A cell-vertex finite volume scheme for solute transport equations in open channel networks

Abstract

A solute particle in a water flow behaves as a stochastic process, which is modeled by a stochastic differential equation. The solute transport equation governing macroscopic dynamics of solute concentration distribution in a locally one-dimensional open channel network is deduced from the Kolmogorov's forward equation associated to the stochastic differential equation. The cell-vertex finite volume method is applied for developing a computational scheme to numerically solve the solute transport equation. A computational domain is divided into a regular mesh, from which a dual mesh is generated. The exact solution to a local two-point boundary value problem is used for evaluating the flux at the interface of each pair of two dual cells. The scheme satisfies the total variation diminishing condition and consistently deals with any singular point such as junctions. The semi-implicit method is applied to temporal integration, and the stability condition for the time increment is presented. A series of test problems is examined in order to verify accuracy and conservative property of the scheme. Sufficiently accurate numerical solutions are obtained for test problems in a one-dimensional interval domain, while solute transport phenomena in an open channel network are correctly reproduced for cases with and without deposition of solute. It is concluded that the cell-vertex finite volume scheme is accurate, stable, and versatile in the numerical analysis of solute transport problems in open channel networks.