Capturing Composite Waves in Non-convex Special Relativistic Hydrodynamics

The non-monotonicity of the local speed of sound in dense matter at baryon number densities much higher than the nuclear saturation density ($n_0 \approx 0.16\,$fm$^{-3}$) suggests the possible existence of a non-convex thermodynamics which will lead to a non-convex dynamics. Here, we explore the rich and complex dynamics that an equation of state (EoS) with non-convex regions in the pressure-density plane may develop as a result of genuinely relativistic effects, without a classical counterpart. To this end, we have introduced a phenomenological EoS, whose parameters can be restricted heeding to causality and thermodynamic stability constraints. This EoS shall be regarded as a toy-model with which we may mimic realistic (and far more complex) EoS of practical use in the realm of Relativistic Hydrodynamics.

ABSTRACTPhase transition in compressible flows involves capillarity effects, described by the Euler–Korteweg (EK) equations with nonconvex equation of state. Far from phase transition, that is, in the two convex parts of the equation of state, the dispersion terms vanish and one should recover the hyperbolic Euler equations of fluid dynamics. However, the solution of EK equations does not converge toward the solution of Euler equations when dispersion tends toward zero while being nonnull: it is a singular limit problem. To avoid this issue in the case of convex equation of state, a Navier–Stokes–Korteweg (NSK) model is considered, whose viscosity is chosen to counterbalance exactly the dispersive terms. In the limit of small viscosity and small dispersion, the Euler model is recovered. Numerically, an extended Lagrangian method is used to integrate the NSK equations so obtained. Doing so allows to use classical numerical schemes of Godunov type with source term. Numerical results for a Riemann problem illustrate the convergence properties with vanishing dispersion.

We present a new special relativistic hydrodynamics (SRHD) code capable of handling coexisting ultra-relativistically hot and non-relativistically cold gases. We achieve this by designing a new algorithm for conversion between primitive and conserved variables in the SRHD solver, which incorporates a realistic ideal-gas equation of state covering both the relativistic and non-relativistic regimes. The code can handle problems involving a Lorentz factor as high as 106 and optimally avoid the catastrophic cancellation. In addition, we have integrated this new SRHD solver into the code gamer (https://github.com/gamer-project/gamer) to support adaptive mesh refinement and hybrid OpenMP/MPI/GPU parallelization. It achieves a peak performance of 7 × 107 cell updates per second on a single Tesla P100 GPU and scales well to 2048 GPUs. We apply this code to two interesting astrophysical applications: (a) an asymmetric explosion source on the relativistic blast wave and (b) the flow acceleration and limb brightening of relativistic jets.

On the computation of the fundamental derivative of gas dynamics using equations of state

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

To study the dynamics of relativistic flows in astrophysical objects such as radio jets, we have developed a new special relativistic hydrodynamic (RHD) code based on the weighted essentially non-oscillatory (WENO) scheme, a high-order finite difference scheme. The code includes different WENO versions, and high-order time integration methods such as the 4th-order accurate Runge-Kutta (RK4) and strong stability preserving RK (SSPRK), as well as the equations of state (EOSs) that closely approximate the EOS of the single-component perfect gas in relativistic regime. Additionally, it is optimized for the reproduction of complex structures in multi-dimensional flows, and implements a modification of eigenvalues for the acoustic modes to effectively control carbuncle instability. As the first application of the code, we have simulated ultra-relativistic jets of FR II radio galaxies, and studied the nonlinear flow structures, such as shocks, velocity shear, and turbulence, through large-scale.

We apply a new way for the construction of efficient non-oscillation shock-capturing schemes, namely artificial wind (AW) scheme, for relativistic hydrodynamics (RHD) simulations in astrophysics. We also discuss physical foundations for computational RHD. An appropriate choice for a realistic equation of state of the matter at relativistic temperatures drastically facilitates the procedure for numerical integration of hydrodynamics equations. The time-consuming iteration approach to solve the coupling equation is avoided. We have developed a 2-D relativistic hydrodynamics code based on the AW scheme and the new RHD. Here we present two applications of the code in astrophysics, namely, numerical simulations of ultra-relativistic jet propagation and relativistic Richtmyer-Meshkov instability.

