Well-Balanced Numerical Schemes for Shallow Water Equations with Horizontal Temperature Gradient
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A class of well-balanced numerical schemes for the one-dimensional shallow water equations with temperature gradient is constructed. The construction of the schemes is based on two steps: the first step to absorb the nonconservative term and the second step to deal with the evolution of the system. Algorithms for computing contact waves which absorb the nonconservative term are developed. Furthermore, to improve the accuracy, the underlying numerical fluxes can be formed as convex combinations of a pair of numerical fluxes of a low and stable scheme and a higher and fast scheme. The schemes are well balanced and can retain the positivity of the water height and the water temperature. Numerical tests show that the schemes are stable and have a good accuracy.
KeywordsShallow water equations Ripa system Conservation law Well-balanced scheme Convergence Accuracy
Mathematics Subject Classification35L65 65M08 76B15
The authors would like to thank the reviewers for their very valuable comments and suggestions. This research is funded by the Vietnam National University HoChiMinh City (VNU-HCM) under Grant No. B2018-28-01.
- 8.Coquel, F., Godlewski, E., Perthame, B., In, Rascle, P.: Some new Godunov and relaxation methods for two-phase flow problems. In: Toro, E. F. (ed.) Godunov Methods (Oxford, 1999), pp. 179–188. Kluwer/Plenum, New York (2001)Google Scholar
- 11.Dubroca, B.: Positively conservative Roe’s matrixfor Euler equations. In: 16th International Conference on Numerical Methods in Fluid Dynamics (Arcachon, 1998), Lecture Notes in Phys., vol. 515, pp. 272–277. Springer, Berlin. https://doi.org/10.1007/BFb0106594
- 24.Qian, S.G., Shao, F.J., Li, G.: High order well-balanced discontinuous Galerkin methods for shallow water flow under temperature fields. Comput. Appl. Math. (2018). https://doi.org/10.1007/s40314-018-0662-y