February 23, 2018

# An Improved Numerical Method For Solving Depth-Resolved Biogeochemical Models

A method of alternating characteristics with application to advection-dominated environmental systems.

### The Science

Scientists at Lawrence Berkeley National Lab (LBNL) propose a numerical integration method, termed the method of alternating characteristics (MAC), to efficiently and accurately solve systems of partial differential equations that arise in modeling environmental processes. They highlight the advantages of MAC with emphasis on advection-dominated environmental systems with biogeochemical reactions.

### The Impact

The proposed method is uniquely suited for solving depth-resolved models of advection-dominated environmental systems with biogeochemical reactions and offers advantages in performance over other numerical integration schemes that often require considerable computational resources.

### Summary

Here, LBNL scientists present a numerical integration method for solving systems of partial differential equations (PDEs) that arise in modeling environmental processes undergoing advection and biogeochemical reactions. The salient feature of these PDEs is that all partial derivatives appear in linear expressions. As a result, the system can be viewed as a set of ordinary differential equations (ODEs), albeit each one along a different characteristic. The proposed method, termed MAC, then consists of alternating between equations and integrating each one step-wise along its own characteristic, thus creating a customized grid on which solutions are computed. Since the solutions of such PDEs are generally smoother along their characteristics, the method offers the potential of using larger time steps while maintaining accuracy and reducing numerical dispersion. The advantages in efficiency and accuracy of the proposed method are demonstrated in two illustrative examples that simulate depth-resolved reactive transport and soil carbon cycling.

### Principal Investigator

William Riley

Lawrence Berkeley National Laboratory

wjriley@lbl.gov

### Program Manager

Daniel Stover

U.S. Department of Energy, Biological and Environmental Research (SC-33)

Environmental System Science

daniel.stover@science.doe.gov

### Funding

This material is based on work supported by the Terrestrial Ecosystem Science Program of the Office of Biological and Environmental Research, within the U.S. Department of Energy (DOE) Office of Science, under contract number DE-AC02-05CH11231. K.G. acknowledges support from the Office of Science Graduate Student Research (SCGSR) program, supported and managed by the DOE Office of Science’s Office of Workforce Development for Teachers and Scientists. The SCGSR program is administered by the Oak Ridge Institute for Science and Education (ORISE) for the DOE. ORISE is managed by Oak Ridge Associated Universities under contract number DE-SC0014664.

### References

Georgiou, K., J. Harte, A. Mesbah, and W. J. Rile.
"A method of alternating characteristics with application to advection-dominated environmental systems."
*Computational Geosciences*
**223 **
851–865
(2018).
https://doi.org/10.1007/s10596-018-9729-5.