Meshless local Petrov–Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation

During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov–Galerkin (MLPG) meth...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 233; no. 10; pp. 2737 - 2754
Main Authors: Mirzaei, Davoud, Dehghan, Mehdi
Format: Journal Article
Language:English
Published: Kidlington Elsevier B.V 15.03.2010
Elsevier
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Meshless local Petrov–Galerkin (MLPG) method
Moving least-squares (MLS) approximation
Numerical analysis
Numerical analysis. Scientific computation
Numerical methods in probability and statistics
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Sine-Gordon (SG) equation
65M15
Meshless local Petrov–Galerkin (MLPG) method
65M Except 65M06
Sine-Gordon (SG) equation
65M12
Moving least-squares (MLS) approximation
65M20
Three-dimensional calculations
Numerical method
Stochastic method
Partial differential equation
Numerical approximation
Sphere
Least squares method
Numerical solution
Meshless local Petrov-Galerkin (MLPG)
Numerical integration
method
Initial value problem
Nonlinear problems
Algorithm
Galerkin-Petrov method
Two dimensional equation
Numerical analysis
Boundary value problem
Applied mathematics
Two-dimensional calculations
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary:During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov–Galerkin (MLPG) method is one of the “truly meshless” methods since it does not require any background integration cells. The integrations are carried out locally over small sub-domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. In this paper the MLPG method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. A time-stepping method is employed to deal with the time derivative and a simple predictor–corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of proposed method to deal with the unsteady non-linear problems in large domains.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2009.11.022