Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

Moving-water equilibria preserving central-upwind schemes for the shallow water equations

Pages: 1643 – 1663



Yuanzhen Cheng (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)

Alexander Kurganov (Department of Mathematics, Tulane University, New Orleans, Louisiana, U.S.A.)


We construct a new second-order moving-water equilibria preserving central-upwind scheme for the one-dimensional Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart.


shallow water equations, central-upwind scheme, well-balanced method, steady-state solutions (equilibria), moving-water and still-water equilibria

2010 Mathematics Subject Classification

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

Published 12 August 2016