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

Abdelkader Mojtabi (1) , Michel Deville (2)

1 IMFT - Institut de mécanique des fluides de Toulouse
2 EPFL - Ecole Polytechnique Fédérale de Lausanne

In 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

Mots clés: Separation of variables; Finite element method; Exponential layer; Symbolic computation; Asymptotic development

Domaines: Mécanique des fluides [physics.class-ph]

Abdelkader Mojtabi, Michel Deville. One-dimensional linear advection–diffusion equation: Analytical and finite element solutions. Computers and Fluids, 2015, 107, pp.189-195. ⟨10.1016/j.compfluid.2014.11.006⟩. ⟨hal-01331727⟩