Solving the advection-diffusion equation with the Eulerian–Lagrangian localized adjoint method on unstructured meshes and non uniform time stepping

• Published in: Journal of computational physics Vol. 208; no. 1; pp. 384 - 402
• Main Authors: Younes, A., Ackerer, P.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.09.2005; Elsevier
• Subjects: ADE; Adjoints; Advection-diffusion equation; Algorithms; Computational techniques; Diffusion; Dispersions; ELLAM; Exact sciences and technology; Finite element method; Forward tracking; Mathematical analysis; Mathematical methods in physics; Mathematical models; Physics; Triangular meshes; Unstructured meshes; Unstructured meshes; ELLAM; ADE; Forward tracking; Triangular meshes; Peclet number; Triangulation; Digital simulation; Adjoint method; Advection diffusion equation; Calculation methods; Galerkin-Petrov method; Finite element method; Algorithms; Mathematical models; Calculation; Diffusion; Lagrangian method; Mathematical physics
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2005.02.019

Eulerian–Lagrangian localized adjoint method (ELLAM) is used to solve the advection diffusion equation (ADE) which is a very common mathematical model in physics. In this work, ELLAM is extended to triangular meshes. Standard integration schemes, which perform well for rectangular grids, are improved to reduce oscillations with unstructured triangulations. Numerical experiments for grid Peclet numbers ranking from 1 to 100 show the efficiency of the developed scheme. A new algorithm is also developed in order to avoid excessive numerical diffusion when using many time steps with the ELLAM. The basic idea of this approach is to keep the same characteristics for all time steps and to interpolate only the concentration variations due to the dispersion process at the end of each time step. Although ELLAM requires a lot of integration points for unstructured meshes, it remains a competitive method when using a single or many time steps compared to explicit discontinuous Galerkin finite element method.