A preliminary investigation on an ELLAM scheme for linear transport equations

We present an Eulerian-Lagrangian localized adjoint method (ELLAM) for linear advection-reaction partial differential equations in multiple space dimensions. We carry out numerical experiments to investigate the performance of the ELLAM scheme with a range of well-perceived and widely used methods in fluid dynamics including the monotonic upstream-centered scheme for conservation laws (MUSCL), the minmod method, the flux-corrected transport method (FCT), and the essentially non-oscillatory (ENO) schemes and weighted essentially non-oscillatory (WENO) schemes. These experiments show that the ELLAM scheme is very competitive with these methods in the context of linear transport PDEs, and suggest/justify the development of ELLAM-based simulators for subsurface porous medium flows and other applications.