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A space-time discontinuous Galerkin method for Boussinesq-type equations. (English) Zbl 1410.76167
Summary: In this work, we propose a new high order accurate, fully implicit space-time discontinuous Galerkin (DG) method for advection-diffusion-dispersion equations, i.e., for so-called Korteweg-de-Vries-type equations. In particular, we focus on Boussinesq-type models for free surface flows, which are used for the modeling of water waves that travel in deep water, where the classical shallow water equations are not valid any more. Our method follows the ideas of the local DG method (LDG) for dispersion equations proposed by J. Yan and C.-W. Shu [SIAM J. Numer. Anal. 40, No. 2, 769–791 (2002; Zbl 1021.65050); J. Sci. Comput. 17, No. 1–4, 27–47 (2002; Zbl 1003.65115)], who used an explicit Runge-Kutta method to integrate their scheme in time. However, such explicit time integrators applied to dispersive equations imply a very severe restriction on the time step, which has to be taken proportional to the cube of the mesh spacing, and which therefore can make even one-dimensional computations prohibitively expensive on fine grids. For the scalar case and with some simplifying assumptions, the scheme presented in this paper can be proven to be unconditionally stable in $$L_{2}$$ norm. Furthermore, our method is based directly on a space-time finite element formulation, which also provides a natural way to discretize third order dispersive terms that contain a mixed space-time derivative. Such terms appear frequently in the context of Boussinesq-type models for free surface flows. We show numerical convergence studies for linear scalar advection-diffusion-dispersion equations and furthermore, we also study the convergence of our method using solitary wave solutions of the nonlinear Boussinesq-type model of P. A. Madson et al. [“A new form of the Boussinesq equations with improved linear dispersion characteristics”, Coast. Eng. 15, No. 4, 371–388 (1991; doi:10.1016/0378-3839(91)90017-b)].

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
STRSCNE
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##### References:
 [1] Abbott, M. B.; McCowan, A. D.; Warren, A. D., Accuracy of short wave numerical models, J. Hydraulic Eng., 110, 10, 1287-1301, (1984) [2] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible -Stokes equations, J. Comput. Phys., 131, 267-279, (1997) · Zbl 0871.76040 [3] Bellavia, S.; Berrone, S., Globalization strategies for Newton-Krylov methods for stabilized FEM discretization of Navier-Stokes equations, J. Comput. Phys., 226, 2, 2317-2340, (2007) · Zbl 1388.76122 [4] Bellavia, S.; Macconi, M.; Morini, B., An affine scaling trust-region approach to bound-constrained nonlinear systems, Appl. Numer. Math., 44, 3, 257-280, (2003) · Zbl 1018.65067 [5] Boussinesq, M. J., Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond, Journal de Mathématiques Pures et Appliquées, 10, 17, 55-108, (1872) · JFM 04.0493.04 [6] Brugnano, L.; Casulli, V., Iterative solution of piecewise linear systems, SIAM J. Sci. Comput., 30, 463-472, (2007) · Zbl 1155.90457 [7] Casulli, V., A semi-implicit finite difference method for non-hydrostatic free-surface flows, Int. J. Numer. Meth. Fluids, 30, 425-440, (1999) · Zbl 0944.76050 [8] Casulli, V.; Stelling, G. S., Semi-implicit subgrid modelling of three-dimensional free-surface flows, Int. J. Numer. Meth. Fluids, 67, 441-449, (2011) · Zbl 1316.76018 [9] Cesenek, J.; Feistauer, M., Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion, SIAM J. Numer. Anal., 50, 3, 1181-1206, (2012) · Zbl 1312.65157 [10] Cockburn, B.; Hou, S.; Shu, C. W., The Runge- Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 190, 545-581, (1990) · Zbl 0695.65066 [11] Cockburn, B.; Lin, S. Y.; Shu, C. W., TVB Runge- Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 1, 90-113, (1989) · Zbl 0677.65093 [12] Cockburn, B.; Shu, C. W., TVB Runge- Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 186, 411-435, (1989) · Zbl 0662.65083 [13] Cockburn, B.; Shu, C. W., The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws, Math. Model. Numer. Anal., 25, 337-361, (1991) · Zbl 0732.65094 [14] Cockburn, B.; Shu, C. W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35, 6, 2440-2463, (1998) · Zbl 0927.65118 [15] Cockburn, B.; Shu, C. W., The Runge- Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 2, 199-224, (1998) · Zbl 0920.65059 [16] de