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

Abstract

This is the further work on compact finite difference schemes for heat equation with Neumann boundary conditions subsequent to the paper, [Sun, Numer Methods Partial Differential Equations (NMPDE) for the boundary condition approximation with the uniform partition. The new obtained scheme is similar to the one given by Liao et al. (NMPDE 22 (2006), 600-616), while the major difference lies in no extension of source terms to outside the computational domain any longer. Compared with ones obtained by Zhao et al. (NMPDE 23 (2007), 949-959) and Dai (NMPDE 27 (2011), 436-446), numerical solutions at all mesh points including two boundary points are computed in our new scheme. The significant advantage of this work is to provide a rigorous analysis of convergence order for the obtained compact difference scheme using discrete energy method. The global accuracy is O(τ 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. The analytical techniques are important and can be successfully used to solve the open problem presented by Sun (NMPDE 25 (2009(NMPDE 25 ( ), 1320(NMPDE 25 ( -1341, where analyzed theoretical convergence order of the scheme by Liao et al. (NMPDE 22 (2006)

A compact finite volume method and its extrapolation for elliptic equations with third boundary conditions

“…Performing integration by parts twice for R i − 1/2 defined by (2.7) and three times for R i, 1 (qu), R i, 2 (qu) defined by (2.10) and (2.11), respectively, we obtain (qu) (4) x i−1 − h 6 3072 (qu) (5) x i−1 + R (1) i,1 (qu),…”

“…In addition, we can easily extend the method to heat equation u t − u xx = f(x, t) with boundary condition (2.2) [17]. Further, when Neumann boundary condition is imposed and the integrals about f(x, t) are discretized by (2.8) and (2.9), then the scheme is exactly the one in [5].…”

“…Discrete energy method is employed in [15] to study the convergence of some schemes and the results show maximum norm convergence orders O(h 2.5 ) and O(h 3.5 ) hold for the schemes in [18] and [8], respectively. Recently, Gao and Sun [5] optimized the discretizing procedures and proposed an improved compact finite difference method with fourth-order accuracy in spatial direction, which was proved rigorously by discrete energy method. Meanwhile, Wang and Wang [17] also presented a similar scheme using finite volume discretizing technique for heat equation with Robin boundary conditions.…”

“…Chen and Wang [3] developed another analysis technique to prove the compact scheme has fourth-order accuracy for steady convection diffusion equations with Robin boundary conditions. We point out the treatments of Neumann or Robin boundary conditions in [3,5,17] are essentially equivalent. Ramos [12] also discussed the finite volume method for one-dimensional reaction diffusion problems, but the scheme is second-order accurate.…”

Fourth‐order compact difference schemes for 1D nonlinear Kuramoto–Tsuzuki equation

“…And recently, Gao and Sun [14] obtained a new compact scheme and provided a rigorous analysis of convergence order. Although the accuracy of the approximate boundary conditions is lower than that in the interior, the authors verified the global convergence order is O(τ 2 + h 4 ).…”