Applied Mathematics and Computation 321 (2018) 49–62
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
Two dimensional Riemann problem for a 2 × 2 system of
hyperbolic conservation laws involving three constant states
Jinah Hwang
a
, Myoungin Shin
c
, Suyeon Shin
b
, Woonjae Hwang
b,∗
a
Department of Mathematics, Korea University, Seoul 02841, Republic of Korea
b
Division of Applied Mathematical Sciences, Korea University, Sejong 30019, Republic of Korea
c
Department of Mathematics, Naval Academy, Changwon 51704, Republic of Korea
a r t i c l e i n f o
MSC:
35L65
35L67
76L15
Keywords:
Riemann problem
Conservation laws
Delta shock
a b s t r a c t
Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional
Riemann problem for Euler equation. To resolve this illuminating conjecture, many re-
searchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann prob-
lem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that
each jump of the initial data projects one planar elementary wave. We classify twelve
topologically distinct solutions and construct analytical and numerical solutions. The com-
puted numerical solutions clearly confirm the constructed analytic solutions.
© 2017 Elsevier Inc. All rights reserved.
1. Introduction
In 1990, Zhang and Zheng [15] conjectured on the structure of a solution for a four quadrant Riemann problem for
two-dimensional (2 − D) gas dynamics system:
ρ
t
+ (ρ u)
x
+ (ρ v)
y
= 0,
(ρ u)
t
+ (ρ u
2
+ p)
x
+ (ρ uv)
y
= 0,
(ρ v)
t
+ (ρ uv)
x
+ (ρ v
2
+ p)
y
= 0,
(1)
for the isentropic flow
p = Aρ
γ
, γ > 1, A > 0,
and for the adiabatic flow
ρ
(
e +
u
2
+ v
2
2
)
t
+
ρ u
(
h +
u
2
+ v
2
2
)
x
+
ρ v
(
h +
u
2
+ v
2
2
)
y
= 0,
e =
p
(γ − 1)ρ
, h = e +
p
ρ
.
They considered one planar elementary wave for each jump in the initial discontinuity. To resolve this conjecture, many
studies have been developed for simplified systems [4].
∗
Corresponding author.
E-mail addresses: jinahwang@korea.ac.kr (J. Hwang), myoungin@navy.ac.kr (M. Shin), angelic52@korea.ac.kr (S. Shin), woonjae@korea.ac.kr (W. Hwang).
https://doi.org/10.1016/j.amc.2017.10.045
0096-3003/© 2017 Elsevier Inc. All rights reserved.