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Solving the equation − u xx − ϵ u yy = f ( x, y, u ) by an O ( h 4 ) finite difference method
Author(s) -
Lungu E.,
Motsumi T.,
Styś T.
Publication year - 2000
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 - 10.1002/1098-2426(200007)16:4<395::aid-num4>3.0.co;2-p
Subject(s) - mathematics , mathematical analysis , partial differential equation , dirichlet boundary condition , matrix (chemical analysis) , finite difference method , boundary value problem , truncation error , algebraic equation , dirichlet problem , nonlinear system , materials science , physics , quantum mechanics , composite material
The semi‐linear equation − u xx − ϵ u yy = f ( x, y, u ) with Dirichlet boundary conditions is solved by an O ( h 4 ) finite difference method, which has local truncation error O ( h 2 ) at the mesh points neighboring the boundary and O ( h 4 ) at most interior mesh points. It is proved that the finite difference method is O ( h 4 ) uniformly convergent as h → 0. The method is considered in the form of a system of algebraic equations with a nine diagonal sparse matrix. The system of algebraic equations is solved by an implicit iterative method combined with Gauss elimination. A Mathematica module is designed for the purpose of testing and using the method. To illustrate the method, the equation of twisting a springy rod is solved. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 395–407, 2000