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multigrid method with eighth-order compact finite difference scheme for Helmholtz equation

PHYSICA SCRIPTA 2020年第5期95卷 055221-055221页

作者： Ghaffar, Fazal Ullah, Saif Badshah, Noor Govt Post Grad Jahanzeb Coll Swat Dept Math Swat Khyber Pakhtunk Pakistan Govt Coll Univ Dept Math Lahore 54000 Pakistan Univ Engn & Technol Peshawar Dept Basic Sci Peshawar Pakistan

Higher-order compact finite difference scheme with multigrid algorithm is applied in this paper for solving one-dimensional and two-dimensional inhomogeneous Helmholtz equations. In two-dimensional case, the suggested scheme has the stencil of twenty one points. An efficient solver multigrid method yields eighth-order accurate approximation on both fine and coarse grids. For the Neumann boundary condition, an eighth-order accurate representation is also developed. The accuracy and efficiency of eighth-order compact difference scheme are exhibited through graphical illustrations and computed results are drafted in tabular form to validate the numerical experiments.

关键词： Helmhlotz equation compact iterative scheme multigrid method uniform grids

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multigrid method based on transformation-free high-order scheme for solving 2D Helmholtz equation on nonuniform grids

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ADVANCES IN DIFFERENCE EQUATIONS 2016年第1期2016卷 1页

作者： Ghaffar, Fazal Badshah, Noor Islam, Saeed Khan, Muhammad Altaf Abdul Wali Khan Univ Dept Math Mardan Pakistan Univ Engn & Technol Dept Basic Sci Peshawar Pakistan

High-order compact difference schemes can achieve higher-order accuracy on uniform grids. However, in some cases these may not achieve the desired accuracy. Therefore, we propose a multigrid method based on high-order compact difference scheme on nonuniform grids. We will use interpolation and restriction operators developed by Ge and Cao (J. Comput. Phys. 230:4051-4070, 2011). The suggested scheme has up to fourth-order accuracy. Lastly, some numerical experiments are given to show the accuracy and performance of the proposed scheme.

关键词： Helmholtz equation compact iterative schemes multigrid method nonuniform grids

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multigrid method for elliptic equations with anisotropic discontinuous coefficients

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2015年第7期55卷 1150-1163页

作者： Zhukov, V. T. Novikova, N. D. Feodoritova, O. B. Russian Acad Sci MV Keldysh Appl Math Inst Moscow 125047 Russia

For difference elliptic equations, an algorithm based on Fedorenko's multigrid method is constructed. The algorithm is intended for solving three-dimensional boundary value problems for equations with anisotropic discontinuous coefficients on parallel computers. Numerical results confirming the performance and parallel efficiency of the multigrid algorithm are presented. These qualities are ensured by using, as a multigrid triad, the standard Chebyshev iteration for coarsest grid equations, Chebyshev-type smoothing explicit iterative procedures, and intergrid transfer operators in problem-dependent form.

关键词： three-dimensional elliptic equations anisotropic discontinuous coefficients multigrid method Chebyshev iteration method parallel implementation

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multigrid method and fourth-order compact difference discretization scheme with unequal meshsizes for 3D poisson equation

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JOURNAL OF COMPUTATIONAL PHYSICS 2010年第18期229卷 6381-6391页

作者： Ge, Yongbin Ningxia Univ Inst Appl Math & Mech Yinchuan 750021 Peoples R China

A fourth-order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three-dimensional (3D) Poisson equation on a cubic domain. Two multgrid methods are developed to solve the resulting sparse linear systems. One is to use the full-coarsening multigrid method with plane Gauss-Seidel relaxation, which uses line Gauss-Seidel relaxation to compute each planewise solution. The other is to construct a partial semi-coarsening multigrid method with the traditional point or plane Gauss-Seidel relaxations. Numerical experiments are conducted to test the computed accuracy of the fourth-order compact difference scheme and the computational efficiency of the multigrid methods with the fourth-order compact difference scheme. (C) 2010 Elsevier Inc. All rights reserved.

关键词： 3D Poisson equation Compact difference scheme Unequal meshsizes multigrid method Partial semi-coarsening

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multigrid method for Poroelasticity Problem by Finite Element method

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Advances in Applied Mathematics and Mechanics 2019年第6期11卷 1339-1357页

作者： Luoping Chen Yanping Chen School of Mathematics Southwest Jiaotong UniversityChengdu 611756SichuanChina School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina

In this paper,we will investigate a multigrid algorithm for poroelasticity problem by a new finite element method with homogeneous boundary conditions in two dimensional *** choose N´ed´elec edge element for the displacement variable and piecewise continuous polynomials for the pressure variable in the model *** constructing multigrid algorithm,a distributive Gauss-Seidel iteration method is *** experiments shows that the finite element method achieves optimal convergence order and the multigrid algorithm is almost uniformly convergent to mesh size h and parameter dt on regular meshes.

关键词： Poroelasticity problem finite element method multigrid method

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multigrid method for anisotropic diffusion equations based on adaptive Chebyshev smoothers

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Mathematical Models and Computer Simulations 2015年第2期7卷 117-127页

作者： Zhukov, V.T. Novikova, N.D. Feodoritova, O.B. Keldysh Institute of Applied Mathematics Russian Academy of Sciences Moscow Russian Federation

We propose an efficient multigrid algorithm for solving anisotropic elliptic difference equations. The algorithm is based on using Chebyshev’s explicit iterations at smoothing stages and in solving coarse-grid equations. We have developed a procedure for adapting smoothers to anisotropy and present examples, which show that adaptation improves the efficiency of the multigrid method and scalability of the parallel code. © 2015, Pleiades Publishing, Ltd.

