B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a
source term, Calcolo 38(4), 201–231 (2001).
hal-00922664.
The scheme implemented here: kinetic formulation, interface densities with
reflections, properties (Theorem 3.2) and closed-form fluxes (Section 4).
M. Ersoy, Perthame–Simeoni kinetic traces and nonconservative interface laws for
DG approximations of the Saint–Venant system, preprint (2026).
hal-05618693.
Discontinuous Galerkin interpretation of the scheme; exact momentum interface
law (Proposition 4.1), tested in this code; the finite volume scheme is the case of
polynomial degree 0.
M. Ersoy and C. Simeoni, Consistency of the kinetic scheme with reflections,
research report, IMATH, Université de Toulon (2013).
E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and
stable well-balanced scheme with hydrostatic reconstruction for shallow water
flows, SIAM J. Sci. Comput. 25(6), 2050–2065 (2004). The hydrostatic
reconstruction used by the Rusanov scheme.
O. Delestre, S. Cordier, F. Darboux and F. James, A limitation of the hydrostatic
reconstruction technique for Shallow Water equations, C. R. Math. Acad. Sci. Paris
350(13–14), 677 (2012), doi:10.1016/j.crma.2012.08.004.
Why the reference scheme struggles with thin flows on slopes.
C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory
shock-capturing schemes, J. Comput. Phys. 77(2), 439–471 (1988). The SSP
Runge-Kutta methods.
H. Takahasi and M. Mori, Double exponential formulas for numerical integration,
Publ. RIMS Kyoto Univ. 9(3), 721–741 (1974). The tanh-sinh quadrature used for
\(II^q_3\).
A. Ritter, Die Fortpflanzung der Wasserwellen, Zeitschrift des Vereines
Deutscher Ingenieure 36 (1892). Dam break on a dry bed.
J. J. Stoker, Water Waves: The Mathematical Theory with Applications,
Interscience (1957). Dam break on a wet bed.
W. C. Thacker, Some exact solutions to the nonlinear shallow-water wave equations,
J. Fluid Mech. (1981), doi:10.1017/S0022112081001882.
Oscillation in a parabolic bowl.
N. Goutal and F. Maurel, Proceedings of the 2nd workshop on dam-break wave
simulation, Électricité de France (1997). Origin of the flows over a bump.
O. Delestre, C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T. N. T. Vo,
F. James and S. Cordier, SWASHES: a compilation of shallow water analytic solutions
for hydraulic and environmental studies, Int. J. Numer. Meth. Fluids (2013),
hal-00628246, arXiv:1110.0288.
The flows over a bump (Section 3.1, same discharges and outlet heights, bump
centred at x = 10) and Thacker's solution (Section 4.2.1).