Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates

Abstract

The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian coordinates. For equations in Lagrangian coordinates an invariant finite-difference scheme is constructed for all cases for which conservation laws exist in the differential model. Such schemes possess the difference analogues of the conservation laws of mass, momentum, energy, the law of center of mass motion for horizontal, inclined and parabolic bottom topographies. Invariant conservative difference scheme is tested numerically in comparison with naive approximation invariant scheme.

arXiv comment30 pages, 12 figures

References in corpus (9)

• Continuous Symmetries of Difference Equations
• Continuous symmetries of Lagrangians and exact solutions of discrete equations
• A heat transfer with a source: the complete set of invariant difference schemes
• Shallow water equations in Lagrangian coordinates: symmetries, conservation laws and its preservation in difference models
• Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space
• One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: symmetry classification, conservation laws, difference schemes
• Lie symmetries of two-dimensional shallow water equations with variable bottom topography
• Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates
• Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations

Cited by in corpus (3)

• Symmetries and conservation laws of the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates
• Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation
• Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates