## Abstract

An energy-stable high-order central finite difference scheme is derived for the two-dimensional shallow water equations. The scheme is mathematically formulated using the semi-discrete energy method for initial boundary value problems described in Olsson (1995, *Math. Comput*. **64**, 1035–1065): after symmetrizing the equations via a change to entropy variables, the flux derivatives are entropy-split enabling the formulation of a semi-discrete energy estimate. We show experimentally that the entropy-splitting improves the stability properties of the fully discretized equations. Thus, the dependence on numerical dissipation to keep the scheme stable for long term time integrations is reduced relative to the original unsplit form, thereby decreasing non-physical damping of solutions. The numerical dissipation term used with the entropy-split equations is in a form which preserves the semi-discrete energy estimate. A random one-dimensional dam break calculation is performed showing that the shock speed is computed correctly for this particular case, however it is an open question whether the correct shock speed will be computed in general

This is a preview of subscription content, log in to check access.

## References

- 1.
Barth T., Charrier P. (2001). Energy stable flux formulas for the discontinuous galerkin discretization of first-order nonlinear conservation laws. NASA Technical Report NAS-01-001.

- 2.
Bova S. W., and Carey, G. F. (1996). An entropy variable formulation and applications for the two dimensional shallow water equations. Int. J. Numerical Methods Fluids 24:29–46.

- 3.
Bova S.W., Carey G.F. (1996). A symmetric formulation and SUPG scheme for the shallow water equations. Adv. Water Res. 19(3):123–131

- 4.
Gerritsen M., Olsson P. (1998). Designing an efficient solution strategy for fluid flows. I. A stable high-order finite difference scheme and sharp shock resolution for the Euler equations. J. Comput. Phys. 129(2):245–262

- 5.
Gerritsen M., Olsson P. (1998). Designing an efficient solution strategy for fluid flows. II. Stable high-order central finite difference schemes on composite adaptive grids with sharp shock resolution. J. Comput. Phys. 147(2):293–317

- 6.
Gerritsen M. (1996). Designing an efficient solution strategy for fluid flows. PhD Thesis, Stanford University

- 7.
Gustafsson B., Olsson P. (1995). Fourth–order difference methods for hyperbolic IBVPs. J. Comput. Phys. 117:300

- 8.
Gustafsson B., Kreiss H.-O., Oliger J.

*Time Dependent Problems and Difference Methods*, New York, Wiley - 9.
Harten A. (1984). On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49:151–164

- 10.
Hauke G. (1998). A symmetric formulation for computing transient shallow water flows. Comput. Methods Appl. Mech. Eng. 163:111–112

- 11.
11. Hughes T.J.R., Franca L.P., Mallat M. (1986). A new finite element formulation for computational fluid dynamics: 1. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54:223–234

- 12.
Kreiss H.-O., Scherer G. (1997). On the existence of energy estimates for difference approximations for hyperbolic systems. Technical Report, Department of Scientific Computing, Uppsala University

- 13.
LeVeque, Yee. (1990). J. Comput. Phys. 86:187–210

- 14.
Oliger J., Sundstrom A. (1978). Theoretical and practical aspects of some initial boundary value problems in fluid dynamics. SIAM J. Appl. Math. 35(3):419–446

- 15.
Olsson, P., and Oliger, J. (1994). Energy and maximum norm estimates for nonlinear conservation laws. Technical Report RIACS 94-01, Research Institute of Advanced Computer Science.

- 16.
Olsson P. (1995). Summation-by-parts, projections and stability. Math. Comput. 64:1035–1065

- 17.
Olsson P. (1995). Summation-by-parts, projections and stability III. Technical Report RIACS 95-06, Research Institute of Advanced Computer Science

- 18.
Shu C.-W. (1988). Total variation diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9(6):1073

- 19.
Stoker, J. J. (1992).

*Water Waves: The Mathematical Theory with Applications*, Wiley-Interscience - 20.
Strand B. (1994). Summation-by-parts for finite difference approximations for

*d*/*d*x. J. Comput. Phys. 110(1):47–67 - 21.
Tadmor E. (1984). Skew-selfadjoint form for systems of conservation Laws. J. Math. Anal. Appl. 103:428–442

- 22.
Tadmor E. (2003). Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems.

*Acta Numerica*451–512. - 23.
Vreugdenhil C. (1994). Numerical Methods for Shallow Water Flow. Kluwer Academic Publishers, Dordrecht

- 24.
Yee H.C., Vinokur M., Djomehri M.J. (2000). Entropy splitting and numerical dissipation. J. Comput. Phys. 162:33–81

- 25.
Yee H.C., Sandham N.D., Djomehri M.J. (1999). Low–dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150:199–238

- 26.
Zhu X. (1996). Composite adaptive grid methods for partial differential equations. PhD thesis, Scientific Computing and Computational Mathematics, Stanford University

## Author information

## Additional information

MSC: 35Q35; 65M12; 65M06

Supported in part by the New Zealand Marsden Fund, grant UOA827

## Rights and permissions

## About this article

### Cite this article

Brown, M., Gerritsen, M. An Energy-Stable High-Order Central Difference Scheme for the Two-Dimensional Shallow Water Equations.
*J Sci Comput* **28, **1–30 (2006). https://doi.org/10.1007/s10915-005-9005-4

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Shallow water equations
- entropy-splitting
- semi-discrete energy estimate
- central finite difference scheme