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 f...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-by- field 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].展开更多
In this paper, we apply arbitrary Riemann solvers, which may not satisfy the Maire＇s requirement, to the Maire＇s node-based Lagrangian scheme developed in [P. H. Maire et al., SIAM J. Sci. Comput, 29 （2007）, 1781-...In this paper, we apply arbitrary Riemann solvers, which may not satisfy the Maire＇s requirement, to the Maire＇s node-based Lagrangian scheme developed in [P. H. Maire et al., SIAM J. Sci. Comput, 29 （2007）, 1781-1824]. In particular, we apply the so-called Multi-Fluid Channel on Averaged Volume （MFCAV） Riemann solver and a Riemann solver that adaptively combines the MFCAV solver with other more dissipative Riemann solvers to the Maire＇s scheme. It is noted that neither of the two solvers satisfies the Maire＇s requirement. Numerical experiments are presented to demonstrate that the application of the two Riemann solvers is successful.展开更多
基金The research is supported by the National Natural Science Foundation of China No.10971132, No.11176015, No.11201436, No.11201435, the Leading Academic Discipline Project of Shanghai Municipal Education Commission, No.J50101, the National Basic Research Program of China（973 program） No.2011CB71064, No.2011CBT1065, and the Fundamental Research Funds for the Central Universities, China University of Geosciences（Wuhan）.
文摘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-by- field 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].
文摘In this paper, we apply arbitrary Riemann solvers, which may not satisfy the Maire＇s requirement, to the Maire＇s node-based Lagrangian scheme developed in [P. H. Maire et al., SIAM J. Sci. Comput, 29 （2007）, 1781-1824]. In particular, we apply the so-called Multi-Fluid Channel on Averaged Volume （MFCAV） Riemann solver and a Riemann solver that adaptively combines the MFCAV solver with other more dissipative Riemann solvers to the Maire＇s scheme. It is noted that neither of the two solvers satisfies the Maire＇s requirement. Numerical experiments are presented to demonstrate that the application of the two Riemann solvers is successful.
