An Operator-Splitting Approach to the Godunov Method for Numerically Solving the Special Relativistic Hydrodynamics Equations in Symmetric Hyperbolic Form

• Published in: Lobachevskii journal of mathematics Vol. 46; no. 1; pp. 112 - 120
• Main Author: Kulikov, I. M.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01.01.2025; Springer Nature B.V
• Subjects: Active galactic nuclei; Algebra; Analysis; Cauchy problems; Exact solutions; Geometry; Godunov method; Hydrodynamic equations; Hydrodynamics; Hyperbolic systems; Linear algebra; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Molecular clouds; Numerical methods; Probability Theory and Stochastic Processes; Problem solving; Relativistic effects; Software; Splitting; computational astrophysics; Godunov method; 2010 Mathematics Subject Classification: 65M08; relativistic hydrodynamics
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080224608294

This paper considers a construction of the Godunov method to numerically solve the equations of special relativistic hydrodynamics. In this method, the equations are presented as a symmetric hyperbolic system decoupled into equations that describe the work of pressure forces and the advective relativistic gas transfer. This makes it possible to solve the Riemann problem by solving problems similar to the ‘‘sound point’’ one in hydrodynamic numerical methods. This operator splitting method allows finding analytical solutions to the problem of relativistic hydrodynamic discontinuity breakdown without using any computational linear algebra tools. The representation of the equations in symmetric hyperbolic form allows determining the hyperbolicity zone of the equations themselves. The construction of the method is verified in test problems and in the problem of interaction of the relativistic wind from active galactic nuclei with the molecular clouds.