A conservative, weakly nonlinear semi-implicit finite volume scheme for the compressible Navier[formula omitted]Stokes equations with general equation of state

The problem of a one-dimensional (1D) cylindrically or spherically symmetric shock wave converging into an inviscid, ideal gas was first investigated by Guderley[Starke kugelige und zylinrische verdichtungsstosse in der nahe des kugelmitterpunktes bzw. Der zylinderachse,” Luftfahrtforschung 19, 302 (1942)]. In the time since, many authors have discussed the practical notion of how Guderley-like flows might be generated. One candidate is a constant velocity, converging “cylindrical or spherical piston,” giving rise to a converging shock wave in the spirit of its classical, planar counterpart. A limitation of pre-existing analyses along these lines is the restriction to flows in materials described by an ideal gas equation of state (EOS) constitutive law. This choice is of course necessary for the direct comparison with the classical Guderley solution, which also features an ideal gas EOS. However, the ideal gas EOS is limited in its utility in describing a wide variety of physical phenomena and, in particular, the shock compression of solid materials. This work is thus intended to provide an extension of previous work to a nonideal EOS. The stiff gas EOS is chosen as a logical starting point due to not only its close resemblance to the ideal gas law but also its relevance to the shock compression of various liquid and solid materials. Using this choice of EOS, the solution of a 1D planar piston problem is constructed and subsequently used as the lowest order term in a quasi-self-similar series expansion intended to capture both curvilinear and nonideal EOS effects. The solution associated with this procedure provides correction terms to the 1D planar solution so that the expected accelerating shock trajectory and nontrivially varying state variable profiles can be obtained. This solution is further examined in the limit as the converging shock wave approaches the 1D curvilinear origin. Given the stiff gas EOS is not otherwise expected to admit a Guderley-like solution when coupled to the inviscid Euler equations, this work thus provides the semianalytical limiting behavior of a flow that cannot be otherwise captured using self-similar analysis.

Background One source of danger to pipelines laid in difficult engineering and geological conditions are groundwater changes, including landslides. The monitoring system of pipelines in such areas must prevent these phenomena, consider the mechanisms and patterns of interaction with the active pipeline ground, supported by modern means of observation and measurement, as well as more sophisticated computational methods. Aims and Objectives to develop a mathematical model of the interaction of the underground water pipelines with soil in the area of longitudinal shear (landslide); to investigate the properties of the differential equation of state of the pipeline in such an area; to develop an algorithm for solving the equation of state in the general case. Methods In deriving the equation of state used the methods of mathematical physics and theoretical mechanics. The state properties is calculated by mathematical analysis. Algorithm for solving the general case is developed using finite difference methods, iterative, successive approximations. Results The developed algorithm is characterized by a feature - in the problem solution it is possible to specify initial and boundary conditions, including the reaction of the soil. There are studied some regularities of the landslide impact on the pipeline. In particular, it was found that the zone of perturbation stress field extends over several hundred meters to each side of the landslide border area.

We give a solution of the Riemann problem in relativistic hydrodynamics in the case of ultrarelativistic equation of state and nonvanishing components of the velocity tangent to the initial discontinuity. Simplicity of the ultrarelativistic equation of state (the pressure being directly proportional to the energy density) allows us to express this solution in analytical terms. The result can be used both to construct and test numerical schemes for relativistic Euler equations in (3+1) dimensions.

97/04852 Coal pulsating combustion model: Xu, W. Ranshao Kexue Yu Jishu., 1996, 2, (3), 260–269. (In Chinese)

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

The hybrid finite element (FE)-boundary element (BE) formulation for electromagnetic diffusion problems provides some advantages in modeling including (1) no truncation error, (2) ease of model generation, and (3) ease in solving problems involving moving conductors. However, the resulting matrix from this formulation loses its symmetry and is not banded, unlike the FE method. This results in significantly lower convergence rates in iterative solution schemes. A novel, split-domain iteration scheme (SDIS) was developed to reduce computation times. The new scheme splits the domain into FE and BE subdomains. Only the matrix associated with the FE subdomain is solved subject to initial guessed potentials along the interface of two subdomains. The BE equation is then used as the corrector to update the prescribed interface potentials. The iteration scheme proceeds with these newly corrected interface potentials until the solution converges within the required prescribed error tolerances. In this paper, detailed descriptions of the newly developed scheme and performance comparisons are presented with a sample railgun problem.

98/03268 Reopening of pulverized coal injection in blast furnace after a pause