A new third‐order finite‐difference method for transient one‐dimensional advection—diffusion

Abstract

SUMMARYThe accuracy of some first-and second-order methods for solving the time-dependent one-dimensional constant-coefficient advection-diffusion equation are compared theoretically on the basis of the dominant error terms in their modified equivalent partial differential equations. A new very stable three-point (in space) third-order implicit method is then developed by combining two second-order methods. The accuracies of the various methods are then compared by means of numerical tests.

Numerical error in groundwater flow and solute transport simulation

“…[29] Our first test problem is the Gauss pulse test, which has been extensively used by Noye in the analysis of finite difference methods [Leonard and Noye, 1990;Noye, 1990Noye, , 1991Noye and Hayman, 1992]. It is a type of contaminant slug problem, similar to those studied by Prickett et al [1981] and Javandel et al [1984].…”

Extension of high‐order compact schemes to time‐dependent problems

“…Given a user-specified µ, the HOC scheme is stable, provided that C satisfies both the left and right inequalities in (14). The scheme is, therefore, unconditionally stable for any positive C and…”

“…1. Here the solid line is the HOC stability limit given in (14), with the left inequality being more restrictive for Re h > 2 √ 3 and the right inequality being more restrictive for Re h < 2 √ 3. The stability regions of the upwind difference scheme (UDS) and central difference scheme (CDS) are included for comparison, and stable choices of C and Re h correspond to points below the respective curves.…”

“…There have been several previous studies related to development of high order schemes for transient problems. For example, Noye has developed and applied several higher-order schemes for time-dependent diffusion and convection-diffusion [14][15][16][17][18][19] and Abarbanel and Kumar [20] have used HOC approaches for the transient Euler equations in 1D and 2D. One of the simplest and most familiar examples involves the forward time, centered space (FTCS) scheme for the onedimensional model diffusion problem ∂φ ∂t = a ∂ 2 φ ∂x 2 .…”

Higher order semi compact scheme to solve transient incompressible Navier-Stokes equations

“…All these investigations are centered on steady flows, even though the basic idea in principle can be applied to unsteady flow computations. Noye [7,8] has developed and applied several higher order schemes for time dependent diffusion and convection-diffusion problems and Abarbanel and Kumar [1] used HOC approach in the transient Euler equations in one and two-dimensional problems. Spotz and Carey [11] also proposed a scheme for time dependent problems similar to the scheme proposed by them for the steady case.…”