Hyperbolic conservation laws with space-dependent flux: I. Characteristics theory and Riemann problem

In the paper, a kind of one-dimensional scalar hyperbolic conservation laws with flux functions dependent on space variable is discussed and analyzed. A better understanding about the behavior of wave propagation of the kind problems is presented. Especially, some sufficient and necessary conditions that ensure the unique physically relevant solution to the Riemann problem are proposed. Because the numerical flux obtained from the Riemannʹs solver is theoretically correct and exact to the problem, it must also be of high resolution in its nature. For comparison, some convincing numerical examples from traffic flow problems are given at the end of the paper.