An unconditionally stable and O (τ 2 + h 4 ) order L ∞ convergent difference scheme for linear parabolic equations with variable coefficients

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