Conservation law with the flux function discontinuous in the space variable. II: Convex-concave type fluxes and generalized entropy solutions
Following one-dimensional nonlinear Cauchy problem is investigated: \[ u_{t}+(H(x)f(u)+(1-H(x))g(u))_{x}=0; \quad u(x,0)=u_{0}(x) \] where \(f\) and \(g\) are Lipschitz continuous and \(H\) is the Heaviside function. Such an equation arises, e.g., in the two phase flows in petroleum reservoirs. The authors develop the concept of generalized entropy solutions for such equations by replacing the Rankine-Hugoniot condition by a generalized Rankine-Hugoniot condition. In a recent work [J. Hyperbolic Differ. Equ. 2, No.~4, 783--837 (2005; Zbl 1093.35045)] the authors have proposed a new entropy framework for the equation of the above type -- the so called ``optimal entropy solutions''. Under the assumption that \( f,g \) have the same type of convexity they have shown the existence of infinitely many stable solutions. As for the weak solutions, they may not exist. A Godunov type of convergent finite difference scheme allows to compute a numerical solution and in the same time serves to prove existence theorems. The last section is devoted to numerical experiments. [For part I see \textit{Adimurthi} and \textit{G. D. Veerappa Gowda}, J. Math. Kyoto Univ. 43, No.~1, 27--70 (2003; Zbl 1063.35114).]
scientific article; zbMATH DE number 5151704
scientific article