Entropy-stable schemes for relativistic hydrodynamics equations
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Entropy-stable schemes for relativistic hydrodynamics equations Deepak Bhoriya and Harish Kumar Abstract. In this article, we propose high-order finite difference schemes for the equations of relativistic hydrodynamics, which are entropy stable. The crucial components of these schemes are a computationally efficient entropy conservative flux and suitable high-order entropy dissipative operators. We first design a higher-order entropy conservative flux. For the construction of appropriate entropy dissipative operators, we derive entropy scaled right eigenvectors. This is then used with ENO-based sign-preserving reconstruction of scaled entropy variables, which results in higher-order entropy-stable schemes. Several numerical results are presented up to fourth order to demonstrate entropy stability and performance of these schemes. Mathematics Subject Classification. 65M08, 65M12. Keywords. Relativistic hydrodynamics, Symmetrization, Entropy stability, Finite difference scheme.
Contents 1. 2. 3.
Introduction Equations of relativistic hydrodynamics Entropy-stable numerical schemes 3.1. Entropy conservative schemes 3.1.1. Entropy conservative flux 3.1.2. Higher-order entropy conservative flux 3.2. Entropy-stable schemes 3.3. High-order diffusion operators 3.4. Semi-discrete entropy stability 3.5. Time discretization 4. Numerical results 4.1. One-dimensional test cases 4.1.1. Accuracy test 4.1.2. Isentropic smooth flows 4.1.3. Riemann problem 1 4.1.4. Riemann problem 2 4.1.5. Riemann problem 3 4.1.6. Density perturbation test case 4.1.7. Blast waves test case 4.2. Two-dimensional test cases 4.2.1. Two-dimensional Riemann problem 1 4.2.2. Two-dimensional Riemann problem 2 4.2.3. Two-dimensional Riemann problem 3 4.2.4. Two-dimensional Riemann problem 4 4.2.5. Two-dimensional Riemann problem 5 0123456789().: V,-vol
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D. Bhoriya and H. Kumar
ZAMP
5. Conclusion Acknowledgements Appendix A. Computation of primitive variables Appendix B. Entropy scaled right eigenvectors References
1. Introduction The equations of relativistic hydrodynamics are used to model astrophysical flows when the fluid is moving with speed comparable to the speed of light, or the flow is under the influence of large gravitational potentials such that relativistic effects cannot be ignored. Some examples are gamma-ray burst, relativistic jets from Galactic sources, core-collapse supernovae and extragalactic jets from active Galactic nuclei (see [5,6,20,27,39]). In this article, we consider the equations of special relativistic hydrodynamics (RHD henceforth) with the ideal equation of state. The system of equations is hyperbolic conservation laws. So, the solutions of the system can exhibit discontinuities, even for the smooth initial data (see [15]). Therefore, we consider weak solutions of the systems which are characterized by the Rankine–Hugoniot jump condition across the discontinuities. However, the weak solutions of the system can be non-unique; hence, an additional criterion in the for
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