finite volume transport scheme on the cubed-sphere. Appl Numer Math 57(9):1021–1032MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cotter CJ, Shipton J (2012) Mixed finite elements for numerical weather prediction. J Comput Phys 231(21):7076–7091MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Doswell CA III (1984) A kinematic analysis of frontogenesis associated with a nondivergent vortex. J Atmos Sci 41(7):1242–1248CrossRefGoogle Scholar
  10. 10.
    Dehghan M, Abbaszadeh M, Khodadadian A, Heitzinger C (2019) Galerkin proper orthogonal decomposition reduced order method (POD-ROM) for solving the generalized Swift–Hohenberg equation. Int J Numer Method Heat Fluid Flow 29(8):2642–2665CrossRefGoogle Scholar
  11. 11.
    Dehghan M, Narimani N (2018) Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction-diffusion equation. Comput Appl Math 37(5):6955–6971MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dehghan M, Narimani N (2018) An element-free Galerkin meshless method for simulating the behavior of cancer cell invasion of surrounding tissue. Appl Math Model 59:500–513MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dehghan M, Mohammadi V (2019) Two-dimensional simulation of the damped Kuramoto–Sivashinsky equation via radial basis function-generated finite difference scheme combined with an exponential time discretization. Eng Anal Bound Elem 107:168–184MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dehghan M (2007) Time-splitting procedures for the solution of the two-dimensional transport equation. Kybernetes 36(5):791–805zbMATHCrossRefGoogle Scholar
  15. 15.
    Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, SingaporezbMATHCrossRefGoogle Scholar
  16. 16.
    Flyer N, Wright GB (2007) Transport schemes on a sphere using radial basis functions. J Comput Phys 226(1):1059–1084MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Flyer N, Wright GB (2009) A radial basis function method for the shallow water equations on a sphere. Proc R Soc Lond A Math Phys Eng Sci 465:1949–1976MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Fornberg B, Lehto E (2011) Stabilization of RBF-generated finite difference methods for convective PDEs. J Comput Phys 230(6):2270–2285MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Fuselier E, Hangelbroek T, Narcowich FJ, Ward JD, Wright GB (2014) Kernel based quadrature on spheres and other homogeneous spaces. Numer. Math. 127(1):57–92MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fuselier E, Wright GB (2015) Order-preserving derivative approximation with periodic radial basis functions. Adv Comput Math 41(1):23–53MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Giraldo FX (2000) Lagrange–Galerkin methods on spherical geodesic grids: the shallow water equations. J Comput Phys 160(1):336–368MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Gross BJ, Trask N, Kuberry P, Atzberger PJ (2019) Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: a generalized moving least-squares (GMLS) approach. arXiv:1905.10469
  23. 23.
    Gu L (2003) Moving kriging interpolation and element-free Galerkin method. Int J Numer Methods Eng 56(1):1–11zbMATHCrossRefGoogle Scholar
  24. 24.
    Jablonowski C, Herzog M, Penner JE, Oehmke RC, Stout QF, Van Leer B, Powell KG (2006) Blockstructured adaptive grids on the sphere: advection experiments. Mon Weather Rev 134(12):3691–3713CrossRefGoogle Scholar
  25. 25.
    Krems M (2007) The Boltzmann transport equation: theory and applications. http://www.mattkrems.com/projects/completedprojects/boltzmann.pdf
  26. 26.
    Khan AU, Hussain ST, Nadeem S (2019) Existence and stability of heat and fluid flow in the presence of nanoparticles along a curved surface by mean of dual nature solution. Appl Math Comput 353:66–81MathSciNetGoogle Scholar
  27. 27.
    Khodadadian A, Heitzinger C (2015) A transport equation for confined structures applied to the OprP, Gramicidin A, and KcsA channels. J Comput Electron 14(2):524–532CrossRefGoogle Scholar
  28. 28.
    Khodadadian A, Heitzinger C (2016) Basis adaptation for the stochastic nonlinear Poisson–Boltzmann equation. J Comput Electron 15(4):1393–1406CrossRefGoogle Scholar
  29. 29.
