# A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws

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## Abstract

**:**

## 1. Introduction

**Definition**

**1**

## 2. Construction of One-Dimensional High-Order ES Scheme

#### 2.1. The EC Schemes

**Lemma**

**1.**

**Proof.**

#### 2.1.1. The EC Schemes for Shallow Water Equations

#### 2.1.2. Entropy Conservative Schemes for Euler Equations

#### 2.2. The High-Order EC Scheme

#### 2.3. High-Order ES Scheme

#### 2.4. The Temporal Discretization

#### 2.5. Summary of the Proposed Scheme

## 3. Extension to a Two-Dimensional System

#### 3.1. The Two-Dimensional SWEs

#### 3.2. The Two-Dimensional Euler Equations

## 4. Numerical Results

#### 4.1. The SWEs

#### 4.1.1. Example 1

#### 4.1.2. Example 2

#### 4.1.3. Example 3

#### 4.1.4. Example 4

#### 4.1.5. Example 5: Circular Dam Break Problem

#### 4.2. The Euler Equations of Gas Dynamics

#### 4.2.1. Testing the Order of Accuracy

#### 4.2.2. Sod Problem

#### 4.2.3. Lax Problem

#### 4.2.4. Shu–Osher Problem

#### 4.2.5. 123 Problem

#### 4.2.6. Modified Shock/Turbulence Interaction

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bradford, S.F.; Sanders, B.F. Finite volume model for shallow water flooding of arbitrary topography. J. Hydraul. Eng.
**2002**, 128, 289–298. [Google Scholar] [CrossRef] - Gottardi, G.; Venutelli, M. Central scheme for the two-dimensional dam-break flow simulation. Adv. Water Resour.
**2004**, 27, 259–268. [Google Scholar] [CrossRef] - Vreugdenhil, C.B. Numerical Methods for Shallow-Water Flow; Springer: Dordrecht, The Netherlands, 1995; pp. 15–25. [Google Scholar]
- Perthame, B.; Simeoni, C. A kinetic scheme for the Saint-Venant system with a source term. Calcolo
**2001**, 38, 201–231. [Google Scholar] [CrossRef] - Xu, K. A well-balanced gas-kinetic scheme for the shallow-water equations with source terms. J. Comput. Phys.
**2002**, 178, 533–562. [Google Scholar] [CrossRef] - Kurganov, A.; Levy, D. Central-upwind schemes for the Saint-Venant system. Math. Model. Numer. Anal.
**2002**, 36, 397–425. [Google Scholar] [CrossRef] [Green Version] - Vukovic, S.; Sopta, L. ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations. J. Comput. Phys.
**2002**, 179, 593–621. [Google Scholar] [CrossRef] - Vukovic, S.; Crnjaric-Zic, N.; Sopta, L. WENO schemes for balance laws with spatially varying flux. J. Comput. Phys.
**2004**, 199, 87–109. [Google Scholar] [CrossRef] - Xing, Y.L.; Shu, C.-W. High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys.
**2005**, 208, 206–227. [Google Scholar] [CrossRef] - Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys.
**2006**, 213, 474–499. [Google Scholar] [CrossRef] - Xing, Y.L.; Shu, C.-W. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms. J. Comput. Phys.
**2006**, 214, 567–598. [Google Scholar] [CrossRef] - Noelle, S.; Xing, Y.; Shu, C.-W. High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys.
**2007**, 226, 29–58. [Google Scholar] [CrossRef] - Li, G.; Lu, C.; Qiu, J. Hybrid well-balanced WENO schemes with different indicators for shallow water equations. J. Sci. Comput.
**2012**, 51, 527–559. [Google Scholar] [CrossRef] [Green Version] - Li, G.; Caleffi, V.; Qi, Z.K. A well-balanced finite difference WENO scheme for shallow water flow model. Appl. Math. Comput.
**2015**, 265, 1–16. [Google Scholar] [CrossRef] [Green Version] - Zhu, Q.; Gao, Z.; Don, W.S.; Lv, X. Well-balanced hybrid compact-WENO scheme for shallow water equations. Appl. Numer. Math.
**2017**, 112, 65–78. [Google Scholar] [CrossRef] - Li, J.J.; Li, G.; Qian, S.G.; Gao, J.M. High-order well-balanced finite volume WENO schemes with conservative variables decomposition for shallow water equations. Adv. Appl. Math. Mech.
**2021**, 13, 827–849. [Google Scholar] - Caleffi, V. A new well-balanced Hermite weighted essentially non-oscillatory scheme for shallow water equations. Int. J. Numer. Methods Fluids
**2011**, 67, 1135–1159. [Google Scholar] [CrossRef] - Russo, G. Central schemes for balance laws. In Proceedings of the VIII International Conference on Nonlinear Hyperbolic Problems, Magdeburg, Germany, 28 February–3 March 2000. [Google Scholar]
- Touma, R.; Khankan, S. Well-balanced unstaggered central schemes for one and two-dimensional shallow water equation systems. Appl. Math. Comput.
**2012**, 218, 5948–5960. [Google Scholar] [CrossRef] - Ern, A.; Piperno, S.; Djadel, K. A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int. J. Numer. Methods Fluids
**2008**, 58, 1–25. [Google Scholar] [CrossRef] [Green Version] - Xing, Y. Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium. J. Comput. Phys.
**2014**, 257, 536–553. [Google Scholar] [CrossRef] - Li, G.; Song, L.N.; Gao, J.M. High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations. J. Comput. Appl. Math.
**2018**, 340, 546–560. [Google Scholar] [CrossRef] - Vignoli, G.; Titarev, V.A.; Toro, E.F. ADER schemes for the shallow water equations in channel with irregular bottom elevation. J. Comput.
**2008**, 227, 2463–2480. [Google Scholar] [CrossRef] - Navas-Montilla, A.; Murillo, J. Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations. J. Comput. Phys.
**2015**, 290, 188–218. [Google Scholar] [CrossRef] - Gassner, G.J.; Winters, A.R.; Kopriva, D.A. A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Appl. Math. Comput.
**2016**, 272, 291–308. [Google Scholar] [CrossRef] [Green Version] - Berthon, C.; Chalons, C. A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comput.
