TY - JOUR

T1 - L1 convergences and convergence rates of approximate solutions for compressible Euler equations near vacuum

AU - Lee, Hsin Yi

AU - Chu, Jay

AU - Hong, John M.

AU - Lin, Ying Chieh

N1 - Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter ν. The solutions ρν and vν of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as ν approaches 0, the solutions ρν and vν converge to the solutions ρ and v, respectively, of pressureless compressible Euler equations in L1 sense. In addition, the L1 convergence rates of these physical quantities in terms of ν are also investigated. The L1 convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of ∂xiρν (i= 0 , 1 , 2) and ∂xjvν (j= 0 , 1 , 2 , 3) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of ν. These theoretic results are also supported by numerical simulations.

AB - In this paper, we study the rarefaction wave case of the regularized Riemann problem proposed by Chu, Hong and Lee in SIMA MMS, 2020, for compressible Euler equations with a small parameter ν. The solutions ρν and vν of such problems stand for the density and velocity of gas flow near vacuum, respectively. We show that as ν approaches 0, the solutions ρν and vν converge to the solutions ρ and v, respectively, of pressureless compressible Euler equations in L1 sense. In addition, the L1 convergence rates of these physical quantities in terms of ν are also investigated. The L1 convergences and convergence rates are proved by two facts. One is to invent an a priori estimate coupled with the iteration method to the high-order derivatives of Riemann invariants so that we obtain the uniform boundedness of ∂xiρν (i= 0 , 1 , 2) and ∂xjvν (j= 0 , 1 , 2 , 3) on the requisite regions. The other is about convexity of characteristic curves, which is used to estimate the distances among characteristic curves in terms of ν. These theoretic results are also supported by numerical simulations.

KW - A priori estimate

KW - Compressible Euler equations

KW - Convergence rate

KW - Hyperbolic systems of conservation laws

KW - Method of characteristics

KW - Regularized Riemann problem

KW - Riemann invariants

KW - Vacuum

UR - http://www.scopus.com/inward/record.url?scp=85081752007&partnerID=8YFLogxK

U2 - 10.1007/s40687-020-00205-8

DO - 10.1007/s40687-020-00205-8

M3 - 期刊論文

AN - SCOPUS:85081752007

SN - 2522-0144

VL - 7

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

IS - 2

M1 - 6

ER -