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In this work, we apply two meshless methods for the numerical solution of thetime-dependent transport equation defined on the sphere in sphericalcoordinates. The first technique, which was introduced by Mirzaei (BITNumerical Mathematics, 54 (4) 1041-1063, 2017) in Cartesian coordinates is ageneralized moving least squares approximation, and the second one, which isdeveloped here, is moving kriging least squares interpolation on the sphere.These methods do not depend on the background mesh or triangulation, and theycan be implemented on the transport equation in spherical coordinates easilyusing different distribution points. Furthermore, the time variable isapproximated by a second-order backward differential formula. The obtainedfully discrete scheme is solved via the biconjugate gradient stabilizedalgorithm with zero-fill incomplete lower-upper (ILU) preconditioner at eachtime step. Three well-known test problems namely solid body rotation, vortexroll-up, and deformational flow are solved to demonstrate our developments.

Generalized moving least squares and moving kriging least squares approximations for solving the transport equation on the sphere