A Lagrangian-Eulerian method with adaptively local ZOOMing approach to solve three-dimensional advection-diffusion transport equations

Abstract

We present a Lagrangian-Eulerian method with adaptively local ZOOMing and Peak/valley Capturing approach (LEZOOMPC), consisting of advection-diffusion decoupling, backward particle tracking, forward particle tracking, adaptively local zooming, peak/valley capturing, and slave point utilization, to solve three-dimensional advection-diffusion transport equations. This approach and the associated computer code, 3DLEZOOMPC, were developed to circumvent the difficulties associated with the Exact Peak Capturing and Oscillation-Free (EPCOF) scheme, developed earlier by the authors, when it was extended from a one-dimensional space to a three-dimensional space. The accurate results of applying EPCOF to solving two one-dimensional benchmark problems under a variety of conditions have shown the capability of this scheme to eliminate all types of numerical errors associated with the advection term and to keep the maximum computational error to be within the prescribed error tolerance. However, difficulties arose when the EPCOF scheme was extended to a multi-dimensional space mainly due to the geometry. To avoid these geometric difficulties, we modified the EPCOF scheme and named the modified scheme LEZOOMPC. LEZOOMPC uses regularly local zooming for rough elements and peak/valley capturing within subelements to resolve the problems of tetrangulation and boundary source as well as to preserve the shape of concentration distribution. In addition, LEZOOMPC employs the concept of 'slave points' to deal with the compatibility problem in the diffusion zooming of the Eulerian step. As a result, not only is the geometrical problem resolved, but also the spirit of EPCOF is retained. Application of 3DLEZOOMPC to solving an advection-decay and a boundary source benchmark problems indicates its capability in solving advection transport problems accurately to within any prescribed error tolerance by using mesh Courant number ranging from 0 to infinity. Demonstration of using 3DLEZOOMPC to solve an advection-diffusion benchmark problem shows how the numerical solution is improved with the increment of the diffusion zooming factors. 3DLEZOOMPC could solve advection-diffusion transport problems accurately by using mesh Peclet numbers ranging from 0 to infinity and very large time-step size. The size of time-step is related to both the diffusion coefficients and mesh sizes. Hence, it is limited only by the diffusion solver. The application of this approach to a two-dimensional space has been demonstrated earlier in the paper entitled 'A Lagrangian-Eulerian method with adaptively local zooming and peak/valley capturing approach to solve two-dimensional advection-diffusion transport equations'.

Modified Mixed Lagrangian–Eulerian Method Based on Numerical Framework of MT3DMS on Cauchy Boundary

“…After the MOC, MMOC, and HMOC included in MT3DMS and previous ELM, such as the LEZOOMPC and Neuman methods, solve the Lagrangian concentrations at all nodes with the method of characteristics from Equations and ; Equation is solved to obtain the final concentration distribution at the next time step, together with the following initial and boundary conditions, by applying FDM (Zheng and Wang ) or FEM (Neuman ; Yeh ; Cheng et al ): where F ( x ) is the initial concentration, f ( t ) is the incoming concentration on the Cauchy boundary, and L is the length of the porous medium.…”

“…Four examples are illustrated here to compare the overall performance of the proposed method with the Neuman approach (Neuman ), the LEZOOMPC approach (Cheng et al ), and, in addition, upstream FDM, MOC, MMOC, HMOC, and TVD included in MT3DMS (Zheng and Wang ) on the Cauchy boundary condition, for mesh Péclet numbers ranging from <1 to infinity and for various Courant numbers.…”

“…Within the numerical framework, following the ELMs of MT3DMS, a sophisticated and advanced approach was introduced to accurately solve solute transport problems without requiring the tracking of a substantial number of real particles. This approach is the Lagrangian–Eulerian method with adaptively local zooming and a peak/valley capturing approach (LEZOOMPC) (Cheng et al ), which employs FEM to calculate the dispersion equation, while MOC, MMOC, and HMOC in MT3DMS use FDM to solve the dispersion equation. Cheng et al () presented the LEZOOMPC consisting of advection–diffusion decoupling, backward and forward particle tracking, adaptive local zooming, a peak/valley‐capturing scheme, and slave point utilization.…”

“…In summary, the Neuman method (, ) and LEZOOMPC (Cheng et al ) assume, when dealing with the Cauchy boundary condition, that the flow regime is advection‐dominated, and thus the method performs well when simulating advection‐dominated flow regimes on a Cauchy boundary. But when simulating dispersion‐dominated flow regimes on a Cauchy boundary, they perform poorly because the advection‐dominant assumption is violated.…”

“…However, the ELMs, including the Neuman approach, LEZOOMPC, as well as the MOC, MMOC, and HMOC included in MT3DMS have been applied mainly under a Dirichlet boundary condition, with few exceptions (Neuman , ; Konikow et al ; Cheng et al ; Lowry and Li ; Herrera et al ). However, in the field, it would rarely be realistic to apply the Dirichlet boundary condition, because it is rare that the concentration of a source at a point will remain fixed in time regardless of changes in the accompanying flow field or in the local concentration gradient (Konikow ).…”