**2016**, 85, 1281–1307. [Google Scholar] [CrossRef] [Green Version] - Yuan, X.H. A well-balanced element-free Galerkin method for the nonlinear shallow water equations. Appl. Math. Comput.
**2018**, 331, 46–53. [Google Scholar] [CrossRef] - Li, G.; Li, J.J.; Qian, S.G.; Gao, J.M. A well-balanced ADER discontinuous Galerkin method based on differential transformation procedure for shallow water equations. Appl. Math. Comput.
**2021**, 395, 125848. [Google Scholar] [CrossRef] - Liu, Y.Y.; Yang, L.M.; Shu, C.; Zhang, H.W. Three-dimensional high-order least square-based finite difference-finite volume method on unstructured grids. Phys. Fluids
**2020**, 32, 123604. [Google Scholar] [CrossRef] - Liu, Y.Y.; Shu, C.; Yang, L.M.; Liu, Y.G.; Liu, W.; Zhang, Z.L. High-order implicit RBF-based differential quadrature-finite volume method on unstructured grids: Application to inviscid and viscous compressible flows. J. Comput. Phys.
**2023**, 478, 111962. [Google Scholar] [CrossRef] - Jitendra; Chaurasiya, V.; Rai, K.N.; Singh, J. Legendre wavelet residual approach for moving boundary problem with variable thermal physical properties. Int. J. Nonlinear Sci. Numer. Simul.
**2022**, 23, 947–956. [Google Scholar] - Chaurasiya, V.; Jain, A.; Singh, J. Numerical study of a non-linear porous sublimation problem with temperature-dependent thermal conductivity and Concentration-Dependent Mass Diffusivity. ASME J. Heat Mass Transf.
**2023**, 145, 072701. [Google Scholar] [CrossRef] - Chaudhary, R.K.; Chaurasiya, V.; Singh, J. Numerical estimation of temperature response with step heating of a multi-layer skin under the generalized boundary condition. J. Therm. Biol.
**2022**, 108, 103278. [Google Scholar] [CrossRef] - Fjordholm, U.S.; Mishra, S.; Tadmor, E. The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput.
**1987**, 49, 91–103. [Google Scholar] - Duan, J.M.; Tang, H.Z. High-order accurate entropy stable finite difference schemes for the shallow water magneto-hydrodynamics. J. Comput. Phys.
**2021**, 431, 110136. [Google Scholar] [CrossRef] - Winters, A.R.; Gassner, G.J. A comparison of two entropy stable discontinuous Galerkin spectral element approximations for the shallow water equations with non-constant topography Author links open overlay panel. J. Comput. Phys.
**2015**, 301, 357–376. [Google Scholar] [CrossRef] [Green Version] - Wintermeyer, N.; Winters, A.R.; Gassner, G.J.; Kopriva, D.A. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. J. Comput. Phys.
**2017**, 340, 200–242. [Google Scholar] [CrossRef] [Green Version] - Wintermeyer, N.; Winters, A.R.; Gassner, G.J.; Warburton, T. An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. J. Comput. Phys.
**2018**, 375, 447–480. [Google Scholar] [CrossRef] [Green Version] - Wen, X.; Don, W.S.; Gao, Z.; Xing, Y.L. Entropy stable and well-balanced discontinuous Galerkin methods for the nonlinear shallow water equations. J. Sci. Comput.
**2020**, 83, 66. [Google Scholar] [CrossRef] - Liu, Q.S.; Liu, Y.Q.; Feng, J.H. The scaled entropy variables reconstruction for entropy stable schemes with application to shallow water equations. Comput. Fluids
**2019**, 192, 104266. [Google Scholar] [CrossRef] - Ismail, F.; Roe, P.L. Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. J. Comput. Phys.
**2009**, 228, 5410–5436. [Google Scholar] [CrossRef] - Lefloch, P.G.; Mercier, J.M.; Rohde, C. Fully discrete entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal.
**2002**, 40, 1968–1992. [Google Scholar] [CrossRef] [Green Version] - Fjordholm, U.S.; Mishra, S.; Tadmor, E. Arbitrarily high-order accurate entropy stable essentially non-oscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal.
**2012**, 50, 544–573. [Google Scholar] [CrossRef] - Fjordholm, U.S.; Mishra, S.; Tadmor, E. ENO reconstruction and ENO interpolation are stable. Found. Comput. Math.
**2013**, 13, 139–159. [Google Scholar] [CrossRef] [Green Version] - Biswas, B.; Dubey, R.K. Low dissipative entropy stable schemes using third order WENO and TVD reconstructions. Adv. Comput. Math.
**2018**, 44, 1153–1181. [Google Scholar] [CrossRef] - Alcrudo, F.; Garcia-Navarro, P. A high-resolution Godunov-type scheme in finite volumes for the 2D shallow-water equations. Int. J. Numer. Methods Fluids
**1993**, 16, 489–505. [Google Scholar] [CrossRef] - Qiu, J.; Shu, C.-W. Hermite WENO schemes and their application as limiters for Runge–Kutta discontinuous Galerkin method: One-dimensional case. J. Comput. Phys.
**2003**, 193, 115–135. [Google Scholar] [CrossRef] - Sod, G. A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys.
**1978**, 27, 1–31. [Google Scholar] [CrossRef] [Green Version] - Shu, C.-W.; Osher, S. Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys.
**1988**, 77, 439–471. [Google Scholar] [CrossRef] [Green Version] - Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction; Springer: Berlin/Heidelberg, Germany, 1999. [Google Scholar]
- Toro, E.F.; Titarev, V.A. TVD fluxes for the high-order ADER schemes. J. Sci. Comput.
**2005**, 24, 285–309. [Google Scholar] [CrossRef] - Jiang, G.S.; Shu, C.-W. Efficient Implementation of weighted ENO Schemes. J. Comput. Phys.
**1996**, 126, 202–228. [Google Scholar] [CrossRef] [Green Version] - Balsara, D.S.; Shu, C.-W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys.
**2000**, 160, 405–452. [Google Scholar] [CrossRef] [Green Version]