关键词： adaptive smoother Chebyshev iterations multigrid method parallel implementation three-dimensional equations of anisotropic diffusion

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multigrid method for a Two Dimensional Fully Nonlinear Black-Scholes Equation with a Nonlinear Volatility Function

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数学研究 2020年第3期53卷 247-264页

作者： Aicha Driouch Hassan Al Moatassime LAMFA UMR 7352 CNRS University of Picardie Jules Verne Amiens France Department of Mathematics Faculty of science and technologyUniversity Cadi Ayyad Marrakesh Morocco

This paper deals with the task of pricing European basket options in the p-resence of transaction costs. We develop a model that incorporates the illiquidity of the market into the classical two-assets Black-Scholes framework. We perform a nu-merical simulation using finite difference method. We consider a nonlinear multigrid method in order to reduce computational costs. The objective of this paper is to inves-tigate a deterministic extension for the Barles' and Soner's model and to demonstrate the effectiveness of multigrid approach to solving a fully nonlinear two dimensional Black-Scholes problem.

关键词： Fully nonlinear equation multigrid method black-scholes equation finite difference method FAS algorithm

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Local grid refinement in multigrid method for point contact problems of polycrystalline anisotropic material under dry and lubricated conditions

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Friction 2022年第12期10卷 2086-2110页

作者： Binbin ZHANG Tristan G.VLOGMAN Predrag ANDRIC Ton C.BOR Cornelis H.VENNER Faculty of Engineering Technology University of TwenteEnschede 7500AEthe Netherlands SKF Research and Technology Development Houten 3992AEthe Netherlands

Predicting rolling bearing fatigue life requires knowledge of the three-dimensional(3D)stress fields in the roller and raceway near the lubricated *** to the increasingly severe operating conditions,the effect of localized features such as surface roughness,subsurface inclusions,and even the crystallographic structure of the material becomes *** such detail requires(locally)extremely dense gridding in simulations,which in 3D is a major *** techniques have been demonstrated to be capable of solving such *** this study,multigrid techniques are shown to further increase the efficiency of the solution by exploiting local grid refinement while maintaining the simplicity of a uniform *** is achieved by employing increasingly finer grids only locally,where the highest resolution is *** are presented for dry contact and elastohydrodynamically lubricated contact cases,circular as well as elliptic,with varying crystallographic structure,and with surface *** results show that the developed algorithm is very well suited for detailed analysis,with also excellent prospects for computational diagnostics involving actual material crystallographic structure from electron backscatter diffraction measurements.

关键词： local grid refinement multigrid method polycrystalline anisotropic material dry and lubricated contact

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Local and parallel multigrid method for semilinear elliptic equations

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APPLIED NUMERICAL MATHEMATICS 2021年 162卷 20-34页

作者： Xu, Fei Huang, Qiumei Jiang, Kun Ma, Hongkun Beijing Univ Technol Fac Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Zhuhai Huafa Investment Holdings Grp Co Ltd Hengqin 519000 Peoples R China Sun Yat Sen Univ Business Sch Guangzhou 510275 Guangdong Peoples R China

This paper presents a new type of local and parallel multigrid method to solve semilinear elliptic equations. The proposed method does not directly solve the semilinear elliptic equations on each layer of the multigrid mesh sequence, but transforms the semilinear elliptic equations into several linear elliptic equations on the multigrid mesh sequence and some low-dimensional semilinear elliptic equations on the coarsest mesh. Furthermore, the local and parallel strategy is used to solve the involved linear elliptic equations. Since solving large-scale semilinear elliptic equations in fine space, which can be fairly time-consuming, is avoided, the proposed local and parallel multigrid scheme will significantly improve the solving efficiency for the semilinear elliptic equations. Besides, compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only requires the Lipschitz continuation property of the nonlinear term. We make a rigorous theoretical analysis of the presented local and parallel multigrid scheme, and propose some numerical experiments to support the theory. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Semilinear elliptic equations Local and parallel Multilevel correction method multigrid method

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A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2017年 336卷 69-86页

作者： Lin, Xue-lei Ng, Michael K. Sun, Hai-Wei Univ Macau Dept Math Macau Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

In this paper, we study a V-cycle multigrid method for linear systems arising from time dependent two-dimensional space-fractional diffusion equations. The coefficient matrices of the linear systems are structured such that their matrix-vector multiplications can be computed efficiently. The main advantage using the multigrid method is to handle the space-fractional diffusion equations on non-rectangular domains, and to solve the linear systems with non-constant coefficients more effectively. The main idea of the proposed multigrid method is to employ two banded splitting iteration schemes as pre-smoother and post-smoother. The pre-smoother and the post-smoother take banded splitting of the coefficient matrix under the x-dominant ordering and the y-dominant ordering, respectively. We prove the convergence of the proposed two banded splitting iteration schemes in the sense of infinity norm. Results of numerical experiments for time dependent two-dimensional space-fractional diffusion equations on rectangular, L-shape and U-shape domains are reported to demonstrate that both computational time and iteration number required by the proposed method are significantly smaller than those of the other tested methods. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Fractional diffusion equation multigrid method Banded-splitting smoother Non-rectangular domain

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