Open Access
February, 2020 Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient
Nguyen Xuan Thanh, Mai Duc Thanh, Dao Huy Cuong
Taiwanese J. Math. 24(1): 179-223 (February, 2020). DOI: 10.11650/tjm/190501


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.


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Nguyen Xuan Thanh. Mai Duc Thanh. Dao Huy Cuong. "Godunov-type Numerical Scheme for the Shallow Water Equations with Horizontal Temperature Gradient." Taiwanese J. Math. 24 (1) 179 - 223, February, 2020.


Received: 23 December 2018; Accepted: 1 May 2019; Published: February, 2020
First available in Project Euclid: 10 May 2019

zbMATH: 07175546
MathSciNet: MR4053844
Digital Object Identifier: 10.11650/tjm/190501

Primary: 35L65 , 65M08 , 76B15

Keywords: Accuracy , Godunov scheme , nonconservative , resonant , shallow water equations with temperature , topography

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

Vol.24 • No. 1 • February, 2020
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