An Eulerian-Lagrangian control volume scheme for two-dimensional unsteady advection-diffusion problems

We develop a mass conservative Eulerian-Lagrangian control volume scheme (ELCVS) for the solution of the transient advection-diffusion equations in two space dimensions. This method uses finite volume test functions over the space-time domain defined by the characteristics within the framework of the class of Eulerian-Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight-node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well-received classes of numerical methods for the equations considered.