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# A spacetime discontinuous Galerkin method for the two-dimensional unsteady convection-dispersion problems

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## Abstract

The unsteady two-dimensional convection-diffusion (CD) equation, which is the governing equation of the unsteady two-dimensional convection-dispersion problem, as the water contamination problems, has a mixed hyperbolic-parabolic character. When the equation has a strong mixed hyperbolic character, the exact solution is nonsmooth. In this case, the conventional numerical methods give approximate solutions which either oscillate or smear out the sharp front of the exact solution. The spacetime discontinuous Galerkin method (STDGM) is an extention of the space discontinuous Galerkin method (SDGM), applying the discontinuity in the time direction, as well as in space. Both these methods are respective modifications of the standard Galerkin finite element method (SGM). In this paper, the STDGM is applied to solve the CD equation, when the Péclet number has extremely high values, which means a strong mixed hyperbolic character. With this method, three artificial diffusion terms are introduced by modifying the test functions of the finite element method. These functions include the discontinuity int, x andy axis. The results obtained from the analytical solution of the problem are used for testing the numerical solution, applying both the space-discontinuous Galerkin method (SDGM) and the STDGM and are presented in diagrams, from which useful observations, comparisons and conclusions can be drawn.

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Karamouzis, D.N. A spacetime discontinuous Galerkin method for the two-dimensional unsteady convection-dispersion problems. Water Resour Manage 6, 35–45 (1992). https://doi.org/10.1007/BF00872186

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### Key words

• Artificial diffusion
• convection-dispersion
• Courant number
• discontinuous Galerkin method
• finite element computational scheme
• mathematical model
• Péclet number
• Petrov-Galerkin
• water contamination