A high‐order finite difference method for 1D nonhomogeneous heat equations

Abstract

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.

High order locally one-dimensional methods for solving two-dimensional parabolic equations

“…In [2], the authors presented a sixth order approximation for the second order derivative together with the constant boundary conditions. To this end, we have…”

“…The (3, 3) Padé approximation [29] is employed to get sixth order accuracy for temporal variable e Z = 120 -60Z + 12Z 2 -Z 3 -1 120 + 60Z + 12Z 2…”

“…Consider the following two-dimensional linear parabolic equation: Many efforts have been made to the development of accurate and stable methods for the numerical solution of (1) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Various high order methods [1-4, 6-9, 11-19] have been proposed.…”

“…They obtained two schemes with third and fifth order accuracy in time, respectively. Liu et al [21] used a similar strategy, (2,2) and (3,3) Padé approximations for the time discretization and C 3 quartic spline approximation for space discretization, to get two higher order difference schemes for the one-dimensional linear hyperbolic equation. Zhang [23] provided a (3, 3) Padé approximation method for solving space fractional Fokker-Planck equations.…”

On the coupling of the homotopy perturbation method and Laplace transformation

“…The main advantage of semi-analytical methods, compared with other methods, is based on the fact that they can be conveniently applied to solve various complicated problems. Several numerical and analytical methods including non-polynomial cubic spline methods, finite difference, the Laplace decomposition method, the homotopy perturbation transform method, variational iteration methods, the Adomian decomposition method and the homogeneous Adomian decomposition method have been developed for solving linear or nonlinear non-homogeneous partial differential equations, see [1][2][3][4][5][6][7]. HPM, ADM, and VIM methods can be used to solve the non-homogeneous variable coefficient partial differential equations with accurate approximation, but this approximation is acceptable only for a small range [7], because, boundary conditions in one dimension are satisfied via these methods.…”