Abstract
The monotonicity-preserving (MP) scheme is an accurate shock-capturing scheme. However, its performance is still inefficient for resolving high-frequency waves. In this paper, to improve the resolution characteristics, an upwind compact interpolation is proposed as a substitute to the original one in the MP scheme, and the coefficients of that were analytically optimized to minimize the dispersion and dissipation errors. Moreover, it was found that the limiting part of the original MP scheme degenerates the accuracy in a high-wavenumber region due to unnecessarily activation. This limitation is improved by applying a new indicator and criterion. The results of the nonlinear wave (N-wave) propagation demonstrate that the proposed scheme guarantees the robustness at the sharp discontinuity. At the same time, the solutions of linear wave propagation prove the excellent resolution of the proposed scheme. We intensively evaluated the performance for the standard and long-time situations of the shock-entropy wave interaction problems. The results prove that the usefulness of proposed scheme is more pronounced in the flow fields involving both of shock and waves.
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Acknowledgements
This work was supported by Space Core Technology Development Program through the National Research Foundation of Korea (NRF), which is funded by the Ministry of Science, ICT & Future Planning [NRF-2014M1A3A3A02034837].
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Appendix
Appendix
In this Appendix, we investigated how the modified MP constraints works with different unwinding interpolations. We solved the long-time shock-entropy wave interaction test problem under the reduced number of grids (N = 750, ≈ 5.0 PPW) condition. In particular, not only the interpolations of MP5 and OMP6 but also that of MP7-LD is considered because it would not result oscillating solutions with the modified MP constraints.
The simulation results are summarized in Fig. 18. First of all, the solution of MP5 with modified MP constraints is almost the same with that the original MP5 (see Fig. 15). From this result, we found that unless the performance of interpolation is enough to resolve high-wavenumber itself, the modified MP constraints does not contribute to improve the accuracy. This is natural because the role of modified MP constraints just helps to avoid unnecessary limiting procedures. Meanwhile, the solution of OMP6 with modified MP constraints seems to be improved compared with the original one; however, there is a large phase error due to lack of dispersion resolution. At the middle of high-frequency region, peaks does not fit at all with the reference solution. On the other hand, MP7-LD shows slightly better behavior of dispersion but failed to predict the peaks of solution.
Solution of long-time shock entropy wave interaction problem (N = 750, ≈ 5.0 PPW)
In summary, we conclude the newly proposed limiting method can be used with any interpolation. However, to obtain a good quality of solution in supersonic flow, the excellent resolving performance of interpolation must be supported.
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Ahn, MH., Lee, DJ. Modified Monotonicity Preserving Constraints for High-Resolution Optimized Compact Scheme. J Sci Comput 83, 34 (2020). https://doi.org/10.1007/s10915-020-01221-0
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DOI: https://doi.org/10.1007/s10915-020-01221-0