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Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

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Abstract

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The work of X. Wen is supported by the China Scholarship Council fellowship. The work of Y. Xing is partially sponsored by NSF Grant DMS-1753581. The work of X. Wen, Z. Gao and W.S. Don is partially supported by the National Natural Science Foundation of China (11871443), Shandong Provincial Natural Science Foundation (ZR2017MA016), and Shandong Provincial Qingchuang Science and Technology Project (2019KJI002). W.S. Don also likes to thank the Ocean University of China for providing the startup funding (201712011) that is used in supporting this work.

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Authors and Affiliations

  1. College of Oceanic and Atmospheric Sciences, Ocean University of China, Qingdao, China

    Xiao Wen

  2. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    Wai Sun Don & Zhen Gao

  3. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Yulong Xing

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Appendix A: Two-Dimensional DG Methods

Appendix A: Two-Dimensional DG Methods

The two-dimensional entropy stable DG methods (4.5), after following the definition of the derivative operators \(D_h^x\) in Definition 3 and the entropy conservative numerical fluxes (4.3) and (4.4), can be expanded as

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \sum _k \left( \frac{\partial h}{\partial t},\omega \right) _{\Omega ^k}+\left( \frac{\partial {\mathbb {P}}m_{1,e}}{\partial x}+\frac{\partial {\mathbb {P}}m_{2,e}}{\partial y},\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left\langle f_S^{(1)}\left( Q_e^+,Q_e\right) -{\mathbb {P}}m_{1,e},\omega n_x\right\rangle _{\partial \Omega ^k} + \left\langle g_S^{(1)}\left( Q_e^+,Q_e\right) -{\mathbb {P}}m_{2,e},\omega n_y\right\rangle _{\partial \Omega ^k} = 0, \\ \displaystyle \\ \displaystyle \sum _k \left( \frac{\partial m_1}{\partial t},\omega \right) _{\Omega ^k} +\left( \frac{1}{2}\frac{\partial }{\partial x}\left( {\mathbb {P}}(m_{1,e}u_e)\right) +\frac{1}{2}u_e\frac{\partial }{\partial x}\left( {\mathbb {P}}m_{1,e}\right) +\frac{1}{2}m_{1,e}\frac{\partial }{\partial x} u_e +gh_e\frac{\partial }{\partial x}\left( {\mathbb {P}}h_e\right) ,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left( gh_e\frac{\partial }{\partial x}b_e,\omega \right) _{\Omega ^k} +\left( \frac{1}{2}\frac{\partial }{\partial y}\left( ({\mathbb {P}}m_{2,e})u_e\right) +\frac{1}{2}u_e\frac{\partial }{\partial y}\left( {\mathbb {P}}m_{2,e}\right) +\frac{1}{2}m_{2,e}\frac{\partial }{\partial y}u_e,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left\langle f_S^{(2)}\left( Q_e^+,Q_e\right) +\hbox {I},\omega n_x\right\rangle _{\partial \Omega ^k} +\hbox {V} +\left\langle g_S^{(2)}\left( Q_e^+,Q_e\right) +\hbox {II},\omega n_y\right\rangle _{\partial \Omega ^k} +\hbox {VI} =0, \\ \displaystyle \\ \displaystyle \sum _k \left( \frac{\partial m_2}{\partial t},\omega \right) _{\Omega ^k} +\left( \frac{1}{2}\frac{\partial }{\partial x}\left( ({\mathbb {P}}m_{1,e})v_e\right) +\frac{1}{2}v_e\frac{\partial }{\partial x}\left( {\mathbb {P}}m_{1,e}\right) +\frac{1}{2}m_{1,e}\frac{\partial }{\partial x}v_e,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \quad +\left( \frac{1}{2}\frac{\partial }{\partial y}\left( {\mathbb {P}}\left( m_{2,e}v_e\right) \right) +\frac{1}{2}v_e\frac{\partial }{\partial y}\left( {\mathbb {P}}m_{2,e}\right) +v\frac{1}{2}m_{2,e}\frac{\partial }{\partial y}v_e +gh_e\frac{\partial }{\partial y} \left( {\mathbb {P}}h_e\right) ,\omega \right) _{\Omega ^k} +\left( gh_e\frac{\partial }{\partial y}b_e,\omega \right) _{\Omega ^k} \\ \displaystyle \\ \displaystyle \qquad +\left\langle f_S^{(3)}\left( Q_e^+,Q_e\right) + \hbox {III},\omega n_y\right\rangle _{\partial \Omega ^k} + \hbox {VII} +\left\langle g_S^{(3)}\left( Q_e^+,Q_e\right) +\hbox {IV},\omega n_y\right\rangle _{\partial \Omega ^k} +\hbox {VIII} =0, \end{array} \right. \end{aligned}$$

with the terms \(\hbox {I}\) - \(\hbox {VIII}\) defined by

$$\begin{aligned} \hbox {I}&=\Phi (m_{1,e},u_e) -\frac{1}{2}gh_e\left( {\mathbb {P}}h_e - \llbracket b_e\rrbracket \right) ,\qquad \hbox {II}=\Phi (m_{2,e},u_e),\\ \hbox {III}&=\Phi (m_{1,e},v_e), \qquad \hbox {IV}=\Phi (m_{2,e},v_e) -\frac{1}{2}gh_e\left( {\mathbb {P}}h_e -\llbracket b_e\rrbracket \right) , \\ \hbox {V}&=\frac{1}{4}\Psi _x(m_{1,e},u_e)+\frac{1}{2}g\Psi _x(h_e,h_e),\qquad \hbox {VI}=\frac{1}{4}\Psi _x(m_{2,e},u_e),\\ \hbox {VII}&=\frac{1}{4}\Psi _y(m_{1,e},v_e),\qquad \hbox {VIII}=\frac{1}{4}\Psi _y(m_{2,e},v_e)+\frac{1}{2}g\Psi _y(h_e,h_e), \end{aligned}$$

where the notations

$$\begin{aligned} \Phi (a,b)= & {} -\frac{1}{2}{\mathbb {P}} (ab) +\frac{1}{4}({\mathbb {P}}a+a)b, \quad \Psi _x(a,b)=\langle {\mathcal {E}}(a),{\mathbb {P}}(b\omega )n_x\rangle _{\partial \Omega ^k}, \quad \Psi _y(a,b)\\= & {} \langle {\mathcal {E}}(a),{\mathbb {P}}(b\omega )n_y\rangle _{\partial \Omega ^k}, \end{aligned}$$

are used. An equivalent form of the DG methods, which uses the vector variables and local matrices to guide the efficient implementation, is available in [9].

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Wen, X., Don, W.S., Gao, Z. et al. Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations. J Sci Comput 83, 66 (2020). https://doi.org/10.1007/s10915-020-01248-3

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