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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

Andreas Hiltebrand 1, and  Siddhartha Mishra 1,

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.
Keywords: shallow water equations, entropy stability, space-time DG, Well-balancedness, bottom topography..
Mathematics Subject Classification: Primary: 65M60, 35L6.
Citation: Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks & Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145
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show all references

References:
[1]

SIAM. J. Sci. Comp., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090.  Google Scholar

[2]

Math. Comp., 75 (2006), 1103-1134. doi: 10.1090/S0025-5718-06-01851-5.  Google Scholar

[3]

Springer, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[4]

Journal of Computational Physics, 230 (2011), 5587-5609. doi: 10.1016/j.jcp.2011.03.042.  Google Scholar

[5]

SIAM J. Numer. Anal., 33 (1996), 1-16. doi: 10.1137/0733001.  Google Scholar

[6]

Ph.D thesis, ETH Zurich, 2014, No. 22279. Google Scholar

[7]

Numerische Mathematik, 126 (2014), 103-151. doi: 10.1007/s00211-013-0558-0.  Google Scholar

[8]

Math. Model. Meth. Appl. Sci., 5 (1995), 367-386. doi: 10.1142/S021820259500022X.  Google Scholar

[9]

Math. Model. Numer. Anal., 35 (2001), 631-645. doi: 10.1051/m2an:2001130.  Google Scholar

[10]

J. Comput. Math., 22 (2004), 230-249.  Google Scholar

[11]

Math. Model. Num. Anal., 36 (2002), 397-425. doi: 10.1051/m2an:2002019.  Google Scholar

[12]

Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

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J. Comput. Phys., 146 (1998), 346-365. doi: 10.1006/jcph.1998.6058.  Google Scholar

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J. Comput. Phys., 213 (2006), 474-499. doi: 10.1016/j.jcp.2005.08.019.  Google Scholar

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