# American Institute of Mathematical Sciences

March  2016, 11(1): 145-162. doi: 10.3934/nhm.2016.11.145

## Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography

 1 Seminar for Applied Mathematics (SAM), Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland, Switzerland

Received  April 2015 Revised  July 2015 Published  January 2016

We describe a shock-capturing streamline diffusion space-time discontinuous Galerkin (DG) method to discretize the shallow water equations with variable bottom topography. This method, based on the entropy variables as degrees of freedom, is shown to be energy stable as well as well-balanced with respect to the lake at rest steady state. We present numerical experiments illustrating the numerical method.
Citation: Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks and Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145
##### References:
 [1] E. Audusse, F. Bouchut, M. O. Bristeau, R. Klien and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM. J. Sci. Comp., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090. [2] M. Castro, J. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with non-conservative products, Math. Comp., 75 (2006), 1103-1134. doi: 10.1090/S0025-5718-06-01851-5. [3] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000. doi: 10.1007/3-540-29089-3_14. [4] U. S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, Journal of Computational Physics, 230 (2011), 5587-5609. doi: 10.1016/j.jcp.2011.03.042. [5] J. M. Greenberg and A. Y. LeRoux, A well-balanced scheme for numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16. doi: 10.1137/0733001. [6] A. Hiltebrand, Entropy-stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock-capturing for Hyperbolic Systems of Conservation Laws, Ph.D thesis, ETH Zurich, 2014, No. 22279. [7] A. Hiltebrand and S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numerische Mathematik, 126 (2014), 103-151. doi: 10.1007/s00211-013-0558-0. [8] J. Jaffre, C. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Model. Meth. Appl. Sci., 5 (1995), 367-386. doi: 10.1142/S021820259500022X. [9] S. Jin, A steady state capturing method for hyperbolic systems with geometrical source terms, Math. Model. Numer. Anal., 35 (2001), 631-645. doi: 10.1051/m2an:2001130. [10] S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math., 22 (2004), 230-249. [11] A. Kurganov and D. Levy, Central-upwind schemes for the St. Vernant system, Math. Model. Num. Anal., 36 (2002), 397-425. doi: 10.1051/m2an:2002019. [12] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [13] R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058. [14] S. Noelle, N. Pankratz, G. Puppo and J. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), 474-499. doi: 10.1016/j.jcp.2005.08.019.

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##### References:
 [1] E. Audusse, F. Bouchut, M. O. Bristeau, R. Klien and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM. J. Sci. Comp., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090. [2] M. Castro, J. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with non-conservative products, Math. Comp., 75 (2006), 1103-1134. doi: 10.1090/S0025-5718-06-01851-5. [3] C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000. doi: 10.1007/3-540-29089-3_14. [4] U. S. Fjordholm, S. Mishra and E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, Journal of Computational Physics, 230 (2011), 5587-5609. doi: 10.1016/j.jcp.2011.03.042. [5] J. M. Greenberg and A. Y. LeRoux, A well-balanced scheme for numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16. doi: 10.1137/0733001. [6] A. Hiltebrand, Entropy-stable Discontinuous Galerkin Finite Element Methods with Streamline Diffusion and Shock-capturing for Hyperbolic Systems of Conservation Laws, Ph.D thesis, ETH Zurich, 2014, No. 22279. [7] A. Hiltebrand and S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of conservation laws, Numerische Mathematik, 126 (2014), 103-151. doi: 10.1007/s00211-013-0558-0. [8] J. Jaffre, C. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws, Math. Model. Meth. Appl. Sci., 5 (1995), 367-386. doi: 10.1142/S021820259500022X. [9] S. Jin, A steady state capturing method for hyperbolic systems with geometrical source terms, Math. Model. Numer. Anal., 35 (2001), 631-645. doi: 10.1051/m2an:2001130. [10] S. Jin and X. Wen, An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math., 22 (2004), 230-249. [11] A. Kurganov and D. Levy, Central-upwind schemes for the St. Vernant system, Math. Model. Num. Anal., 36 (2002), 397-425. doi: 10.1051/m2an:2002019. [12] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253. [13] R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058. [14] S. Noelle, N. Pankratz, G. Puppo and J. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213 (2006), 474-499. doi: 10.1016/j.jcp.2005.08.019.
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