Saint-Venant, M., Theorie du mouvement non permanent des eaux, avec application aux crues des rivieres et a l’introduction de marees dans leurs lits, Technical Report, (1871), Académie des Sciences, Paris · JFM 03.0482.04 [17] Dumbser, M., Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput. Fluids, 39, 60-76, (2010) · Zbl 1242.76161 [18] Dumbser, M.; Casulli, V., A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations, Appl. Math. Comput., 219, 15, 8057-8077, (2013) · Zbl 1366.76050 [19] Dumbser, M.; Schwartzkopff, T.; Munz, C. D., Arbitrary high order finite volume schemes for linear wave propagation, Computational Science and High Performance Computing II, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), 129-144, (2006), Springer [20] Engsig-Karup, A. P.; Hesthaven, J. S.; Bingham, H. B.; Warburton, T., DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations, Coastal Eng., 55, 3, 197-208, (2008) [21] Eskilsson, C.; Sherwin, S. J., Spectral/hp discontinuous Galerkin methods for modelling 2D Boussinesq equations, J. Comput. Phys., 212, 2, 566-589, (2006) · Zbl 1084.76058 [22] Eskilsson, C.; Sherwin, S. J.; Bergdahl, L., An unstructured spectral/hp element model for enhanced Boussinesq-type equations, Coastal Eng., 53, 11, 947-963, (2006) [23] Feistauer, M.; Kucera, V.; Najzar, K.; Prokopová, J., Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numerische Mathematik, 117, 2, 251-288, (2011) · Zbl 1211.65125 [24] Gassner, G.; Lörcher, F.; Munz, C. D., A discontinuous Galerkin scheme based on a space-time expansion II. viscous flow equations in multi dimensions, J. Sci. Comput., 34, 260-286, (2008) · Zbl 1218.76027 [25] Gassner, G.; Lörcher, F.; Munz, C. D., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, J. Comput. Phys., 224, 1049-1063, (2007) · Zbl 1123.76040 [26] Hartmann, R.; Houston, P., An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations, J. Comput. Phys., 227, 9670-9685, (2008) · Zbl 1359.76220 [27] Jiang, G.; Shu, C. W., On a cell entropy inequality for discontinuous Galerkin methods, Math. Comput., 62, 531-538, (1994) · Zbl 0801.65098 [28] Klaij, C. M.; van der Vegt, J. J.W.; van der Ven, H., Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations, J. Comput. Phys., 217, 2, 589-611, (2006) · Zbl 1099.76035 [29] Levy, D.; Shu, C. W.; Yan, J., Local discontinuous Galerkin methods for nonlinear dispersive equations, J. Comput. Phys., 196, 2, 751-772, (2004) · Zbl 1055.65109 [30] Madsen, P. A.; Murray, R.; Sørensen, O. R., A new form of the Boussinesq equations with improved linear dispersion characteristics, Coastal Eng., 15, 4, 371-388, (1991) [31] Madsen, P. A.; Sørensen, O. R., A new form of the Boussinesq equations with improved linear dispersion characteristics. part 2. A slowly-varying bathymetry, Coastal Eng., 18, 3-4, 183-204, (1992) [32] Reed, W. H.; Hill, T. R., Triangular mesh methods for the neutron transport equation, Technical Report, (1973), University of California, Los Alamos Scientific Laboratory [33] Saad, Y.; Schultz, M. H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on scientific and statistical computing, 7, 3, 856-869, (1986) · Zbl 0599.65018 [34] Stroud, A. H., Approximate calculation of multiple integrals, (1971), Prentice-Hall Inc. Englewood Cliffs, New Jersey · Zbl 0379.65013 [35] Tavelli, M.; Dumbser, M., A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes, Appl. Math. Comput., 234, 623-644, (2014) · Zbl 1298.76120 [36] van der Vegt, J. J.W.; van der Ven, H., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I. general formulation, J. Comput. Phys., 182, 2, 546-585, (2002) · Zbl 1057.76553 [37] van der Ven, H.; van der Vegt, J. J.W., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: II. efficient flux quadrature, Comput. Meth. Appl. Mech. Eng., 191, 41-42, 4747-4780, (2002) · Zbl 1099.76521 [38] Veeramony, J.; Svendsen, I. A., The flow in surf-zone waves, Coastal Eng., 39, 2-4, 93-122, (2000) [39] Xu, Y.; Shu, C. W., A local discontinuous Galerkin method for the Camassa- Holm equation, SIAM J. Numer. Anal., 46, 4, 1998-2021, (2008) · Zbl 1173.65063 [40] Yan, J.; Shu, C. W., A local discontinuous Galerkin method for KdV type equations, SIAM J. Numer. Anal., 40, 2, 769-791, (2002) · Zbl 1021.65050 [41] Yan, J.; Shu, C. W., Local discontinuous Galerkin methods for partial differential equations with higher order derivatives, J. Sci. Comput., 17, 1-4, 27-47, (2002) · Zbl 1003.65115
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