Concentration of source terms in generalized Glimm scheme for initial-boundary problem of nonlinear hyperbolic balance laws

Ying-Chieh Lin

Department of Applied Mathematics

National University of Kaohsiung

linyj@nuk.edu.tw

    In this talk, we consider the initial-boundary value problem for a nonlinear hyperbolic system of balance laws with sources `a_x g` and `a_t h`. We assume that the boundary data satisfy a inear or smooth nonlinear relation. Generalized Riemann and boundary Riemann problems are provided with the variation of `a` concentrated on a thin `T`-shaped region of each grid. We generalize Goodman’s boundary interaction estimates and introduce a new version of Glimm scheme to construct the approximation solutions and its stability is proved by considering two types of conditions on `a`. The global existence of entropy solutions is established. Under some sampling condition, we find the entropy solutions converge to its boundary values in `L_{\text{loc}}^1` as `x\rightarrow 0^+` and the boundary values satisfy the boundary condition almost everywhere in `t`.

Keyword: nonlinear balance laws, initial-boundary value problem, Riemann problem, generalized Glimm scheme, concentration of source, wave interaction estimates, entropy solutions.