A third‐order semi‐implicit finite difference method for solving the one‐dimensional convection‐diffusion equation

Abstract

SUMMARYA third-order semi-implicit five-point finite difference method is developed to solve the one-dimensional convection-diffusion equation, using the 'weighted' modified equation method. It is shown to have a large stability region, to be very accurate and computationally fast.

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

“…On the other hand, the present (9; 5) scheme gives a remarkably accurate solution (with pulse height 0.166325 against the analytical value of 0.166667), free from di usion or anti-di usion, yielding a pulse which is indistinguishable from the exact one as seen from Figure 2(b). It will be worthwhile to compare the CPU time-wise e ciency of the present (9; 5) scheme with the (9; 9) [5] scheme and the nine point scheme of Noye and Tan [6,7] time. The CPU time ratio of the Noye and Tan scheme and (9; 9) [5] scheme to that of FTCS scheme is 447.12 and 5.257, respectively.…”

“…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.…”

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

“…Several higher order implicit schemes for the one dimensional (1-D) time dependent convection-di usion problems were developed by Noye and Tan [12]. Later on, they also developed a nine-point scheme of third order spatial and second order temporal accuracy for the 2-D convection-di usion equations with constant coe cients [13].…”

A high-order compact ADI method for solving three-dimensional unsteady convection-diffusion problems

“…Recently, due to their high order, compactness and high resolution, we have seen increasing popularity for high-order compact (HOC) difference methods in computational fluid dynamics. After deriving several high-order implicit schemes for unsteady 1-D convection-diffusion equations [10], Noye and Tan proposed a compact nine-point HOC implicit scheme for unsteady 2-D convection-diffusion equations with constant coefficients [11]. The scheme is third-order accurate in space and second-order accurate in time and has a large zone of stability.…”