An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

Abstract

Highly nonlinear advection-dispersion-reaction equations govern numerous transport phenomena. Robust, accurate, and efficient algorithms to solve these equations hold the key to the success of applying numerical models to field problems. This paper presents the development and verification of a computational algorithm to approximate the highly nonlinear transport equations of reactive chemical transport and multiphase flow. The algorithm was developed based on the Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture (LEZOOMPC) scheme. It consists of both backward and forward node tracking, rough element determination, peak/valley capturing, and adaptive local grid refinement. A second-order tracking was implemented to accurately and efficiently track all fictitious particles. Shanks' method was introduced to deal with slowly converging case. The accuracy and efficiency of the algorithm were verified with the Burger equation for a variety of cases. The robustness of the algorithm to achieve convergent solutions was demonstrated by highly nonlinear reactive contaminant transport and multiphase flow problems.

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

“…Also, it should be noted that in the advection‐dominated case, the TVD solutions are not quite comparable to the proposed method, but they can produce similar accurate solutions as the proposed method by employing finer grid sizes by a factor of 2. This result has already been reported in previous works (Yeh et al ; Zhang et al ), which demonstrated that TVD algorithms cannot solve the problem satisfactorily using the same coarse grid and time step sizes used in the LEZOOMPC algorithm, which has a similar fundamental algorithm as the proposed method, that is., an adaptive local grid refinement and peak/valley capture algorithm.…”

“…Comparing TVD with the proposed method, the former exhibits peak‐clipping and valley‐elevation errors in Case 3A. To avoid these errors, the TVD needs much finer grid sizes and smaller time step sizes, as Zhang et al () and Yeh et al () have pointed out. So, in the TVD, grid sizes and time step sizes are reduced by a factor of 5 and 2.4, respectively, to match well with the exact solution (not shown).…”