Original Article

Environmental Fluid Mechanics

, Volume 14, Issue 4, pp 711-742

First online:

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

  • Kaveh ZamaniAffiliated withDepartment of Civil and Environmental Engineering, University of California, Davis
  • , Fabián A. BombardelliAffiliated withDepartment of Civil and Environmental Engineering, University of California, Davis Email author 

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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