Numerical Solution of Reactive Transport Equations by Discontinuous Galerkin Method

Prof. Dr. Bülent Karasözen, Middle East Technical University, Ankara (Turkey)

1 Jun 2017, 17:00–18:30; Location: S4|10-1

Protection and sustainable management of water resources is one of the key problems in environmental engineering. In this aspect, accurate modeling and simulating coupled ground and surface water flows is a necessity. Chemically reactive components such as dissolved minerals, colloids, or contaminants are transported by advection and diffusion over long distances through some highly heterogeneous porous media are described by advection diffusion reaction (ADR) equations. Reactive transport modeling is characterized by material flow, transport, and reactions at multiple spatial and time scales. The permeability in heterogeneous porous and fractured media typically varies over orders of magnitude in space and time. Therefore efficient and accurate numerical methods are needed to solve the ADR equations which resolve the wide range in flow velocities and reaction rates to predict spreading in space and mixing of reactive solutes. In the non-stationary case, resolution of spatial layers is more critical since the nature of sharp layers may vary as time progresses, and it is highly possible that there occur also temporal layers in addition to the spatial one.

We consider the following prototype ADR equation as a semi-linear PDE:

\[ \begin{align*}   \frac{\partial u_i}{\partial t}-\epsilon_i\Delta u_i + \beta_i\cdot\nabla u_i + r_i(\vec{u}) &= f_i & \hbox{ in } \; \Omega_i\times (0,T], \\    u_i(x,t) &= g^D_i  & \hbox{on } \; \Gamma_{D,i}\times (0,T], \\ \epsilon_i\frac{\partial u_i}{\partial \vec{n}}(x,t) &= g^N_i  & \hbox{on } \; \Gamma_{N,i}\times (0,T],\\ u_i(x,0) &= u_i^0 & \hbox{ in } \; \Omega_i, \quad  i = 1,\ldots, m \end{align*} \]

The ADR equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) and backward Euler in time.  Using the elliptic reconstruction technique, we derive a posteriori error estimators for the  full discretized system in space and time. We show that our adaptive algorithm is robust in diffusion parameter, similar to the linear non- stationary ADR equations, by demonstrating both the spatial and temporal effectivity indices and convergence rates for a wide range of diffusion parameters including the advection domination case. We also point out that our adaptive algorithm is capable of catching not  only the spatial layers but also the temporal layers.

Joint work with Murat Uzunca,  Department of Industrial Engineering, University of Turkish Aeronautical Association, Ankara, Turkey

Category: CE Seminar


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