Solution of the Advection-diffusion Equation using the Differential Quadrature Method

Abstract

Differential Quadrature Method (DQM) to integrate the one-dimensional Advection-diffusion Equation (ADE) is presented. This method was applied to two examples and the results were compared with the performance of the Explicit Finite Difference Method (EFDM) and Implicit Finite Differences Method (IFDM). Based on the comparison with the exact solution, and both the explicit and implicit finite difference solutions, it was concluded that the DQM provides similar results but less grid points; besides the results are converged quickly. A numerical comparison for a case in which the exact solution is known, DQM gives closer results to the exact values than EFDM and IFDM.

Differential Quadrature Method for Linear Long Wave Propagation in Open Channels

“…A comparison was conducted taking into account the DQM, FVM, FDM, actual measurements, and nonlinear FDM solution performed by Sivapalan et al (1997 Determining the weight coefficients is the most crucial step in the use of DQM. In wave propagation problems in open channels, by making use of Legendre polynomials for weight coefficients and Chebyshev-Gauss-Lobatto points for the numerical discretization, the results can be obtained closer to an analytical solution (Kaya, 2010).…”

“…A recent study (Fung, 2001) has emphasized the application principles of DQM in problems we may face in the area of fluid mechanics and heat transfer. Kaya (2010) has investigated the use of DQM on the solution of Advection Diffusion Equation. Shu et al (2003) are presented local radial basis function-based differential quadrature method.…”

Generalized collocation method for linear and nonlinear convection-diffusion models

“…The Burgers equation is the c a.s. x y t , , Concentration at (a) t=0, (b) 0.5, and (c) 1 (x is the spatial coordinate, t is time, and c is a generic concentration described by model (Kaya, 2010). The spatial domain is normalized in the unit square.)…”

“…Moreover, the generalized collocation method has a simple algorithmic structure and is easy to implement. These properties explain why such a method can be of interest for researchers and engineers, along side other numerical approaches proposed to deal with convection-diffusion models (e.g., Satofuka, 1983;Yeh, 1990, Stefanovic andStefan, 2001;Kanney et al, 2003;Bascia and Tucciarelli, 2004;Herrera et al, 2004;Tsai et al, 2004;Simpson and Landman, 2007;Kaya, 2010).…”

An Efficient Scheme of Differential Quadrature Based on Upwind Difference for Solving Two-dimensional Heat Transfer Problems

“…The heat transfer (convection and diffusion) problems attract many researchers interest and they have wide applications, for example, in energy system, which includes the solar collector, nuclear reactor, heat transfer, natural convective motion of fluid flow…etc. Most numerical simulations for these problems can be currently carried out by using finite differences method(FDM) and finite elements method(FEM) [1,3,6,7,10,11,14,16,17,18,22]. These techniques for solving numerically these problems may be very complex and require a large number of grid points to obtain accurate solutions.…”