Scientific Investigations Report 2009-5034


Prepared in cooperation with
South Florida Water Management District

Development and Implementation of a Transport Method for the Transport and Reaction Simulation Engine (TaRSE) based on the Godunov-Mixed Finite Element Method

By Andrew I. James1,2, James W. Jawitz1, and Rafael Muņoz-Carpena1

1University of Florida, Gainesville
2Soil and Water Engineering Technology, Inc., University of Florida, Gainesville


      Select an option:

      Purpose and Scope
      Methods and Model Development
      Previous Work
            Exact Solutions
            Non-Operator-Splitting Numerical Models
            Operator-Splitting Methods
Numerical Scheme Development
      Transport Equations
            Discretization and Calculation
                  Discretization for the Advective Step
                  Calculation of the Numerical Advective Flux
            Hybridized Mixed Finite Element for Dispersion
            Simplification of the Solution Matrices
            Notes on the Linear System
Verification of the Algorithm
      One-Dimensional Problem
      Two-Dimensional Problem
References Cited
Appendix 1: Transport Model Structure and Application Details
Appendix 2: Example Input Files


A model to simulate transport of materials in surface water and ground water has been developed to numerically approximate solutions to the advection-dispersion equation. This model, known as the Transport and Reaction Simulation Engine (TaRSE), uses an algorithm that incorporates a time-splitting technique where the advective part of the equation is solved separately from the dispersive part. An explicit finite-volume Godunov method is used to approximate the advective part, while a mixed-finite element technique is used to approximate the dispersive part. The dispersive part uses an implicit discretization, which allows it to run stably with a larger time step than the explicit advective step. The potential exists to develop algorithms that run several advective steps, and then one dispersive step that encompasses the time interval of the advective steps. Because the dispersive step is computationally most expensive, schemes can be implemented that are more computationally efficient than non-time-split algorithms. This technique enables scientists to solve problems with high grid Peclet numbers, such as transport problems with sharp solute fronts, without spurious oscillations in the numerical approximation to the solution and with virtually no artificial diffusion.

Suggested Citation:

James, A.I., Jawitz, J.W., and Muņoz-Carpena, Rafael, 2009, Development and Implementation of a Transport Method for the Transport and Reaction Simulation Engine (TaRSE) based on the Godunov-Mixed Finite Element Method: U.S. Geological Survey Scientific Investigations Report 2009-5034, 40 p.

For more information, contact:

Rafael Muņoz-Carpena at

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