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’. © 1998 John Wiley & Sons, Ltd.