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

Abstract

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.