Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

Abstract

The advection-diffusion equation has a long history as a benchmark for numerical methods. Taylor-Galerkin methods are used together with the type of splines known as B-splines to construct the approximation functions over the finite elements for the solution of time-dependent advection-diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor-Galerkin methods have been constructed by using both linear and quadratic B-spline shape functions. Results shown by the method are found to be in good agreement with the exact solution.

Exponential B-Spline Solution of Convection-Diffusion Equations

“…When velocity field is complex, changing in time and transport process cannot be analytically calculated, and then numerical approximations to the convection equation are indispensable [4]. They are also important in many branches of engineering and applied science.…”

“…Recently, Spline and B-spline functions together with some numerical techniques have been used in getting the numerical solution of the differential equations. Kadalbajoo and Arora [4] used Taylor-Galerkin methods together with the type of splines known as Bsplines to construct the approximation functions over the finite elements for the solution of time-dependent advection-diffusion problems. Mittal and Jain [1] discussed collocation method based on redefined cubic B-splines basis functions for solving convection-diffusion equation with Dirichlet's type boundary conditions.…”

“…They find their applications in water transfer in soils, heat transfer in draining film, spread of pollutants in rivers, dispersion of tracers in porous media. They are also widely used in studying the spread of solute in a liquid flowing through a tube, long range transport of pollutants in the atmosphere, flow in porous media and many others [1][2][3][4].…”

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].…”