A LAGRANGIAN-EULERIAN METHOD WITH ADAPTIVELY LOCAL ZOOMING APPROACH TO SOLVE 3-DIMENSIONAL ADVECTION-DIFFUSION TRANSPORT-EQUATIONS

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 tr acking, adaptively local zooming, peak/valley capturing, and slave poi nt utilization, to solve three-dimensional advection-diffusion transpo rt equations. This approach and the associated computer code, 3DLEZOOM PC, were developed to circumvent the difficulties associated with the Exact Peak Capturing and Oscillation-Free (EPCOF) scheme, developed ea rlier by the authors, when it was extended from a one-dimensional spac e 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 t ypes of numerical errors associated with the advection term and to kee p the maximum computational error to be within the prescribed error to lerance. However, difficulties arose when the EPCOF scheme was extende d to a multi-dimensional space mainly due to the geometry. To avoid th ese geometric difficulties, we modified the EPCOF scheme and named the modified scheme LEZOOMPC. LEZOOMPC uses regularly local zooming for r ough elements and peak/valley capturing within subelements to resolve the problems of tetrangulation and boundary source as well as to prese rve the shape of concentration distribution. In addition, LEZOOMPC emp loys the concept of 'slave points' to deal with the compatibility prob lem in the diffusion zooming of the Eulerian step. As a result, not on ly is the geometrical problem resolved, but also the spirit of EPCOF i s retained. Application of 3DLEZOOMPC to solving an advection-decay an d a boundary source benchmark problems indicates its capability in sol ving advection transport problems accurately to within any prescribed error tolerance by using mesh Courant number ranging from 0 to infinit y. Demonstration of using 3DLEZOOMPC to solve aa advection-diffusion b enchmark problem shows how the numerical solution is improved with the increment of the diffusion zooming factors. 3DLEZOOMPC could solve ad vection-diffusion transport problems accurately by using mesh Peclet n umbers ranging from 0 to infinity and very large time-step size. The s ize of time-step is related to both the diffusion coefficients and mes h sizes. Hence, it is limited only by the diffusion solver. The applic ation of this approach to a two-dimensional space has been demonstrate d earlier in the paper entitled 'A Lagrangian-Eulerian method with ada ptively local zooming and peak/valley capturing approach to solve two- dimensional advection-diffusion transport equations'. (C) 1998 John Wi ley & Sons, Ltd.