Taiwanese Journal of Mathematics

Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient

Nguyen Xuan Thanh, Mai Duc Thanh, and Dao Huy Cuong

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We present a Godunov-type scheme for the shallow water equations with horizontal temperature gradient and variable topography. First, the exact solutions of the Riemann problem in a computational form are given, where algorithms for computing these solutions are described. Second, a Godunov-type scheme is constructed relying on exact solutions of the local Riemann problems. Computing algorithms for the scheme are given. The scheme is shown to be well-balanced and preserve the positivity of the water height. Numerical tests show that the scheme is convergent with a good accuracy, even for the resonant phenomenon, where the exact solutions contain several distinct waves propagating with the same shock speed. Furthermore, the scheme also provides us with good results for the solution of the wave interaction problem.

Article information

Taiwanese J. Math., Volume 24, Number 1 (2020), 179-223.

Received: 23 December 2018
Accepted: 1 May 2019
First available in Project Euclid: 10 May 2019

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Zentralblatt MATH identifier

Primary: 35L65: Conservation laws 65M08: Finite volume methods 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

shallow water equations with temperature topography nonconservative Godunov scheme accuracy resonant


Thanh, Nguyen Xuan; Thanh, Mai Duc; Cuong, Dao Huy. Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient. Taiwanese J. Math. 24 (2020), no. 1, 179--223. doi:10.11650/tjm/190501. https://projecteuclid.org/euclid.twjm/1557475461

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