Riemann problem for the 2D scalar conservation law involving linear fluxes with discontinuous coefficients

This paper is devoted to the four-constant Riemann problem for the two-dimensional (2D) scalar conservation laws involving linear fluxes with discontinuous coefficients. First, under the assumption that each discontinuity ray of initial data outside of the origin emits exactly one elementary wave, by studying the pointwise interactions occurring at the interaction points of waves, we completely solve this Riemann problem in the self-similar plane with 21 nontrivial and different geometric structures. Second, when each discontinuity ray of initial data outside of the origin emits two different kinds of contact discontinuities, by studying the pointwise interactions, we construct an interesting kind of spiral structure in the self-similar plane.