The meshless local Petrov-Galerkin method based on moving Kriging interpolation for solving the time fractional Navier-Stokes equations

Springerplus. 2016 Apr 6;5:417. doi: 10.1186/s40064-016-2047-2. eCollection 2016.


In this paper, we present a numerical scheme used to solve the nonlinear time fractional Navier-Stokes equations in two dimensions. We first employ the meshless local Petrov-Galerkin (MLPG) method based on a local weak formulation to form the system of discretized equations and then we will approximate the time fractional derivative interpreted in the sense of Caputo by a simple quadrature formula. The moving Kriging interpolation which possesses the Kronecker delta property is applied to construct shape functions. This research aims to extend and develop further the applicability of the truly MLPG method to the generalized incompressible Navier-Stokes equations. Two numerical examples are provided to illustrate the accuracy and efficiency of the proposed algorithm. Very good agreement between the numerically and analytically computed solutions can be observed in the verification. The present MLPG method has proved its efficiency and reliability for solving the two-dimensional time fractional Navier-Stokes equations arising in fluid dynamics as well as several other problems in science and engineering.

Keywords: Meshless local Petrov–Galerkin method; Moving Kriging interpolation; Quadrature formula; Time fractional Navier–Stokes equations.