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A new highly accurate discretization for three‐dimensional singularly perturbed nonlinear elliptic partial differential equations
Author(s) -
Mohanty R.K.,
Singh Swarn
Publication year - 2006
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.20160
Subject(s) - mathematics , discretization , partial differential equation , nonlinear system , context (archaeology) , elliptic partial differential equation , dirichlet boundary condition , boundary value problem , convergence (economics) , finite difference method , partial derivative , boundary (topology) , dirichlet problem , mathematical analysis , finite difference , paleontology , physics , quantum mechanics , economics , biology , economic growth
We employ a new fourth‐order compact finite difference formula based on arithmetic average discretization to solve the three‐dimensional nonlinear singularly perturbed elliptic partial differential equation ε( u xx + u yy + u zz ) = f ( x , y , z , u , u x , u y , u z ), 0 < x , y , z < 1, subject to appropriate Dirichlet boundary conditions prescribed on the boundary, where ε > 0 is a small parameter. We also describe new fourth‐order methods for the estimates of (∂ u /∂ x ), (∂ u /∂ y ), and (∂ u /∂ z ), which are quite often of interest in many physical problems. In all cases, we require only a single computational cell with 19 grid points. The proposed methods are directly applicable to solve singular problems without any modification. We solve three test problems numerically to validate the proposed derived fourth‐order methods. We compare the advantages and implementation of the proposed methods with the standard central difference approximations in the context of basic iterative methods. Numerical examples are given to verify the fourth‐order convergence rate of the methods. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006