Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some part
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Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions Mehdi Dehghan1 · Mostafa Abbaszadeh1
Received: 1 June 2015 / Accepted: 23 November 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
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 timediscrete 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
Communicated by Pascal Frey.
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Mehdi Dehghan [email protected]; [email protected] Mostafa Abbaszadeh [email protected]
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Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran 15914, Iran
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M. Dehghan, M. Abbaszadeh
solved and numerical simulations are reported which confirm the efficiency of the proposed schemes. Keywords Variational multiscale element-free Galerkin (EFG) · Moving Kriging interpolation · Burger’s and Sod’s shock tube equations · Advection–reaction–diffusion and Kuramoto–Sivashinsky equations · Boussinesq and shallow water equations Mathematics Subject Classification 65M06 · 65M60
1 Introduction Partial differential equations (PDEs) are useful tools for explaining physical phenomena. In other words, many phen
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