A Lagrangian-Eulerian method with adaptively local ZooMing and Peak/Valley capturing approach to solve two-dimensional advection-diffusion transport equations

A Langrangian-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, is presented to solve two-dimensional advection-diffusion transport equations. this approach and the associated computer code, 2DLEZOOMPC, were developed to circumvent the difficulties associated with the EPCOF scheme, developed earlir by the authors, when it was extended from a one-dimensional space to a multi-dimensional space. In EPCOF, all the nodes, including global nodes and fine-grid nodes, of the previous time are forward tracked for both determining rough elements and exactly capuring peaks and valleys. After kicking off those unnecessary nodes, a subset of the forward-tracked nodes are activated to preserve the shape of spatial distribution of the quantity of interest (e.g. concentration in the solute transport). The accurate results of applying EPCOF to solving two one-dimensional bench-mark problems under a variety of conditions have shown the capability of this scheme to eliminate all types of numerical errors associated with the advection therm and to keep the maximum computational error to be within the prerscribed error tolerance. However, difficulties arose when the EPCOF scheme was extended to a multi-dimensional space mainly due to the geometric difference between a one-dimensional space and a multi-dimensional space. To avoid these geometric difficulties, we modified the EPCOF cheme 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 triangulation 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 associated with the diffusion zooming in a multi-dimensional space. As a resule, not only is the geometrical problem resolved, but also the spirit of EPCOF is retained. Application of 2DLEZOOMPC to solving three two-dimensional bench-mark problems indicates it yields extremely accurate results for all the test cases. 2DLEZOOMPCcould solve advection-diffusion transport problems accurately to within any prescribed error tolerance by using mesh Peclet numbers ranging from 0 to ∞ and very large time-step sizes as well as coarse global grid sizes. The size of time-step is related to both the diffusion coefficients and mesh sizes. Hence, it is limited only by the diffusion solver. Extension of this approach to a tree-dimensional space will contain only implementation complexity but neither conceptual nor implementation difficulties. Details of the three-dimensional computer code, 3DLEZOOMPC, is to be presented in the companion paper.