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My collaborators and I just published a new set of equations to simulate groundwater flow and reactive transport.
This work started in 2018, when I was a postdoc at AICS RIKEN in Japan. I came across a paper by the numerical analyst Hiroaki Nishikawa, wherein the one-dimensional diffusion equation was transformed into an advection equation. The approach was called "first order hyperbolic system approach".
Nishikawa showed that using this transformation leads to faster numerical solutions. At that time, I was working on the so-called zero-inertia equation; a highly non-linear equation that describes water flow at the surface of earth. The form of the zero-inertia equation is very similar to the diffusion equation (in fact, the zero-inertia equation is a non-linear diffusion equation). My plan was to transform the zero-inertia equation into a "first order hyperbolic system", which would allow me to solve the equation more efficiently. Since they were so similar, it should be easy to accomplish.
The work became messy very soon. There were many technical problems. The most upsetting one was that the equation became formally non-conservative, which necessitates using more complex solution strategies. After a month, I realized that I needed help. I emailed Adrian Navas Montilla, a computational fluid mechanician at Universidad de Zaragoza, for help. He kindly agreed to take a look at the equations and help me figure out their numerical solution.
We gave up the zero-inertia equation pretty quickly. It was too non-linear and cumbersome to work with. Instead we focused on the acoustic equation and the heat equation. First, we needed to learn how to solve non-conservative systems. We derivedn augmented Roe solutions for such systems.
We summarized our work in a preprint at arXiv.org:
Augmented resolution of linear hyperbolic systems under nonconservative form
Once that technology worked, we took a slightly less non-linear diffusion equation: the Boussinesq equation for groundwater flow. This worked better than the zero-inertia equation, and gave nice results. We finally extended our approach to also cover the general advection-diffusion-reaction equation that simulates contaminant spreading in fluids.
Our accepted paper is found here:
We apply Cattaneo's relaxation approach to the one-dimensional coupled Boussinesq groundwater flow and advection-diffusion-reaction equations, commonly used in engineering applications to simulate contaminant transport in the subsurface. The diffusion-type governing equations are reformulated as a hyperbolic system, augmented by an equation that can be interpreted as a momentum balance. The hyperbolization enables an efficient unified computation of the primary variable and its gradients, for example piezometric head and unit discharge in the Boussinesq equation. An augmented Roe scheme is used to solve the hyperbolic system. The hyperbolized system of equations is studied in a set of steady state and transient test cases with idealized geometry. These test cases confirm the equivalence of the hyperbolic system to its original formulation. The larger time step size of the hyperbolic equation is verified theoretically by means of a stability analysis and numerically in the test cases. Finally, a reach-scale application of flow and transport across a river meander is considered. This application case shows that the performance of the hyperbolic relaxation approach holds for more realistic groundwater flow and transport problems, relevant to water resources management.
The groundwater flow equation assumes that water in the soil behaves similar to ink mixing in water. It "diffuses" from high pressure to low pressure regions. The numerical solution of such equations requires small time steps to ensure correct results. In contrast, the equation describing the movement of a particle transported by water flow can be solved with much larger time steps. We call this type of equation an advection equation. In this paper, we overcome the small time steps of the groundwater flow equation by transforming it such that it looks like an advection equation. We carry out a series of numerical tests to ensure that the modified equation is equivalent to the original one. We further verify that the time step is indeed larger when our modified equation is used.
Wed Mar 3 21:22:55 PST 2021