## Taiwanese Journal of Mathematics

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

#### Abstract

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

Source
Taiwanese J. Math., Advance publication (2019), 45 pages.

Dates
First available in Project Euclid: 10 May 2019

https://projecteuclid.org/euclid.twjm/1557475461

Digital Object Identifier
doi:10.11650/tjm/190501

#### Citation

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., advance publication, 10 May 2019. doi:10.11650/tjm/190501. https://projecteuclid.org/euclid.twjm/1557475461

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