Communications in Mathematical Sciences

Volume 21 (2023)

Number 3

Adaptive central-upwind scheme on triangular grids for the Saint–Venant system

Pages: 671 – 708



Yekaterina Epshteyn (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Thuong Nguyen (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)


In this work, we develop a robust adaptive well-balanced and positivity-preserving centralupwind scheme on unstructured triangular grids for shallow water equations. The numerical method is an extension of the scheme from [Liu et al., J. Comput. Phys., 374:213–236, 2018]. As a part of the adaptive central-upwind algorithm, we obtain a local a posterior error estimator for the efficient mesh refinement strategy. The accuracy, high-resolution and efficiency of new adaptive central-upwind scheme are demonstrated on a number of challenging tests for shallow water models.


Saint–Venant system of shallow water equations, central-upwind scheme, well-balanced and positivity-preserving scheme, adaptive algorithm, weak local residual error estimator, unstructured triangular grid

2010 Mathematics Subject Classification

35L65, 65M08, 76M12, 86-08, 86A05

The work of Yekaterina Epshteyn was partially supported by NSF DMS-2207207.

Received 25 June 2021

Received revised 1 June 2022

Accepted 8 July 2022

Published 27 February 2023