High-Order Finite Difference Schemes for Solving the Advection-Diffusion Equation

Abstract

Up to tenth-order finite difference schemes are proposed in this paper to solve one-dimensional advection-diffusion equation. The schemes based on high-order differences are presented using Taylor series expansion. To obtain the solutions, up to tenth-order finite difference schemes in space and a fourth-order Runge-Kutta scheme in time have been combined. The methods are implemented to solve two problems having exact solutions. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of the current methods. The techniques are seen to be very accurate in solving the advection-diffusion equation for 5 Pe ≤. The produced results are also seen to be more accurate than some available results given in the literature.

Fourth-order compact finite difference method for solving two-dimensional convection–diffusion equation

“…In terms of the above reasons, a compact finite difference scheme is desired to solve lots of differential equations numerically [13][14][15][16]. One can compute more accurate solutions using limited grid sizes through developing high-order compact finite difference schemes.…”

“…In recent years, the high accuracy compact difference method has attracted more and more attention; see [18][19][20][21][22]. Using a Taylor series expansion, Sari et al [14] developed a tenth-order finite difference scheme, proposed to solve one-dimensional advection-diffusion equation. Gurarslan et al [16] presented a sixth-order compact difference scheme in space and a fourth-order Runge-Kutta scheme in time to produce numerical solutions of the one-dimensional advection-diffusion equation, it has been seen to be very accurate in solving the contaminant transport equation for Pe ≤ 5.…”

Accurate Simulation of Contaminant Transport Using High-Order Compact Finite Difference Schemes

“…Remarkable research studies have been conducted in order to solve advection-dispersion equation numerically like method of characteristic with Galerkin method [2], finite difference method [3][4][5], high-order finite element techniques [6], high-order finite difference methods [7][8][9][10][11][12][13][14][15][16][17][18][19][20], green element method [21], cubic B-spline [22], cubic Bspline differential quadrature method [23], method of characteristics integrated with splines [24][25][26], Galerkin method with cubic B-splines [27], Taylor collocation and Taylor-Galerkin methods [28], B-spline finite element method [29], least squares finite element method (FEMLSF and FEMQSF) [30], lattice Boltzman method [31], Taylor-Galerkin B-spline finite element method [32], and meshless method [33,34].…”

Numerical Solution of Advection-Diffusion Equation Using a Sixth-Order Compact Finite Difference Method

“…For that reason, several alternative methods are proposed in the literature for solving the ADE with high accuracy [11]. These include method of characteristic with Galerkin method (MOCG) [11], finite difference method [12][13][14], high-order finite element techniques [15], high-order finite difference methods [16][17][18][19][20][21][22][23][24], Green-element method [25], cubic B-spline [26], cubic B-spline differential quadrature method (CBSDQM) [27], method of characteristics integrated with splines (MOCS) [28][29][30], Galerkin method with cubic B-splines (CBSG) [31], Taylor-Collocation (TC) and Taylor-Galerkin (TG) methods [32], B-spline finite element method [33], Least squares finite element method (FEMLSF and FEMQSF) [34], Lattice Boltzmann method [35], Taylor-Galerkin B-spline finite element method [36], and meshless method [37,38].…”