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A fourth‐order difference method for elliptic equations with nonlinear first derivative terms
Author(s) -
Jain M. K.,
Jain R. K.,
Mohanty R. K.
Publication year - 1989
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.1690050203
Subject(s) - mathematics , order (exchange) , convergence (economics) , mathematical analysis , nonlinear system , dirichlet boundary condition , finite difference , derivative (finance) , finite difference method , elliptic curve , boundary value problem , dirichlet distribution , finite difference coefficient , differential equation , dirichlet problem , second derivative , finite element method , mixed finite element method , physics , finance , quantum mechanics , financial economics , economics , thermodynamics , economic growth
Abstract We present a nine‐point fourth‐order finite difference method for the nonlinear second‐order elliptic differential equation Au xx + Bu yy = f ( x, y, u, u x , u y ) on a rectangular region R subject to Dirichlet boundary conditions u ( x, y ) = g ( x, y ) on ∂ R . We establish, under appropriate conditions O( h 4 )‐convergence of the finite difference scheme. Numerical examples are given to illustrate the method and its fourth‐order convergence.