Glimm Interaction Functional for Bgk Schemes

Under the assumptions that the matrix A(u) = DF(u) is strictly hyperbolic and its eigenvalues are genuinely non linear or linearly degenerate, the global existence of a solution u : [0,+∞) 7→ R for small BV initial data has first been proved in [18]. The uniqueness of solutions and their Lipschitz dependence in L w.r.t. the initial data has been established in a series of papers [9, 11, 12, 13, 14, 15]. See [10, 17] for a general introduction to the theory of hyperbolic systems in one space dimension. In the two papers [6, 4], a different approach has been used. Instead of constructing approximate solutions of the hyperbolic system (1.1) and studying their properties, the authors consider two approximations: