Environmental Fluid Mechanics

, Volume 14, Issue 4, pp 711–742 | Cite as

Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers

  • Kaveh Zamani
  • Fabián A. Bombardelli
Original Article


All numerical codes developed to solve the advection–diffusion-reaction (ADR) equation need to be verified before they are moved to the operational phase. In this paper, we initially provide four new one-dimensional analytical solutions designed to help code verification; these solutions are able to handle the challenges of the scalar transport equation including nonlinearity and spatiotemporal variability of the velocity and dispersion coefficient, and of the source term. Then, we present a solution of Burgers’ equation in a novel setup. Proposed solutions satisfy the continuity of mass for the ambient flow, which is a crucial factor for coupled hydrodynamics-transport solvers. By the end of the paper, we solve hypothetical test problems for each of the solutions numerically, and we use the derived analytical solutions for code verification. Finally, we provide assessments of results accuracy based on well-known model skill metrics.


Advection–diffusion-reaction equation Advection–diffusion-equation (ADE) Analytical solution Nonlinear partial differential equation (NPDE)  Scalar transport equation Verification and validation Model skill assessment 



This work was possible thanks to funding provided in part by the California Department of Water Resources (DWR), with Dr. J. Anderson, Dr. F. Chung and T. Smith as Program Managers. Authors are grateful to Drs E. Ateljevich and J. Anderson from DWR, who provided help during the completion of this work. The Authors would also like to thank Profs. T. R. Ginn, R. D. Guy and M. Hafez for reading an early revision of the manuscript and for providing notable suggestions to improve it. Dr. P. J. Roache provided the authors with invaluable insights in the distinction between code verification and solution verification methods.


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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of California, DavisDavisUSA

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