Numerical resolution of advection-diffusion equations on connected graphs requires careful treatment in implementing internal boundary conditions at junctions. This study proposes a simple conforming Petrov-Galerkin finite element method with compact stencil, referred to as CPGFEM2, which effectively solves the non-conservative advection-diffusion equations on connected graphs. The CPGFEM2 utilizes the fitting technique in the spatial discretization so that the junctions are consistently handled as implicit internal boundary conditions. A selective lumping algorithm in conjunction with the local  -scheme is applied in the temporal discretization. The CPGFEM2 is verified through several test problems. The CPGFEM2 is successfully applied to numerical analysis of conservative solute transport in an existing agricultural drainage system, showing its applicability to real problems.

Understanding transport processes in surface water systems involving farmlands is a fundamental research topic in environmental hydraulics. Analysis of water flows in agricultural drainage canals, ubiquitous in the surface water systems in Japan, is an essential step to assess their environmental and ecological states. This paper carries out numerical analysis of steady water flows in a vegetated agricultural drainage canal that collects runoff from paddy fields. Water flows in the canal are described on the basis of the shallow water theory, which leads to the governing equations of the water surface elevation and discharge. Two shallow water models, the dynamic wave model and diffusion wave model, are applied to analyze steady flows in the canal. The parameters to specify the drag force caused by aquatic vegetation in the canal are determined so that the observed water surface profile is accurately reproduced using the dynamic wave model. The resulting dynamic wave model with the estimated parameters reasonably well reproduces the observed water surface profile. The computational results also suggest the validity of the diffusion wave model in simulating the steady flows in the canal.

An extended mathematical model governing shallow water flows in vegetated open channels, referred to as the 1-D extended shallow water equations (1-D ESWEs), is presented in this paper as a physically more consistent alternative to the conventional 1-D SWEs. Emerged and submerged aquatic plants in channels are considered in the 1-D ESWEs as momentum sinks with appropriately defined water fractions. The 1-D ESWEs improve the double-counting problem on the momentum losses in the 1-D SWEs originating from the additivity assumption of the wall friction and vegetation drag forces without considering the water fraction. The 1-D ESWEs are applied to steady and unsteady numerical simulation of water flows in an agricultural drainage canal in Japan. The computational results demonstrate advantages of the 1-D EWSEs over the 1-D SWEs, reducing overestimation of the water depth. Impacts of the vegetation modeling on the flows in the canal are also assessed through the unsteady simulation.