Improved Entropy-Ultra-Bee Scheme for the Euler System of Gas Dynamics (Journal of Computational Mathematics, 2018)
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 could be applied only to the second characteristic field of the system and the computation in the other two nonlinear fields would 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 neither efficient nor 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 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 those of the previous Entropy-Ultra-Bee scheme in [13].