Compact difference schemes for heat equation with Neumann boundary conditions (II)

O((2) + h(4)) for interior mesh point approximation and O((2) + h(3)) for the boundary condition approximation with the uniform partition. O((2) + h(4)) in discrete maximum norm, although the spatial approximation order at the Neumann boundary is one lower than that for interior mesh points. O((2) + h(3.5)) while the numerical accuracy is O((2) + h(4)), and convergence order of theoretical analysis for the scheme by Zhao et al. (NMPDE 23 (2007), 949-959) is O((2) + h(2.5)), while the actual numerical accuracy is O((2) + h(3)). Following the procedure used for the new obtained difference scheme in this work, convergence orders of these two schemes can be proved rigorously to be O((2) + h(4)) and O((2) + h(3)), respectively. Meanwhile, extension to the case involving the nonlinear reaction term is also discussed, and the global convergence order O((2) + h(4)) is proved. A compact ADI difference scheme for solving two-dimensional case is derived. Finally, several examples are given to demonstrate the numerical accuracy of new obtained compact difference schemes. (c) 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013