Jm. Greenberg et Ay. Leroux, A WELL-BALANCED SCHEME FOR THE NUMERICAL PROCESSING OF SOURCE TERMS IN HYPERBOLIC-EQUATIONS, SIAM journal on numerical analysis, 33(1), 1996, pp. 1-16
In a variety of physical problems one encounters source terms that are
balanced by internal forces and this balance supports multiple steady
state solutions that are stable. Typical of these are gravity-driven
flows such as those described by the shallow water equations over a no
nuniform ocean bottom. (1.10) h(t)+(hu)(x) = 0 and (hu)(t)+(hu(2)+gh(2
)/2)(x)+ga(x)(x)h = 0; Many classic numerical schemes cannot maintain
these steady solutions or achieve them in the long time limit with an
acceptable level of accuracy because they do not preserve the proper b
alance between the source terms and internal forces. We propose here a
numerical scheme, adapted to a scalar conservation law, that preserve
s this balance and that can, it is hoped, be extended to more general
hyperbolic systems. The proof of convergence of this scheme toward the
entropy solution is given and some numerical tests are reported.