Hyperbolic conservation laws with space-dependent fluxes: II. General study of numerical fluxes
[For part I see ibid. 156, No.~1, 1--21 (2003; Zbl 1031.35104).] The authors consider the initial value problem of the following scalar conservation equation \[ \begin{cases} u_t+f(u(x,t),a(x))_x=0,\quad & x\in\mathbb{R}=(-\infty,+\infty),\;t\geq 0\\ u(x,0)=u_0(x),\end{cases}\tag{1} \] where \(a(x)\) is the known function and \(u(x,t)\) is the unknown function. Some generalized fluxes of the Godunov and Engquist-Osher types are verified to enjoy more theoretical properties without the restrictions of the previous works. Moreover, the authors show that the resulting numerical solutions are characterized by an \(L^\infty\) contraction. Numerical examples of the authors show the convergence to exact solutions and very good resolution of discontinuities. The numerical results are compared to some ``simple'' generalization of the Godunov, the Engquist-Osher, and the local Lax-Friedrichs fluxes.
scientific article; zbMATH DE number 2137366