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


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18


  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 98:253–264

    MathSciNet  MATH  Article  Google 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):524

    Article  Google 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):206–215

    Article  Google Scholar 

  4. 4.

    Atkinson K, Han W (2012) Spherical harmonics and approximations on the unit sphere: an introduction, vol 2044. Springer Science & Business Media, Berlin

    Google 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):726–750

    Article  Google 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):5019–5044

    MathSciNet  MATH  Article  Google 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):1021–1032

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Cotter CJ, Shipton J (2012) Mixed finite elements for numerical weather prediction. J Comput Phys 231(21):7076–7091

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Doswell CA III (1984) A kinematic analysis of frontogenesis associated with a nondivergent vortex. J Atmos Sci 41(7):1242–1248

    Article  Google 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):2642–2665

    Article  Google 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):6955–6971

    MathSciNet  MATH  Article  Google 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 59:500–513

    MathSciNet  Article  Google 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 107:168–184

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Dehghan M (2007) Time-splitting procedures for the solution of the two-dimensional transport equation. Kybernetes 36(5):791–805

    MATH  Article  Google Scholar 

  15. 15.

    Fasshauer GE (2007) Meshfree approximation methods with MATLAB, vol 6. World Scientific, Singapore

    Google Scholar 

  16. 16.

    Flyer N, Wright GB (2007) Transport schemes on a sphere using radial basis functions. J Comput Phys 226(1):1059–1084

    MathSciNet  MATH  Article  Google 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 465:1949–1976

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Fornberg B, Lehto E (2011) Stabilization of RBF-generated finite difference methods for convective PDEs. J Comput Phys 230(6):2270–2285

    MathSciNet  MATH  Article  Google 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):57–92

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Fuselier E, Wright GB (2015) Order-preserving derivative approximation with periodic radial basis functions. Adv Comput Math 41(1):23–53

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Giraldo FX (2000) Lagrange–Galerkin methods on spherical geodesic grids: the shallow water equations. J Comput Phys 160(1):336–368

    MathSciNet  MATH  Article  Google 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):1–11

    MATH  Article  Google 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):3691–3713

    Article  Google Scholar 

  25. 25.

    Krems M (2007) The Boltzmann transport equation: theory and applications.

  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 353:66–81

    MathSciNet  Google 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):524–532

    Article  Google Scholar 

  28. 28.

    Khodadadian A, Heitzinger C (2016) Basis adaptation for the stochastic nonlinear Poisson–Boltzmann equation. J Comput Electron 15(4):1393–1406

    Article  Google 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):937–949

    MathSciNet  MATH  Article  Google 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):1235–1266

    MathSciNet  MATH  Article  Google 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):887–901

    Article  Google 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):609–628

    MathSciNet  MATH  Article  Google 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):2129–2151

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    LeVeque RJ (1996) High-resolution conservative algorithms for advection in incompressible flow. SIAM J Numer Anal 23(2):627–665

    MathSciNet  MATH  Article  Google 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):1070

    Article  Google 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 614

  37. 37.

    Mirzaei D (2017) Direct approximation on spheres using generalized moving least squares. BIT Numer Math 57(4):1041–1063

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Mirzaei D, Schaback R, Dehghan M (2012) On generalized moving least squares and diffuse derivatives. IMA J Numer Anal 32(3):983–1000

    MathSciNet  MATH  Article  Google 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):493–516

    MathSciNet  MATH  Article  Google 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 345:919–950

    Article  Google 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 537:122738

    MathSciNet  Article  Google 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):1445–1486

    Article  Google Scholar 

  43. 43.

    Nair RD, Thomas SJ, Loft RD (2005) A discontinuous Galerkin global shallow water model. Mon Weather Rev 123(4):876–888

    Article  Google 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):699–711

    Article  Google 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):8868–8887

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Parvizi M, Eslahchi MR, Dehghan M (2015) Numerical solution of fractional advection–diffusion equation with a nonlinear source term. Numer Algorithms 68:601–629

    MathSciNet  MATH  Article  Google 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):125808

    Article  Google 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):2336–2360

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Salehi R, Dehghan M (2013) A generalized moving least square reproducing kernel method. J Comput Appl Math 249:120–132

    MathSciNet  MATH  Article  Google 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 117–143

    Google 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.

    Article  Google 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 366:170–190

    MathSciNet  MATH  Article  Google 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 372:616–639

    MathSciNet  MATH  Article  Google 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 375:1121–1137

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Sleijpen GL, Fokkema DR (1993) BiCGstab (l) for linear equations involving unsymmetric matrices with complex spectrum. Electron Trans Numer Anal 1:11–32

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Smolarkiewicz PK (1982) The multi-dimensional Crowley advection scheme. Mon Weather Rev 110(12):1968–1983

    Article  Google Scholar 

  57. 57.

    Staniforth A, Côté J, Pudykjewicz J (1987) Comments on Swolarkiewicz’s deformational flow. Mon Weather Rev 115(4):894–900

    Article  Google 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):1898–1922

    Article  Google 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 78:012074

    Article  Google Scholar 

  60. 60.

    Wendland H (2004) Scattered data approximation, vol 17. Cambridge University Press, Cambridge

    Google Scholar 

  61. 61.

    Wendland H (2001) Moving least squares approximation on the sphere, Mathematical methods for curves and surfaces. Vanderbilt Univ. Press, Nashville

    Google 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):211–224

    MathSciNet  MATH  Article  Google Scholar 

  63. 63.

    Womersley R, Sloan I (2007) Interpolation and cubature on the 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):2649–2664

    Google Scholar 

Download references


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

Author information



Corresponding author

Correspondence to Mehdi Dehghan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mohammadi, V., Dehghan, M., Khodadadian, A. et al. Numerical investigation on the transport equation in spherical coordinates via generalized moving least squares and moving kriging least squares approximations. Engineering with Computers (2019).

Download citation


  • 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