Abstract
In this paper, a four-quadrant Riemann problem for a 2×2 system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave at each initial discontinuity. We include the existence of two elementary waves at each initial discontinuity and classify 14 topologically distinct solutions. For each case, we construct an analytic solution and compute the numerical solution.
Highlights
They found that the delta function is supported on a discontinuity, which was motivated by some numerical calculations of the Lax-Friedrich scheme, which was later called a delta shock
The results show extremely interesting structures of wave interactions of the Riemann problem (RP), and the numerical solution is remarkably coincident with the constructed analytic solution
A delta shock is formed at the positive η-axis and negative η-axis, and the rarefaction-contact and shock-contact are formed at the negative η-axis and positive ξ-axis, respectively
Summary
Shen et al [24] constructed ten solutions for the system (9): ut + (u2 ) x + (u2 )y = 0, ρt + (ρu) x + (ρu)y = 0 This equation can be considered to be a simplified gas-dynamics-like model because it can be derived from 2-D isentropic Euler equations [25]. Hwang et al [26] considered a 2-D RP with three constant initial data for the system (9) Our interest is to classify and construct the analytic solution for a 2 × 2 hyperbolic system (9) in the case of initial four-quadrant data without the assumption ( H ).
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