    Khodadadian A, Parvizi M, Abbaszadeh M, Dehghan M, Heitzinger C (2019) A multilevel Monte Carlo finite element method for the stochastic Cahn–Hilliard–Cook equation. Comput Mech 64(4):937–949MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kumar K, Wheeler MF, Wick T (2013) Reactive flow and reaction-induced boundary movement in a thin channel. SIAM J Sci Comput 35(6):1235–1266MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Lauritzen PH, Skamarock WC, Prather M, Taylor M (2012) A standard test case suite for two-dimensional linear transport on the sphere. Geosci Model Dev 5(3):887–901CrossRefGoogle Scholar
  32. 32.
    Läuter M, Handorf D, Rakowsky N, Behrens J, Frickenhaus S, Best M, Dethloff K, Hiller W (2007) A parallel adaptive barotropic model of the atmosphere. J Comput Phys 223(2):609–628MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Lehto E, Shankar V, Wright GB (2017) A radial basis function (RBF) compact finite difference (FD) scheme for reaction-diffusion equations on surfaces. SIAM J Sci Comput 39(5):2129–2151MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    LeVeque RJ (1996) High-resolution conservative algorithms for advection in incompressible flow. SIAM J Numer Anal 23(2):627–665MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Li X, Khan AU, Khan MR, Nadeem S, Khan SU (2019) Oblique stagnation point flow of nanofluids over stretching/shrinking sheet with Cattaneo-Christov heat flux model: existence of dual solution. Symmetry 11(9):1070CrossRefGoogle Scholar
  36. 36.
    MacLaren J, Malkinski L, Wang J (2000) First principles based solution to the Boltzmann transport equation for co/cu/co spin valves. In: MRS online proceedings library archive, vol 614Google Scholar
  37. 37.
    Mirzaei D (2017) Direct approximation on spheres using generalized moving least squares. BIT Numer Math 57(4):1041–1063MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Mirzaei D, Schaback R, Dehghan M (2012) On generalized moving least squares and diffuse derivatives. IMA J Numer Anal 32(3):983–1000MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Mohammadi V, Mirzaei D, Dehghan M (2019) Numerical simulation and error estimation of the time-dependent Allen–Cahn equation on surfaces with radial basis functions. J Sci Comput 79(1):493–516MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Mohammadi V, Dehghan M (2019) Simulation of the phase field Cahn–Hilliard and tumor growthmodels via a numerical scheme: element-free Galerkin method. Comput Methods Appl Mech Eng 345:919–950CrossRefGoogle Scholar
  41. 41.
    Muhammad N, Nadeem S, Issakhov A (2020) Finite volume method for mixed convection flow of Ag-ethylene glycol nanofluid flow in a cavity having thin central heater. Phys A Stat Mech Appl 537:122738MathSciNetCrossRefGoogle Scholar
  42. 42.
    Nair RD, Côté J, Staniforth A (1999) Cascade interpolation for semi-Lagrangian advection over the sphere. Q J R Meteorol Soc 125(556):1445–1486CrossRefGoogle Scholar
  43. 43.
    Nair RD, Thomas SJ, Loft RD (2005) A discontinuous Galerkin global shallow water model. Mon Weather Rev 123(4):876–888CrossRefGoogle Scholar
  44. 44.
    Nair RD, Jablonowski C (2008) Moving vortices on the sphere: a test case for horizontal advection problems. Mon Weather Rev 136(2):699–711CrossRefGoogle Scholar
  45. 45.
    Nair RD, Lauritzen PH (2010) A class of deformational flow test cases for linear transport problems on the sphere. J Comput Phys 229(23):8868–8887MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Parvizi M, Eslahchi MR, Dehghan M (2015) Numerical solution of fractional advection–diffusion equation with a nonlinear source term. Numer Algorithms 68:601–629MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Rizwana R, Hussain A, Nadeem S (2019) MHD oblique stagnation point flow of copper–water nanofluid with variable properties. Phys Scr 94(12):125808CrossRefGoogle Scholar
  48. 48.
