On a monotonicity preserving Eulerian-Lagrangian localized adjoint method for advection-diffusion equations

Eulerian-Lagrangian localized adjoint methods (ELLAMs) provide a general approach to the solution of advection-dominated advection-diffusion equations allowing large time steps while maintaining good accuracy. Moreover, the methods can treat systematically any type of boundary condition and are mass conservative. However, all ELLAMs developed so far suffer from non-physical oscillations and are usually implemented on structured grids. In this paper, we propose a finite volume ELLAM which incorporates a novel correction step rendering the method monotone while maintaining conservation of mass. The method has been implemented on fully unstructured meshes in two space dimensions. Numerical results demonstrate the applicability of the method for problems with highly non-uniform flow fields arising from heterogeneous porous media.