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Title: Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions

Abstract

Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolationmore » and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.« less

Authors:
;  [1]
  1. Amirkabir University of Technology, Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences (Iran, Islamic Republic of)
Publication Date:
OSTI Identifier:
22769257
Resource Type:
Journal Article
Journal Name:
Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 37; Journal Issue: 3; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; BOUNDARY CONDITIONS; COMPUTERIZED SIMULATION; FINITE ELEMENT METHOD; INTERPOLATION; LEAST SQUARE FIT; NUMERICAL ANALYSIS; PARTIAL DIFFERENTIAL EQUATIONS; VARIATIONAL METHODS

Citation Formats

Dehghan, Mehdi, and Abbaszadeh, Mostafa. Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions. United States: N. p., 2018. Web. doi:10.1007/S40314-017-0546-6.
Dehghan, Mehdi, & Abbaszadeh, Mostafa. Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions. United States. doi:10.1007/S40314-017-0546-6.
Dehghan, Mehdi, and Abbaszadeh, Mostafa. Sun . "Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions". United States. doi:10.1007/S40314-017-0546-6.
@article{osti_22769257,
title = {Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions},
author = {Dehghan, Mehdi and Abbaszadeh, Mostafa},
abstractNote = {Solving partial differential equations with discontinuous solutions is an important challenging problem in numerical analysis. To this end, there are some methods such as finite volume method, discontinuous Galerkin approach and particle technique that are able to solve these problems. In the current paper, the moving Kriging element-free Galerkin method has been combined with the variational multiscale algorithm to obtain acceptable and high-resolution solutions. For testing this technique, we select some PDEs with discontinuous solution such as Burgers’, Sod’s shock tube, advection–reaction–diffusion, Kuramoto–Sivashinsky, Boussinesq and shallow water equations. First, we obtain a time-discrete scheme by approximating time derivative via finite difference technique. Then we introduce the moving Kriging interpolation and also obtain their shape functions. We use the element-free Galerkin method for approximating the spatial derivatives. This method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the classical element-free Galerkin method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, we employ the shape functions of moving Kriging interpolation and radial point interpolation technique which have the mentioned property. Also, in the element-free Galekin method, we do not use any triangular, quadrangular or other type of meshes. The element-free Galerkin method is a global method while finite element method is a local one. This technique employs a background mesh for integration which makes it different from the truly mesh procedures. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed schemes.},
doi = {10.1007/S40314-017-0546-6},
journal = {Computational and Applied Mathematics},
issn = {0101-8205},
number = 3,
volume = 37,
place = {United States},
year = {2018},
month = {7}
}