Two Implicit Meshless Finite Point Schemes for the Two-Dimensional Distributed-Order Fractional Equation

2019 ◽  
Vol 19 (4) ◽  
pp. 813-831
Author(s):  
Rezvan Salehi
Keyword(s):  
Reproducing Kernel ◽  
Point Method ◽  
Least Square ◽  
Computationally Efficient ◽  
Finite Point ◽  
Moving Least Square ◽  
Point Collocation ◽  
Finite Point Method ◽  
Distributed Order ◽  
Fractional Equation

AbstractIn this paper, the distributed-order time fractional sub-diffusion equation on the bounded domains is studied by using the finite-point-type meshless method. The finite point method is a point collocation based method which is truly meshless and computationally efficient. To construct the shape functions of the finite point method, the moving least square reproducing kernel approximation is employed. Two implicit discretisation of order{O(\tau)}and{O(\tau^{1+\frac{1}{2}\sigma})}are derived, respectively. Stability and{L^{2}}norm convergence of the obtained difference schemes are proved. Numerical examples are provided to confirm the theoretical results.

scholarly journals Coupling of Point Collocation Meshfree Method and FEM for EEG Forward Solver

2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Chany Lee ◽  
Jong-Ho Choi ◽  
Ki-Young Jung ◽  
Hyun-Kyo Jung
Keyword(s):  
Reproducing Kernel ◽  
Kernel Method ◽  
Current Source ◽  
Meshfree Method ◽  
Forward Problem ◽  
Least Square ◽  
Head Model ◽  
Source Modeling ◽  
Moving Least Square ◽  
Point Collocation

For solving electroencephalographic forward problem, coupled method of finite element method (FEM) and fast moving least square reproducing kernel method (FMLSRKM) which is a kind of meshfree method is proposed. Current source modeling for FEM is complicated, so source region is analyzed using meshfree method. First order of shape function is used for FEM and second order for FMLSRKM because FMLSRKM adopts point collocation scheme. Suggested method is tested using simple equation using 1-, 2-, and 3-dimensional models, and error tendency according to node distance is studied. In addition, electroencephalographic forward problem is solved using spherical head model. Proposed hybrid method can produce well-approximated solution.

A SHALLOW WATER MODEL BY FINITE POINT METHOD

2013 ◽  
Vol 11 (01) ◽  
pp. 1350047 ◽  
Author(s):  
CHINAPAT BUACHART ◽  
WORSAK KANOK-NUKULCHAI ◽  
ENRIQUE ORTEGA ◽  
EUGENIO OÑATE
Keyword(s):  
Shallow Water ◽  
Local Approximation ◽  
Point Method ◽  
Least Square ◽  
Qr Factorization ◽  
Water Model ◽  
Approximation Procedure ◽  
Finite Point ◽  
Finite Point Method ◽  
Characteristic Based Split

In this paper, "finite point method" (FPM) is presented for modeling 2D shallow water flow problem. The method is based on the use of a weighted least-square approximation procedure, incorporating QR factorization and an iterative adjustment of local approximation parameters. The stabilization of the convective term in this present work is derived from the approximate Riemann solver proposed by Roe. The present method is shown to produce competitive accuracy in the comparisons with the analytical solutions and the well-known Galerkin characteristic-based split (CBS) algorithm.

A Miniaturized Electron Beam Column Simulation by the Fast Moving Least Square Reproducing Kernel Point Collocation Method

2003 ◽  
Vol 42 (Part 1, No. 6B) ◽  
pp. 3842-3848 ◽  
Author(s):  
Do Wan Kim ◽  
Yongsik Kim ◽  
Young Chul Kim ◽  
Ho Seob Kim ◽  
Seungjoon Ahn ◽  
...  
Keyword(s):  
Electron Beam ◽  
Collocation Method ◽  
Fast Moving ◽  
Reproducing Kernel ◽  
Least Square ◽  
Moving Least Square ◽  
Point Collocation

scholarly journals A finite point algorithm for soil water-salt movement equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fenhong Li ◽  
Gang Hu ◽  
Thabet Abdeljawad ◽  
Muhammad Abbas
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Soil Water ◽  
Point Method ◽  
Square Function ◽  
Finite Point ◽  
Moving Least Square ◽  
Finite Point Method ◽  
Water Salt ◽  
Element Method

AbstractIn this paper, we propose the meshless finite point method for solving a type of fluid flow problem. The moving least square function is combined with the collocation method to treat nonlinear one- and two-dimensional soil water-salt movement equations. An adaptive windward scheme is used to stabilize the numerical solution in regions with a large gradient change. Numerical examples with the comparison among the proposed method, finite element method and characteristic finite element method show that the meshless finite point method is more accurate and is used to eliminate the numerical oscillation phenomenon.

scholarly journals Real-Time Computer Simulation of Three Dimensional Elastostatics Using the Finite Point Method

2011 ◽  
Vol 110-116 ◽  
pp. 2740-2745
Author(s):  
Kirana Kumara P. ◽  
Ashitava Ghosal
Keyword(s):  
Real Time ◽  
Graphics Processing Unit ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Point Method ◽  
Processing Unit ◽  
Finite Point ◽  
Deformable Solids ◽  
Finite Point Method ◽  
Linear Elastostatics

Real-time simulation of deformable solids is essential for some applications such as biological organ simulations for surgical simulators. In this work, deformable solids are approximated to be linear elastic, and an easy and straight forward numerical technique, the Finite Point Method (FPM), is used to model three dimensional linear elastostatics. Graphics Processing Unit (GPU) is used to accelerate computations. Results show that the Finite Point Method, together with GPU, can compute three dimensional linear elastostatic responses of solids at rates suitable for real-time graphics, for solids represented by reasonable number of points.

Upwinding Meshfree Point Collocation Method for Steady MHD Flow

2010 ◽  
Author(s):  
Xinghui Cai ◽  
Guanghui Su ◽  
Suizheng Qiu
Keyword(s):  
Magnetic Field ◽  
Collocation Method ◽  
Mhd Flow ◽  
Least Square ◽  
Good Convergence ◽  
Moving Least Square ◽  
Point Collocation ◽  
Coupled Equations ◽  
Meshless Collocation ◽  
Mls Approximation

In this paper, a meshfree point collocation method, with a upwinding scheme, is presented to obtain the numerical solution of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight pipe of rectangular section with insulated walls. The moving least-square (MLS) approximation is employed to construct shape functions in conjunction with the framework of point collocation method. Computations have been carried out for different applied magnetic field orientations and different Hartmann numbers from 5 to 1,000,000. As the adaptive upwinding local support domain is introduced in the meshless collocation method, numerical results show that the method can compute MHD problems not only at low and moderate values but also at high values of the Hartmann number with high accuracy and good convergence.

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