Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics

Abstract

This paper studies the physical-constraints-preserving (PCP) Lagrangian finite volume schemes for one-and two-dimensional special relativistic hydrodynamic (RHD) equations. First, the PCP property (i.e. preserving the positivity of the rest-mass density and the pressure and the bound of the velocity) is proved for the first-order accurate Lagrangian scheme with the HLLC Riemann solver and forward Euler time discretization. The key is that the intermediate states in the HLLC Riemann solver are shown to be admissible or PCP when the HLLC wave speeds are estimated suitably. Then, the higher-order accurate schemes are proposed by using the high-order accurate strong stability preserving (SSP) time discretizations and the scaling PCP limiter as well as the WENO reconstruction. Finally, several one-and two-dimensional numerical experiments are conducted to demonstrate the accuracy and the effectiveness of the PCP Lagrangian schemes in solving the special RHD problems involving strong discontinuities, or large Lorentz factor, or low rest-mass density or low pressure etc.

High-Order Accurate Entropy Stable Finite Difference Schemes for One- and Two-Dimensional Special Relativistic Hydrodynamics

“…Example 4.8 (Accuracy test). It is a 2D relativistic isentropic vortex problem to test the accuracy, whose detailed construction can be found in [24]. The initial rest-mass density and pressure are and the initial velocities are…”

An analytical solution of the isentropic vortex problem in the special relativistic magnetohydrodynamics

“…B = 0 or g = 0), which is helpful for us to deal with the RMHD case. It is worth noting that the analytical solution of the 2D isentropic vortex for this case has been provided in [8] without the detailed derivation and used in [5,7].…”

“…For the RHD and RMHD equations, an isentropic vortex problem is constructed in [2], where an ordinary differential equation (ODE) should be integrated numerically to obtain the initial solutions at each given grid point, which is not convenient. In [8], the analytic solution of the isentropic vortex problem with the algebraic expression is given for the RHD equations and has been used to test the accuracy of the high-order accurate entropy conservative and stable schemes in [5,7]. This note aims at deriving an analytical solution of the isentropic vortex problem with explicit algebraic expressions for the 2D and 3D RMHD equations (1.1)-(1.3).…”

“…It can be verified from the above expression of f that the velocities v 1 and v 2 vanish rapidly as r increases. To sum up, (2.2)-(2.3) and (3.2)-(3.4) gives algebraic expressions of the analytical solution of the 2D isentropic vortex problem in the RHDs, which have been provided in[8]. The case of σ < 0 and ς < 0 may be similarly discussed by changing (σ, ς) as (−σ, −ς).…”

Geometric Quasilinearization Framework for Analysis and Design of Bound-Preserving Schemes

“…Therefore, the design and analysis of bound-preserving schemes involving nonlinear constraints are highly nontrivial, even for first-order schemes; cf. [49,2,48,39,26,34,36,57,51].…”