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.