One-dimensional linear advection–diffusion equation: Analytical and finite element solutions

Abstract

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15920 a b s t r a c tIn this paper, a time dependent one-dimensional linear advection-diffusion equation with Dirichlet homogeneous boundary conditions and an initial sine function is solved analytically by separation of variables and numerically by the finite element method. It is observed that when the advection becomes dominant, the analytical solution becomes ill-behaved and harder to evaluate. Therefore another approach is designed where the solution is decomposed in a simple wave solution and a viscous perturbation. It is shown that an exponential layer builds up close to the downstream boundary. Discussion and comparison of both solutions are carried out extensively offering the numericist a new test model for the numerical integration of the Navier-Stokes equation.

On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations

“…with periodic boundaries. According to Mojtabi et al [30], the analytical solution to (5.4) is ρ * (x, t) = sin (π (x − ut)) exp −µπ 2 t .…”

Study of one‐dimensional contaminant transport in soils using fractional calculus

“…Mojtabi and Deville have found analytical and numerical solutions with their comparisons for advection-diffusion equations in one-dimension using boundary conditions of Dirichlet homogeneous type. 10 Grant and Wilkinson 11 have solved homogeneous ADE equation with advective fluid velocity and variable diffusion tensor using absorbing boundary. More impactful work related to the advectiondiffusion phenomenon can be seen in the references.…”

High Order Finite Difference Schemes for Solving Advection-Diffusion Equation

“…Advection Diffusion Equation arises in a wide range of science and industry such as biology, astrophysics, aerospace sciences, environmental problems and particularly in fluid dynamics and transport problems. So this equation has been the focus of many studies [1][2][3][4][5][6][7][8][9][10][11][12][13].…”

“…(5). Journal of Computer Science & Computational Mathematics, Volume 7, Issue 2, June 2017 DOI: 10.20967/jcscm.2017.02.004Therefore at the ( , ) ℎ grid point, the partial differential equation actually being solved bythe formula(3This equation may be written in a more convenient form by the following procedure, which converts all time derivatives except into the spatial derivatives. For elimination term∆ we differentiate with respect to time from equation(6) and then multiplying by −…”