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

Previous Article
The Escalator Boxcar Train method for a system of age-structured equations
NHM Home
This Issue
Next Article
One-dimensional aggregation equation after blow up: Existence, uniqueness and numerical simulation

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

Abstract
Related Papers

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

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. 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. doi: 10.1090/S0025-5718-06-01851-5.
[3]	C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Springer, (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. 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. 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, (2014).
[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. 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. 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. 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.
[11]	A. Kurganov and D. Levy, Central-upwind schemes for the St. Vernant system,, Math. Model. Num. Anal., 36 (2002), 397. doi: 10.1051/m2an:2002019.
[12]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (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. 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. doi: 10.1016/j.jcp.2005.08.019.*

show all references

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. 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. doi: 10.1090/S0025-5718-06-01851-5.
[3]	C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics,, Springer, (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. 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. 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, (2014).
[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. 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. 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. 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.
[11]	A. Kurganov and D. Levy, Central-upwind schemes for the St. Vernant system,, Math. Model. Num. Anal., 36 (2002), 397. doi: 10.1051/m2an:2002019.
[12]	R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (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. 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. doi: 10.1016/j.jcp.2005.08.019.*

[1]	Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212
[2]	Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[3]	Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257
[4]	François Bouchut, Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 739-758. doi: 10.3934/dcdsb.2010.13.739
[5]	Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. Conference Publications, 2005, 2005 (Special) : 355-364. doi: 10.3934/proc.2005.2005.355
[6]	Zhaoyang Yin. Well-posedness, blowup, and global existence for an integrable shallow water equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 393-411. doi: 10.3934/dcds.2004.11.393
[7]	Zhen-Hui Bu, Zhi-Cheng Wang. Curved fronts of monostable reaction-advection-diffusion equations in space-time periodic media. Communications on Pure & Applied Analysis, 2016, 15 (1) : 139-160. doi: 10.3934/cpaa.2016.15.139
[8]	Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673
[9]	Eddaly Guerra, Héctor Sánchez-Morgado. Vanishing viscosity limits for space-time periodic Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 331-346. doi: 10.3934/cpaa.2014.13.331
[10]	Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707
[11]	Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607
[12]	Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353
[13]	Ying Hu, Shanjian Tang. Nonlinear backward stochastic evolutionary equations driven by a space-time white noise. Mathematical Control & Related Fields, 2018, 8 (3&4) : 739-751. doi: 10.3934/mcrf.2018032
[14]	Min Niu, Bin Xie. Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 2989-3009. doi: 10.3934/dcdsb.2018296
[15]	Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209
[16]	Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015
[17]	Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 593-608. doi: 10.3934/dcdsb.2010.13.593
[18]	Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 799-818. doi: 10.3934/dcdsb.2010.13.799
[19]	Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
[20]	Daniel Guo, John Drake. A global semi-Lagrangian spectral model for the reformulated shallow water equations. Conference Publications, 2003, 2003 (Special) : 375-385. doi: 10.3934/proc.2003.2003.375

2018 Impact Factor: 0.871

American Institute of Mathematical Sciences