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An unconditionally stable and O (τ 2 + h 4 ) order L ∞ convergent difference scheme for linear parabolic equations with variable coefficients
Author(s) -
Sun ZhiZhong
Publication year - 2001
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.1030
Subject(s) - mathematics , extrapolation , mathematical analysis , dirichlet boundary condition , constant coefficients , order (exchange) , neumann boundary condition , mathematical physics , partial differential equation , boundary value problem , finance , economics
M. K. Jain, R. K. Jain, and R. K. Mohanty presented a finite difference scheme of O (τ 2 + τ h 2 + h 4 ) for solving the one‐dimensional quasilinear parabolic partial differential equation, u xx = f ( x, t, u, u t , u x ), with Dirichlet boundary conditions. The method, when applied to a linear constant coefficient case, was shown to be unconditionally stable by the Von Neumann method. In this article, we prove that the method, when applied to a linear variable coefficient case, is unconditionally stable and convergent with the convergence order O (τ 2 + h 4 ) in the L ∞ ‐norm. In addition, we obtain an asymptotic expansion of the difference solution, with which we obtain an O (τ 4 + τ 2 h 4 + h 6 ) order accuracy approximation after extrapolation. And last, we point out that the analysis method in this article is efficacious for complex equations. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:619–631, 2001