Communications in Mathematical Sciences

Volume 18 (2020)

Number 3

High-order fully well-balanced Lagrange-projection scheme for shallow water

Pages: 781 – 807



Tomás Morales de Luna (Departamento de Matemáticas, Universidad de Córdoba, Spain)

Manuel J. Castro Díaz (Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Spain)

Christophe Chalons (Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, CNRS, Université Paris-Saclay, Versailles, France)


In this work we propose a novel strategy to define high-order fully well-balanced Lagrange-projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following [M.J. Castro et al., in Handbook of Numerical Analysis, 18:131175, 2017], we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme.


shallow water, finite volume method, high-order well-balanced schemes, Lagrangian formulation

2010 Mathematics Subject Classification

35L60, 35L65, 74G15, 74S10

Copyright © 2020 T. Morales de Luna, M.J. Castro Díaz and C. Chalons

This research has been partially supported by the Spanish Government and FEDER through the coordinated Research projects MTM 2015-70490-C2-1-R and MTM 2015-70490-C2-2-R, as well as by the coordinated Research projects RTI2018-096064-B-C21 and RTI2018-096064-B-C22. The authors were partially supported by the Innovative Training Networks (ITN) grant 642768 (ModCompShock). The third author was partially supported by the ANR project ACHYLLES (ANR-14-CE25-0001-03).

Received 14 June 2019

Accepted 30 November 2019

Published 30 June 2020