Abstract
We present two high-order hybridizable discontinuous Galerkin (HDG) formulations combined with high-order diagonally implicit Runge-Kutta schemes to solve one-phase and two-phase flow problems through porous media. The HDG method is locally conservative and allows reducing the size of the global systems due to the hybridization procedure, and the pressure, the saturation and the velocity converge with order P + 1 in L 2-norm, with P being the polynomial degree. In addition, an element-wise post-process can be applied to obtain a convergence rate of P + 2 in L 2-norm for the pressure and saturation. To achieve these rates of convergence the temporal errors should be small enough. For this purpose we combine HDG with high-order diagonally implicit Runge-Kutta (DIRK) temporal schemes. Finally, we present four examples dealing with 2D and 3D problems, and high-order structured and unstructured meshes.
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References
Arbogast, T., Juntunen, M., Pool, J., Wheeler, M.F.: A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocity and continuous capillary pressure. Comput. Geosci. 17(6), 1055–1078 (2013)
Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution, vol. 2. Springer Science & Business Media, Netherlands (2012)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, England (2016)
Chen, Z., Huan, G., Ma, Y.: Computational Methods for Multiphase Flows in Porous Media, vol. 2. SIAM, Philadelphia (2006)
Corey, A.: Hydraulic properties of porous media. Colorado State University, Hydraulic Papers (3) (1964)
Costa-Solé, A., Ruiz-Gironés, E., Sarrate, J.: An HDG formulation for incompressible and immiscible two-phase porous media flow problems. Int. J. Comput. Fluid D. 33(4), 137–148 (2019)
Donaldson, E., Chilingarian, G., Yen, T.: Enhanced oil recovery, II: processes and operations, vol. 17. Elsevier, Amsterdam/New York (1989)
Ern, A., Mozolevski, I., Schuh, L.: Discontinuous Galerkin approximation of two-phase flows in heterogeneous porous media with discontinuous capillary pressures. Comput. Methods Appl. Mech. Eng. 199(23–24), 1491–1501 (2010)
Fabien, M.S., Knepley, M.G., Rivière, B.M.: A hybridizable discontinuous Galerkin method for two-phase flow in heterogeneous porous media. Int. J. Numer. Meth. Eng. 116(3), 161–177 (2018)
Gargallo-Peiró, A., Roca, X., Peraire, J., Sarrate, J.: Optimization of a regularized distortion measure to generate curved high-order unstructured tetrahedral meshes. Int. J. Numer. Meth. Eng. 103(5), 342–363 (2015)
Gargallo-Peiró, A., Roca, X., Peraire, J., Sarrate, J.: A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Eng. 106(13), 1100–1130 (2016). Nme.5162
Jamei, M., Ghafouri, H.: A novel discontinuous Galerkin model for two-phase flow in porous media using an improved IMPES method. Int. J. Numer. Method H. 26(1), 284–306 (2016)
Jamei, M., Raeisi Isa Abadi, A., Ahmadianfar, I.: A Lax-Wendroff-IMPES scheme for a two-phase flow in porous media using interior penalty discontinuous Galerkin method. Numer. Heat Tr. B.-Fund. 75(5), 325–346 (2019)
Kennedy, C.A., Carpenter, M.H.: Diagonally implicit Runge-Kutta methods for ordinary differential equations a review. Technical Report NASA/TM-2016-219173, NASA (2016)
Kirby, R., Sherwin, S., Cockburn, B.: To CG or to HDG: a comparative study. J. Sci. Comput. 51(1), 183–212 (2012)
Klieber, W., Rivière, B.: Adaptive simulations of two-phase flow by discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 196(1–3), 404–419 (2006)
Montlaur, A., Fernandez-Mendez, S., Huerta, A.: High-order implicit time integration for unsteady incompressible flows. Int. J. Numer. Meth. Fl. 70(5), 603–626 (2012)
Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations. J. Comput. Phys. 228(9), 3232–3254 (2009)
Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection–diffusion equations. J. Comput. Phys. 228(23), 8841–8855 (2009)
Persson, P.O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, p. 112 (2006)
Roca, X., Ruiz-Gironés, E., Sarrate, J.: EZ4U: mesh generation environment (2010)
Ruiz-Gironés, E., Roca, X., Sarrate, J.: High-order mesh curving by distortion minimization with boundary nodes free to slide on a 3D CAD representation. Comput. Aided Des. 72, 52–64 (2016)
Sevilla, R., Huerta, A.: Tutorial on hybridizable discontinuous Galerkin (HDG) for second-order elliptic problems. In: Advanced Finite Element Technologies, pp. 105–129. Springer, Cham (2016)
Warburton, T.: An explicit construction of interpolation nodes on the simplex. J. Eng. Math. 56(3), 247–262 (2006)
Acknowledgements
This work has been supported by FEDER and the Spanish Government, Ministerio de Economía y Competitividad grant project contract CTM2014-55014-C3-3-R, Ministerio de Ciencia Innovación y Universidades grant project contract PGC2018-097257-B-C33, and the grant BES-2015-072833.
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Costa-Solé, A., Ruiz-Gironés, E., Sarrate, J. (2021). One-Phase and Two-Phase Flow Simulation Using High-Order HDG and High-Order Diagonally Implicit Time Integration Schemes. In: Asensio, M.I., Oliver, A., Sarrate, J. (eds) Applied Mathematics for Environmental Problems. SEMA SIMAI Springer Series(), vol 6. Springer, Cham. https://doi.org/10.1007/978-3-030-61795-0_4
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