Formation of Delta Standing Wave for a Scalar Conservation Law with a Linear Flux Function Involving Discontinuous Coefficients

The aim of this paper is to study the formation of delta standing wave for a scalar conservation law with a linear flux function involving discontinuous coefficients. In order to deal with it, we approximate the discontinuous coefficients by piecewise affine ones and then apply the method of characteristics to construct a global solution to the original equation by approximation. Finally, it is proved rigorously that the delta standing wave can be obtained in the limit of the approximate solution as the perturbation parameter tends to zero. In contrast to the classical method of vanishing viscosity, it can be seen clearly here how the delta standing wave forms naturally along the characteristics.