报告摘要
This talk mainly reviews two high-order accurate structure-preserving finite difference schemes for the special relativistic hydrodynamics (RHD). The first is the physical-constraints-preserving (PCP) scheme, which preserves the positivity of the rest-mass density and the pressure and the bounds of the fluid velocity and is built on the local Lax-Friedrichs (LxF) splitting, the WENO reconstruction, the PCP flux limiter, and the high-order strong stability preserving time discretization. The key to developing such scheme is to prove the convexity and other properties of the admissible state set and to discover a concave function with respect to the conservative vector. The second is the entropy stable (ES) scheme, whose semi-discrete version satisfies the entropy inequality. The key is to technically construct the affordable entropy conservative (EC) flux of the semi-discrete second-order accurate EC schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the EC flux is derived, the dissipation term can be added to give the semi-discrete ES schemes satisfying the semi-discrete entropy inequality. The WENO reconstruction for the scaled entropy variables and the previous time discretization are implemented to obtain the fully-discrete high-order “ES” schemes. The performance of the proposed schemes has been demonstrated by numerical experiments. By the way, we also briefly review other relative works on the structure-preserving schemes for the special RHDs. Those works have been further to the general equation of state and the special relativistic magnetohydrodynamics etc., see our papers listed below for details.
References
1. Wu, K.L. and Tang, H.Z., “High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics”, J. Comput. Phys., Vol. 298, 2015, pp. 539-564.
2. Wu, K.L. and Tang, H.Z., “Physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state”, Astrophys. J. Suppl. ser., Vol. 228, 2017, 3.
3. Wu, K.L. and Tang, H.Z., “Admissible states and physical constraints preserving numerical schemes for special relativistic magnetohydrodynamics”, Math. Mod. and Meth. Appl. Sci., Vol. 27, 2017, pp. 1871-1928.
4. Wu, K.L. and Tang, H.Z., “On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state”, Z. Angew. Math. Phys., Vol. 69, 2018, 84.
5. Ling, D., Duan, J.M. and Tang, H.Z., “Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics”, J. Comput. Phys., Vol. 396, 2019, pp. 507-543.
6. Ling, D. and Tang, H.Z., “Genuinely multidimensional physical-constraints-preserving finite volume schemes for the special relativistic hydrodynamics”, submitted to Commun. Comput. Phys., March 4, 2023. arXiv: 2303.02686.
7. Duan, J.M. and Tang, H.Z., “High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics”, Adv. Appl. Math. Mech., Vol. 12, 2020, pp. 1-29.
8. Duan, J.M. and Tang, H.Z., “High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics”, J. Comput. Phys., Vol. 421, 2020, 109731.
9. Duan, J.M. and Tang, H.Z., “Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics”, J. Comput. Phys., Vol. 426, 2021, 109949.
