×

zbMATH — the first resource for mathematics

Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers. (English) Zbl 1058.76046
Summary: We report here on our numerical study of two-dimensional Riemann problem for compressible Euler equations. Compared with relatively simple 1-D configurations, the 2-D case consists of a plethora of geometric wave patterns that pose a computational challenge for high-resolution methods. The main feature in the present computations of these 2-D waves is the use of Riemann-solvers-free central schemes presented by Kurganov et al. This family of central schemes avoids intricate and time-consuming computation of the eigensystem of the problem, and hence offers a considerably simpler alternative to upwind methods. The numerical results illustrate that, despite their simplicity, the central schemes are able to recover with comparable high resolution, various features observed in earlier, more expensive computations.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Schulz-Rinne, SIAM J Math Anal 24 pp 76– (1993)
[2] Schulz-Rinne, SIAM J Sci Comp 14 pp 1394– (1993)
[3] Chang, Discrete Contin Dynam Systems 1 pp 555– (1995)
[4] Zhang, SIAM J Math Anal 21 pp 593– (1990)
[5] Lax, SIAM J Sci Comp 19 pp 319– (1998)
[6] Chang, Discrete Contin Dynam Systems 6 pp 419– (2000)
[7] Lax, Comm Pure Appl Math 7 pp 159– (1954)
[8] Godunov, Mat Sb 47 pp 271– (1959)
[9] Shock waves and reaction diffusion-equations (2nd ed.), Grundleheren Series 258, Springer-Verlag, New York, 1994. · Zbl 0807.35002
[10] Nessyahu, J Comp Phys 87 pp 408– (1990)
[11] Liu, Numerische Mathematik 79 pp 397– (1998)
[12] Jiang, SIAM J Sci Comp 19 pp 1892– (1998)
[13] and Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, (Ed.), Lecture Notes in Math 1697, Springer, New York, 1997.
[14] Kurganov, J Comp Phys 160 pp 214– (2000)
[15] Kurganov, SIAM J Sci Comp 22 pp 1461– (2000)
[16] Kurganov, Numerische Mathematik 88 pp 683– (2001)
[17] Kurganov, SIAM J Sci Comp 23 pp 707– (2001)
[18] Kurganov, Math Model Numer Anal 34 pp 1259– (2000)
[19] Jin, CPAM 48 pp 235– (1995)
[20] Liu, J Comput Phys 142 pp 304– (1998)
[21] Harten, J Comp Phys 49 pp 357– (1983)
[22] Harten, J Comp Phys 71 pp 231– (1987)
[23] Conservation laws: stability of numerical approximations and nonlinear regularization, Ph.D. Thesis, Tel-Aviv University, Israel, 1997.
[24] Osher, Math Comp 50 pp 19– (1988)
[25] van Leer, J Comp Phys 32 pp 101– (1979)
[26] Levy, Appl Numer Math 33 pp 407– (2000)
[27] Levy, SIAM J Sci Comp 22 pp 656– (2000)
[28] Liu, SIAM J Numer Anal 33 pp 760– (1996)
[29] Shu, J Comp Phys 77 pp 439– (1988)
[30] Shu, SIAM J Sci Comp 6 pp 1073– (1988)
[31] Kurganov, J Comp Phys 160 pp 720– (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.