Dr. Dawson has authored or co-authored over 200 technical articles in the areas of numerical analysis, numerical methods and parallel computing, with applications to flow and transport in porous media, and shallow water systems. In 2001, he was elected Chair of the Society for Industrial and Applied Mathematics Activity Group on Geosciences. He has served on numerous conference organizing committees, review panels, and editorial boards. He is currently managing editor of Computational Geosciences. In 2011, he was given the Institute for Computational Engineering and Sciences Distinguished Research Excellence Award. He received the Society for Industrial and Applied Mathematics Geosciences Career Prize in 2013. He was named a Fellow of the Society for Industrial and Applied Mathematics in 2016.

In 2024, Dawson received The University of Texas President's Research Impact Award, which recognizes university researchers whose scholarly or creative endeavors have changed lives and the way we look at and understand the world. Dawson's work has been instrumental in developing a simulation code that is used worldwide in coastal ocean modeling and hurricane storm surge predictions. A video highlighting Dawson's work was created for the award ceremony.