The reproducing kernel particle Petrov–Galerkin method for solving two-dimensional nonstationary incompressible Boussinesq equations

2019 ◽  
Vol 106 ◽  
pp. 300-308 ◽  
Author(s):  
Mostafa Abbaszadeh ◽  
Mehdi Dehghan
Keyword(s):  
Galerkin Method ◽  
Reproducing Kernel ◽  
Boussinesq Equations ◽  
Two Dimensional ◽  
Reproducing Kernel Particle

The meshless method for two-dimensional space-time fractional dispersion equation based on reproducing kernel particle method

2017 ◽  
Vol 06 (02) ◽  
pp. 1750015
Author(s):  
Rongjun Cheng ◽  
Fengxin Sun ◽  
Jufeng Wang
Keyword(s):  
Dispersion Equation ◽  
Reproducing Kernel ◽  
Convergence Rates ◽  
Dimensional Space ◽  
Particle Method ◽  
Equation System ◽  
Approximation Schemes ◽  
Two Dimensional ◽  
Reproducing Kernel Particle Method ◽  
Reproducing Kernel Particle

The two-dimensional space fractional dispersion equation (SFDE) is obtained from the standard dispersion equation by replacing the two second-order space derivatives by the Riemann–Liouville fractional derivatives. A numerical analysis of the two-dimensional SFDE is presented based on the reproducing kernel particle method (RKPM). The final algebraic equation system is obtained by employing Galerkin weak form and functional minimization procedure. The Riemann–Liouville operator is discretized by the shifted Grünwald formula. The fully-discrete approximation schemes for SFDE are established using center difference method and RKPM and the shifted Grünwald formula. Numerical simulations for SFDE with known exact solution were presented in the format of the tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error estimate of RKPM for SFDE has been analyzed, which shows that this method has reasonable convergence rates in spatial and temporal discretizations.

scholarly journals Penyelesaian Numerik Advection Equation 1 Dimensi dengan EFG-DGM

2016 ◽  
Vol 22 (1) ◽  
pp. 51
Author(s):  
Kresno Wikan Sadono
Keyword(s):  
Galerkin Method ◽  
Reproducing Kernel ◽  
Time Integration ◽  
Particle Method ◽  
Advection Equation ◽  
Reproducing Kernel Particle Method ◽  
Domain Space ◽  
Reproducing Kernel Particle ◽  
The One ◽  
Element Free

Differential equation can be used to model various phenomena in science and engineering. Numerical method is the most common method used in solving DE. Numerical methods that popular today are finite difference method (FDM), finite element method (FEM) dan discontinuous Galerkin method (DGM), which the method includes mesh based. Lately, the developing methods, that are not based on a mesh, which the nodes directly spread in domain, called meshfree or meshless. Element free Galerkin method (EFG), Petrov-Galerkin meshless (MLPG), reproducing kernel particle method (RKPM) and radial basis function (RBF) fall into the category meshless or meshfree. Time integration generally use an explicit Runge Kutta 4th order, Newmark- , HHT- , Wilson-  dll. This research was carried out numerical simulations DE, by combining the EFG method to solve the domain space and time integration with DGM methods. EFG using the complete order polynomial 1, and DGM used polynomial order 1. The equation used advection equation in one dimension. EFG-DGM comparison with analytical results also performed. The simulation results show the method EFG-DGM match the one-dimensional advection equations well.

Meshless Method for the Numerical Solution of a Kind of Linear Hyperbolic Equations

2011 ◽  
Vol 101-102 ◽  
pp. 586-590
Author(s):  
Hai Na Sun ◽  
Rong Jun Cheng ◽  
Hong Xia Ge
Keyword(s):  
Numerical Solution ◽  
Reproducing Kernel ◽  
Hyperbolic Equations ◽  
Particle Method ◽  
Two Dimensional ◽  
Reproducing Kernel Particle Method ◽  
Hyperbolic Problems ◽  
Linear Hyperbolic Equation ◽  
Discrete Equations ◽  
Reproducing Kernel Particle

The present paper deals with the numerical solution of two-dimensional linear hyperbolic equation using the meshless reproducing kernel particle method (RKPM). A variational method is used to obtain the discrete equations and the essential boundary conditions that are enforced by the penalty method. The effectiveness RKPM for two-dimensional hyperbolic problems is investigated by two numerical examples in this paper.

The Meshless Reproducing Kernel Particle Method for Two-Dimensional Linear Hyperbolic Equations

2011 ◽  
Vol 365 ◽  
pp. 73-76
Author(s):  
Hai Na Sun ◽  
Rong Jun Cheng
Keyword(s):  
Penalty Method ◽  
Reproducing Kernel ◽  
Hyperbolic Equations ◽  
Particle Method ◽  
Two Dimensional ◽  
Reproducing Kernel Particle Method ◽  
Hyperbolic Problems ◽  
Discrete Equations ◽  
Reproducing Kernel Particle ◽  
Linear Hyperbolic Equations

The meshless reproducing kernel particle method (RKPM) is used to find the numerical solution of a kind of hyperbolic equations. A variational method is used to obtain the discrete equations and the essential boundary conditions are enforced by the penalty method. The effectiveness RKPM for two-dimensional hyperbolic problems is investigated by numerical example in this paper.

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