Volume 35, Issue 2
Improved Entropy-Ultra-Bee Scheme for the Euler System of Gas Dynamics

Rongsan Chen & Dekang Mao

J. Comp. Math., 35 (2017), pp. 121-151.

Published online: 2017-04

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  • Abstract

The Entropy-Ultra-Bee scheme was developed for the linear advection equation and extended to the Euler system of gas dynamics in [13]. It was expected that the technology be applied only to the second characteristic field of the system and the computation in the other two nonlinear fields be implemented by the Godunov scheme. However, the numerical experiments in [13] showed that the scheme, though having improved the wave resolution in the second field, produced numerical oscillations in the other two nonlinear fields. Sophisticated entropy increaser was designed to suppress the spurious oscillations by increasing the entropy when there are waves in the two nonlinear fields presented. However, the scheme is then not efficient neither robust with problem-related parameters. The purpose of this paper is to fix this problem. To this end, we first study a 3 × 3 linear system and apply the technology precisely to its second characteristic field while maintaining the computation in the other two fields be implemented by the Godunov scheme. We then follow the discussion for the linear system to apply the Entropy-Ultra-Bee technology to the second characteristic field of the Euler system in a linearlized field-byfield fashion to develop a modified Entropy-Ultra-Bee scheme for the system. Meanwhile a remark is given to explain the problem of the previous Entropy-Ultra-Bee scheme in [13]. A reference solution is constructed for computing the numerical entropy, which maintains the feature of the density and flats the velocity and pressure to constants. The numerical entropy is then computed as the entropy cell-average of the reference solution. Several limitations are adopted in the construction of the reference solution to further stabilize the scheme. Designed in such a way, the modified Entropy-Ultra-Bee scheme has a unified form with no problem-related parameters. Numerical experiments show that all the spurious oscillations in smooth regions are gone and the results are better than that of the previous Entropy-Ultra-Bee scheme in [13].

  • Keywords

Entropy-Ultra-Bee scheme Step-reconstruction Characteristic field Reference solution

  • AMS Subject Headings

65M06 35L65.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

rchen@cug.edu.cn (Rongsan Chen)

dkmao@staff.shu.edu.cn (Dekang Mao)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-121, author = {Chen , Rongsan and Mao , Dekang }, title = {Improved Entropy-Ultra-Bee Scheme for the Euler System of Gas Dynamics}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {2}, pages = {121--151}, abstract = { The Entropy-Ultra-Bee scheme was developed for the linear advection equation and extended to the Euler system of gas dynamics in [13]. It was expected that the technology be applied only to the second characteristic field of the system and the computation in the other two nonlinear fields be implemented by the Godunov scheme. However, the numerical experiments in [13] showed that the scheme, though having improved the wave resolution in the second field, produced numerical oscillations in the other two nonlinear fields. Sophisticated entropy increaser was designed to suppress the spurious oscillations by increasing the entropy when there are waves in the two nonlinear fields presented. However, the scheme is then not efficient neither robust with problem-related parameters. The purpose of this paper is to fix this problem. To this end, we first study a 3 × 3 linear system and apply the technology precisely to its second characteristic field while maintaining the computation in the other two fields be implemented by the Godunov scheme. We then follow the discussion for the linear system to apply the Entropy-Ultra-Bee technology to the second characteristic field of the Euler system in a linearlized field-byfield fashion to develop a modified Entropy-Ultra-Bee scheme for the system. Meanwhile a remark is given to explain the problem of the previous Entropy-Ultra-Bee scheme in [13]. A reference solution is constructed for computing the numerical entropy, which maintains the feature of the density and flats the velocity and pressure to constants. The numerical entropy is then computed as the entropy cell-average of the reference solution. Several limitations are adopted in the construction of the reference solution to further stabilize the scheme. Designed in such a way, the modified Entropy-Ultra-Bee scheme has a unified form with no problem-related parameters. Numerical experiments show that all the spurious oscillations in smooth regions are gone and the results are better than that of the previous Entropy-Ultra-Bee scheme in [13].}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1603-m2015-0338}, url = {http://global-sci.org/intro/article_detail/jcm/9767.html} }
TY - JOUR T1 - Improved Entropy-Ultra-Bee Scheme for the Euler System of Gas Dynamics AU - Chen , Rongsan AU - Mao , Dekang JO - Journal of Computational Mathematics VL - 2 SP - 121 EP - 151 PY - 2017 DA - 2017/04 SN - 35 DO - http://doi.org/10.4208/jcm.1603-m2015-0338 UR - https://global-sci.org/intro/article_detail/jcm/9767.html KW - Entropy-Ultra-Bee scheme KW - Step-reconstruction KW - Characteristic field KW - Reference solution AB - The Entropy-Ultra-Bee scheme was developed for the linear advection equation and extended to the Euler system of gas dynamics in [13]. It was expected that the technology be applied only to the second characteristic field of the system and the computation in the other two nonlinear fields be implemented by the Godunov scheme. However, the numerical experiments in [13] showed that the scheme, though having improved the wave resolution in the second field, produced numerical oscillations in the other two nonlinear fields. Sophisticated entropy increaser was designed to suppress the spurious oscillations by increasing the entropy when there are waves in the two nonlinear fields presented. However, the scheme is then not efficient neither robust with problem-related parameters. The purpose of this paper is to fix this problem. To this end, we first study a 3 × 3 linear system and apply the technology precisely to its second characteristic field while maintaining the computation in the other two fields be implemented by the Godunov scheme. We then follow the discussion for the linear system to apply the Entropy-Ultra-Bee technology to the second characteristic field of the Euler system in a linearlized field-byfield fashion to develop a modified Entropy-Ultra-Bee scheme for the system. Meanwhile a remark is given to explain the problem of the previous Entropy-Ultra-Bee scheme in [13]. A reference solution is constructed for computing the numerical entropy, which maintains the feature of the density and flats the velocity and pressure to constants. The numerical entropy is then computed as the entropy cell-average of the reference solution. Several limitations are adopted in the construction of the reference solution to further stabilize the scheme. Designed in such a way, the modified Entropy-Ultra-Bee scheme has a unified form with no problem-related parameters. Numerical experiments show that all the spurious oscillations in smooth regions are gone and the results are better than that of the previous Entropy-Ultra-Bee scheme in [13].
Rongsan Chen & Dekang Mao . (2020). Improved Entropy-Ultra-Bee Scheme for the Euler System of Gas Dynamics. Journal of Computational Mathematics. 35 (2). 121-151. doi:10.4208/jcm.1603-m2015-0338
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