A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

Abstract

SUMMARYA class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two-dimensional unsteady convection-di usion equation with variable convection coe cients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter and fourth order accurate in space. For 0:56 61, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They e ciently capture both transient and steady solutions of linear and nonlinear convectiondi usion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection-di usion problem and three ows of varying complexities governed by the two-dimensional incompressible Navier-Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, e cient and accurate.

An improved (9,5) higher order compact scheme for the transient two-dimensional convection–diffusion equation

“…We also carry out error analysis wherever analytical solutions are available. Results are compared with those obtained by the (9; 9) and (9; 5) scheme of Reference [5], and well established results from various other schemes.…”

“…On the other hand, the higher order accuracy of the HOC methods combined with the compactness of the di erence stencil yields highly accurate numerical solutions on relatively coarser grids with greater computational e ciency. Of these, Kalita et al [5] developed a class of HOC schemes for the unsteady 2-D convection-di usion equation with variable coe cients which requires either nine or ÿve points at the nth and (n+1)th time levels. Accordingly, they are termed as (9; 5), (9; 9) or (5; 9) schemes.…”

“…Various schemes have been developed for problems based on Equation (1). Quite recently, HOC ÿnite di erence methods [2,3,[5][6][7][8][9][11][12][13][14][15] have become quite popular as against the other lower order accurate schemes which require high mesh reÿnement and hence are computationally ine cient. On the other hand, the higher order accuracy of the HOC methods combined with the compactness of the di erence stencil yields highly accurate numerical solutions on relatively coarser grids with greater computational e ciency.…”

Iterative methods and high-order difference schemes for 2D elliptic problems with mixed derivative

“…The fourth-order compact schemes have also been used in the numerical simulation of incompressible Navier-Stokes equations with good results [1,12,13,14]. Extension of high-order compact schemes to time-dependent problems are reported in [15,16,17].…”

Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation

“…In the past two decades, HOC schemes have been well studied. Various fourth order compact (FOC) finite difference schemes have been developed for Poisson equations, convection-diffusion equations, and Navier-Stokes equations [6] [7] [8] [9].…”