An exact peak capturing and Oscillation‐Free Scheme to solve advection‐dispersion transport equations

Abstract

An exact peak capturing and essentially oscillation‐free (EPCOF) algorithm, consisting of advection‐dispersion decoupling, backward method of characteristics, forward node tracking, and adaptive local grid refinement, is developed to solve transport equations. This algorithm represents a refinement of LEZOOM, developed earlier by the senior author. In LEZOOM, a predetermined number of evenly spaced, hidden nodes was zoomed for a sharp front element, while in the EPCOF scheme, a subset of forwardly tracked nodes is zoomed. The number and location of this subset were automated. As a result, the peaks and valleys are captured exactly; and the ancillary problems of spurious oscillation, numerical dispersion, and phase errors are alleviated. Means of checking accumulated mass balance errors are provided. Application of the algorithm to two one‐dimensional benchmark problems under a variety of conditions indicated that it completely eliminated peak clipping, spurious oscillation, phase error, and numerical dispersion. It yielded identical results, within the error tolerance, to exact solutions for all 19 test cases. Accumulated mass balance errors are extremely small for all 19 cases. The EPCOF scheme could solve the advective transport problems exactly, within any prescribed error tolerance, using mesh Peclet numbers ranging from 0 to infinity and very large mesh Courant numbers. The size of mesh Courant number is limited only by the accuracy requirement of the dispersion solver. Extension of this approach to multidimensional problems does not pose any conceptual difficulty and should alleviate the grid orientation trouble associated with such problems.