    Rosales RR, Seibold B, Shirokoff D, Zhou D (2017) Unconditional stability for multistep ImEx schemes: theory. SIAM J Numer Anal 55(5):2336–2360MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Salehi R, Dehghan M (2013) A generalized moving least square reproducing kernel method. J Comput Appl Math 249:120–132MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Schaback R (2017) Error analysis of nodal meshless methods. In: Griebel M, Schweitzer MA (eds) Meshfree methods for partial differential equations VIII. Springer, Berlin, pp 117–143zbMATHCrossRefGoogle Scholar
  51. 51.
    Shahzadi I, Ahsan N, Nadeem S (2019) Analysis of bifurcation dynamics of streamlines topologies for pseudoplastic shear thinning fluid: biomechanics application. Phys A Stat Mech Appl.  https://doi.org/10.1016/j.physa.2019.122502 CrossRefGoogle Scholar
  52. 52.
    Shankar V, Wright GB (2018) Mesh-free semi-Lagrangian methods for transport on a sphere using radial basis functions. J Comput Phys 366:170–190MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Shankar V, Fogelson AL (2018) Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations. J Comput Phys 372:616–639MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Shipton J, Gibson T, Cotter C (2018) Higher-order compatible finite element schemes for the nonlinear rotating shallow water equations on the sphere. J Comput Phys 375:1121–1137MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Sleijpen GL, Fokkema DR (1993) BiCGstab (l) for linear equations involving unsymmetric matrices with complex spectrum. Electron Trans Numer Anal 1:11–32MathSciNetzbMATHGoogle Scholar
  56. 56.
    Smolarkiewicz PK (1982) The multi-dimensional Crowley advection scheme. Mon Weather Rev 110(12):1968–1983CrossRefGoogle Scholar
  57. 57.
    Staniforth A, Côté J, Pudykjewicz J (1987) Comments on Swolarkiewicz’s deformational flow. Mon Weather Rev 115(4):894–900CrossRefGoogle Scholar
  58. 58.
    St-Cyr A, Jablonowski C, Dennis JM, Tufo HM, Thomas SJ (2008) A comparison of two shallow-water models with nonconforming adaptive grids. Mon Weather Rev 136(6):1898–1922CrossRefGoogle Scholar
  59. 59.
    Taylor M, Edwards J, Thomas S, Nair RD (2007) A mass and energy conserving spectral element atmospheric dynamical core on the cubed-sphere grid. J Phys Conf Ser 78:012074CrossRefGoogle Scholar
  60. 60.
    Wendland H (2004) Scattered data approximation, vol 17. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  61. 61.
    Wendland H (2001) Moving least squares approximation on the sphere, Mathematical methods for curves and surfaces. Vanderbilt Univ. Press, NashvillezbMATHGoogle Scholar
  62. 62.
    Williamson DL, Drake JB, Hack JJ, Jakob R, Swarztrauber PN (1992) A standard test set for numerical approximations to the shallow water equations in spherical geometry. J Comput Phys 102(1):211–224MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Womersley R, Sloan I (2007) Interpolation and cubature on the sphere. http://web.maths.unsw.edu.au/~rsw/Sphere
  64. 64.
    Zerroukat M, Wood N, Staniforth A (2004) SLICE-S: a semi-Lagrangian inherently conserving and efficient scheme for transport problems on the sphere. Q J R Meteorol Soc J Atmos Sci Appl Meteorol Phys Oceanogr 130(602):2649–2664Google Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Vahid Mohammadi
    • 1
  • Mehdi Dehghan
    • 1
    Email author
  • Amirreza Khodadadian
    • 2
  • Thomas Wick
    • 2
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and Computer SciencesAmirkabir University of TechnologyTehranIran
  2. 2.Institute of Applied MathematicsLeibniz University of HannoverHanoverGermany

Personalised recommendations