Abstract
In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.
Highlights
The two-dimensional scalar conservation law is given by Citation: Hwang, J.; Shin, S.; Shin, M.; Hwang, W
In 1983, Wagner [6] constructed a solution for the four-quadrant Riemann problem (RP) of a 2-D scalar conservation law with convex f = g
We consider a four-quadrant RP for system (3) without the assumption that each jump of the initial data projects exactly one planar elementary wave
Summary
The two-dimensional scalar conservation law is given by Citation: Hwang, J.; Shin, S.; Shin, M.; Hwang, W. Riemann Problem for a 2 × 2 System. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in ut + f (u) x + g(u)y = 0, where u( x, y, t) is a conserved quantity, and f and g are nonlinear fluxes. In 1975, Guckenheimer [5] initiated the construction of a solution for the two-dimensional (2-D) Riemann problem (RP) by developing an interesting example called the Guckenheimer structure. In 1983, Wagner [6] constructed a solution for the four-quadrant RP of a 2-D scalar conservation law with convex f = g. Lindquist showed that the Riemann solutions are piecewise smooth when f = g [7] and outlined the construction method [8]
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