Construction of the 2d Riemann Solutions for a Nonstrictly Hyperbolic Conservation Law

Abstract

Abstract. In this note, we consider the Riemann problem for a twodimensional nonstrictly hyperbolic system of conservation laws. Without the restriction that each jump of the initial data projects one planar elementary wave, six topologically distinct solutions are constructed by applying the generalized characteristic analysis method, in which the delta shock waves and the vacuum states appear. Moreover we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct global solutions.

Four-Quadrant Riemann Problem for a 2 × 2 System Involving Delta Shock

“…This new shock satisfies the equations in (22) in 0 ≤ u ≤ u 3 . Similarly, this curved shock has the same form as (23), and continues to C(2u 3 , 2u 3 ). In addition, J e3 formed at point B ends at the singular point D(u 3 , u 3 ) and satisfies…”

“…Under the fundamental assumption of u = v, Sun [23] constructed six topologically distinct solutions for a nonstrictly hyperbolic system (8) :…”

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

“…This assumption helps us to convert the second equation of Equation () to an equivalent conservative form. Such an assumption for gas dynamics equations has been well received in the literature, where the researchers obtained a simplified hyperbolic system of gas dynamics by assuming that the velocities in both the directions are equal; see previous works 34,35,45 …”

“…Such an assumption for gas dynamics equations has been well received in the literature, where the researchers obtained a simplified hyperbolic system of gas dynamics by assuming that the velocities in both the directions are equal; see previous works. 34,35,45 Under the above assumption, one can differentiate the second equation of ( 15) with respect to x and 𝑦, respectively, and add them to obtain a system of conservation laws of the form…”