The Riemann problem of the Burgers equation with a discontinuous source term

In this paper we are concerned with the Riemann problem of the Burgers equation with a discontinuous source term, motivated by studying the propagation of singular waves in radiation hydrodynamics. By calculating the representation of solutions, we construct the global entropy solution to this Riemann problem, in which one needs to pay attention to the effects of the discontinuous source term on the propagation of the Riemann waves. It turns out that the discontinuity of the source term has clear influences on the shock or rarefaction waves generated by the initial Riemann data, and produces some new and interesting phenomena such as the appearance of weak discontinuities, the appearance and absorption of new shocks, artificial “vacuums”, and different types of asymptotic behavior of shocks. These new waves and phenomena shall be analyzed in this paper.