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A high‐order finite difference method for 1D nonhomogeneous heat equations
Author(s) -
Lin Yuan,
Gao Xuejun,
Xiao MingQing
Publication year - 2009
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.20345
Subject(s) - mathematics , ode , discretization , boundary value problem , partial differential equation , crank–nicolson method , algebraic equation , heat equation , finite difference method , mathematical analysis , ordinary differential equation , finite difference , matrix (chemical analysis) , grid , differential equation , geometry , physics , materials science , nonlinear system , quantum mechanics , composite material
In this article a sixth‐order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth‐order finite difference approximation scheme for a two‐point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels‐Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second‐order Crank‐Nicolson scheme as well as Sun‐Zhang's recent fourth‐order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009