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

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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.

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Acknowledgements

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

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Correspondence to Mehdi Dehghan.

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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) doi:10.1007/s00366-019-00881-3

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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