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The combination of collocation, finite difference, and multigrid methods for solution of the two‐dimensional wave equation
Author(s) -
Dehghan Mehdi,
Mohebbi Akbar
Publication year - 2008
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20295
Subject(s) - multigrid method , mathematics , collocation (remote sensing) , collocation method , discretization , finite difference method , partial differential equation , orthogonal collocation , wave equation , finite difference , mathematical analysis , grid , differential equation , ordinary differential equation , geometry , computer science , machine learning
Abstract In this article, we apply compact finite difference approximations of orders two and four for discretizing spatial derivatives of wave equation and collocation method for the time component. The resulting method is unconditionally stable and solves the wave equation with high accuracy. The solution is approximated by a polynomial at each grid point that its coefficients are determined by solving a linear system of equations. We employ the multigrid method for solving the resulted linear system. Multigrid method is an iterative method which has grid independently convergence and solves the linear system of equations in small amount of computer time. Numerical results show that the compact finite difference approximation of fourth order, collocation and multigrid methods produce a very efficient method for solving the wave equation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008