A Family of Eulerian–Lagrangian Localized Adjoint Methods for Multi-dimensional Advection-Reaction Equations

A numerical modeling of multicomponent compressible flows in porous media with multiple wells by an Eulerian-Lagrangian method

“…To develop a numerical scheme that reflects the advective and reactive nature of the transport equation, we use the idea of the Eulerian-Lagrangian localized adjoint method [4,22] to choose the test functions z(y, θ) to satisfy the hyperbolic part of the adjoint equation. In particular, we include the sink term in the governing transport equations in the adjoint equation to define the test functions…”

“…Sincen i = n i at production wells, a treatment of sinks as a reaction term seems more natural [22]. Let y = r(θ ; x, t k ) be the characteristic defined by the initial-value problem of the differential equation…”

“…Hence, numerical means have to be used to track the characteristics. An Euler or Runge-Kutta quadrature with a possibly use of micro time steps in tracking is often used [22].…”

“…But they are expensive to solve per time step and are very diffusive. Moreover, they often have spurious grid orientation effect [6,8,19,22,24].…”

“…They naturally cancel the majority of the temporal error with the spatial error due to the advection, which was intended in some Eulerian methods with partial success [2,11]. Thus, they generate accurate numerical solutions even if large time steps and coarse spatial grids are used, and are very competitive with many numerical methods [8,22]. Furthermore, they eliminate the excessive numerical diffusion present in upstream-weighted, largescale numerical simulators in industrial production.…”