Cells | ${\mathit{L}}^{\mathit{\infty}}\phantom{\rule{0.277778em}{0ex}}\mathbf{Error}$ | Order | ${\mathit{L}}^{1}\phantom{\rule{0.277778em}{0ex}}\mathbf{Error}$ | Order | ${\mathit{L}}^{2}\phantom{\rule{0.277778em}{0ex}}\mathbf{Error}$ | Order |
---|---|---|---|---|---|---|

25 | 7.1557 × 10${}^{-9}$ | 5.4250 × 10${}^{-9}$ | 4.6434 × 10${}^{-9}$ | |||

50 | 1.4620 × 10${}^{-9}$ | 2.29 | 5.0198 × 10${}^{-10}$ | 3.43 | 5.4977 × 10${}^{-10}$ | 3.08 |

100 | 3.2688 × 10${}^{-11}$ | 5.48 | 1.1626 × 10${}^{-11}$ | 5.43 | 1.2667 × 10${}^{-11}$ | 5.44 |

200 | 1.2699 × 10${}^{-12}$ | 4.69 | 5.8475 × 10${}^{-13}$ | 4.31 | 5.2894 × 10${}^{-13}$ | 4.58 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Z.; Zhou, X.; Li, G.; Qian, S.; Niu, Q.
A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws. *Mathematics* **2023**, *11*, 2604.
https://doi.org/10.3390/math11122604

**AMA Style**

Zhang Z, Zhou X, Li G, Qian S, Niu Q.
A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws. *Mathematics*. 2023; 11(12):2604.
https://doi.org/10.3390/math11122604

**Chicago/Turabian Style**

Zhang, Zhizhuang, Xiangyu Zhou, Gang Li, Shouguo Qian, and Qiang Niu.
2023. "A New Entropy Stable Finite Difference Scheme for Hyperbolic Systems of Conservation Laws" *Mathematics* 11, no. 12: 2604.
https://doi.org/10.3390/math11122604