Ultra-relativistic Euler Equations Research Articles

Radially symmetric solutions of the ultra-relativistic Euler equations in several space dimensions

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers.

The modified Rusanov scheme for solving the ultra-relativistic Euler equations

We consider the ultra-relativistic Euler equations for an ideal gas, which are described in terms of the pressure p and the spatial part u∈R3 of the dimensionless four-velocity. We also numerically investigate the ultra-relativistic Euler equations using the modified Rusanov scheme compared with classical Rusanov scheme. Our scheme consists of predictor and corrector stage. The predictor stage contains a parameter of control (αi+12n) is responsible for the numerical diffusion of this scheme. In order to control this parameter, we use a strategy depending on limiter theory and using Riemann invariants. The corrector stage recovers the balance conservation equation. The scheme can compute the numerical flux corresponding to the real state of solution without relying on Riemann problem solvers. The numerical results show the high resolution of the proposed finite volume scheme (modified Rusanov) and confirm its capability to provide accurate simulations for the ultra-relativistic Euler equations under regimes with strong shocks and rarefactions.

Radially symmetric solutions of the ultra-relativistic Euler equations

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$, the spatial part $\underline{u} \in \R^3$ of the dimensionless four-velocity and the particle density $n$. Radially symmetric solutions of these equations are studied. Analytical solutions are presented for the linearized system. For the original nonlinear equations we design and analyze a numerical scheme for simulating radially symmetric solutions in three space dimensions. The good performance of the scheme is demonstrated by numerical examples. In particular, it was observed that the method has the capability to capture accurately the pressure singularity formation caused by shock wave reflections at the origin.

A new variation for the relativistic Euler equations

The Glimm scheme is one of the so famous techniques for getting solutions of the general initial value problem by building a convergent sequence of approximate solutions. The approximation scheme is based on the solution of the Riemann problem. In this paper, we use a new strength function in order to present a new kind of total variation of a solution. Based on this new variation, we use the Glimm scheme to prove the global existence of weak solutions for the nonlinear ultra-relativistic Euler equations for a class of large initial data that involve the interaction of nonlinear waves.

Self-similar solutions of the radially symmetric relativistic Euler equations

The study of radially symmetric motion is important for the theory of explosion waves. We construct rigorously self-similar entropy solutions to Riemann initial-boundary value problems for the radially symmetric relativistic Euler equations. We use the assumption of self-similarity to reduce the relativistic Euler equations to a system of nonlinear ordinary differential equations, from which we obtain detailed structures of solutions besides their existence. For the ultra-relativistic Euler equations, we also obtain the uniqueness of the self-similar entropy solution to the Riemann initial-boundary value problems.

Conserved schemes with high pressure ratio, high particle density ratio and self-similar method

The paper proposes a numerical investigation for three conserved schemes when applied to the nonlinear ultra-relativistic Euler equations with an initial high pressure ratio and with an initial high particle density ratio. The results show that two of these schemes work efficiently and the third one may present inaccurate results even applied over a very delicate mesh. Several problems of the relativistic Euler system have self-similar solutions which can be solved by more efficient techniques. We also introduce a representation for self-similar ultra-relativistic Euler equations, which can be solved by the proposed schemes in this article and does not need to reconstruct schemes. Numerical tests are given for the one-dimensional shock tube problems to deal with the problem under consideration. These tests display that increasing the order of accuracy does not help much in upgrading the results. It is also indicated that one-dimensional results solved by self-similar ultra-relativistic Euler equations are almost identical to the exact solutions.

Numerical solution of special ultra-relativistic Euler equations using central upwind scheme

Abstract This article is concerned with the numerical approximation of one and two-dimensional special ultra-relativistic Euler equations. The governing equations are coupled first-order nonlinear hyperbolic partial differential equations. These equations describe perfect fluid flow in terms of the particle density, the four-velocity and the pressure. A high-resolution shock-capturing central upwind scheme is employed to solve the model equations. To avoid excessive numerical diffusion, the considered scheme avails the specific information of local propagation speeds. By using Runge-Kutta time stepping method and MUSCL-type initial reconstruction, we have obtained 2nd order accuracy of the proposed scheme. After discussing the model equations and the numerical technique, several 1D and 2D test problems are investigated. For all the numerical test cases, our proposed scheme demonstrates very good agreement with the results obtained by well-established algorithms, even in the case of highly relativistic 2D test problems. For validation and comparison, the staggered central scheme and the kinetic flux-vector splitting (KFVS) method are also implemented to the same model. The robustness and efficiency of central upwind scheme is demonstrated by the numerical results.

Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations

This paper presents second-order accurate genuine BGK schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present schemes are derived from the analytical solution of the Anderson–Witting model, which is given for the first time and includes the “genuine” particle collisions in the gas transport process. The proposed schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

Global solutions for the ultra-relativistic Euler equations

A front tracking scheme for the ultra-relativistic Euler equations is introduced. This scheme is based on piecewise constant approximations to the front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. We study the interaction estimates of the generalized shocks (entropy and nonentropy shocks) of the ultra-relativistic Euler equations and the outcoming asymptotic Riemann solution. Moreover we use a new function to measure the strengths of the waves of the ultra-relativistic Euler equations. This function has the important implication that the strength is non increasing for the interactions of the generalized shocks. This enables us to define a new kind of total variation of a solution. The main application of this scheme, is proving the global existence of weak solutions for the ultra relativistic Euler equations.

Solutions of the Ultra-Relativistic Euler Equations in Riemann Invariants

In this paper we introduce a new technique for constructing solutions of the ultra-relativistic Euler equations. The Riemann invariants are formulated. We also give some applications of the Riemann invariants. We firstly study the geometric properties of the solution in Riemann invariants coordinates. The other application of Riemann invariants, representing the ultra-relativistic Euler equations in diagonal form, which admits the existence of global smooth solution for the ultra-relativistic Euler equations.

A new front tracking scheme for the ultra-relativistic Euler equations

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure p, the spatial part u∈R3 of the dimensionless four-velocity and the particle density n. Two schemes for these equations are presented in one space dimension, namely a front tracking and a cone-grid scheme. A new front tracking technique for the ultra-relativistic Euler equations is introduced, which gives weak solutions. The front tracking method is based on piecewise constant approximations to Riemann solutions, called front tracking Riemann solutions, where continuous rarefaction waves are approximated by finite collections of discontinuities, so-called non-entropy shocks. This method can be used for analytical as well as for numerical purposes. A new unconditionally stable cone-grid scheme is also derived in this paper, which is based on the Riemann solution for the ultra-relativistic Euler equations. Both schemes are compared by two numerical examples, where explicit solutions are known.

The interaction of waves for the ultra-relativistic Euler equations

The interactions between nonlinear waves for the ultra-relativistic Euler equations for an ideal gas are studied. These equations are described in terms of the pressure p and the spatial part u∈R3 of the dimensionless four-velocity. A new function, which measures the strengths of the waves of the ultra-relativistic Euler equations is presented, and sharp estimates for these strengths are derived. The interpretation of the strength for the Riemann solution is given. This function has the important implication that the strength is non increasing for the interactions of waves for the system. This study of interaction estimates also allows to determine the type of the outgoing Riemann solutions.

The space–time CE/SE method for solving ultra-relativistic Euler equations

This paper reports the application of space–time conservation element and solution element (CE/SE) method for solving one- and two-dimensional special ultra-relativistic Euler equations. For a sufficiently large internal energy of fluid particles the rest-mass energy of the fluid can be ignored. Then, the fluid flow can be modeled by ultra-relativistic Euler equations consisting a pair of coupled first-order non-linear hyperbolic partial differential equations. The governing equations describe the flow of a perfect fluid in terms of the particle density ρ, the spatial part of the four-velocity u and the pressure p. The CE/SE method is capable to accurately captures the sharp propagating wavefront of relativistic fluid without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building block of the suggested method. The method differs from the previous techniques because of global and local flux conservation in a space–time domain without resorting to interpolation or extrapolation. In order to reveal the efficiency and performance of the approach, several numerical test cases are presented in this manuscript. For validation, the results of current method are compared with other finite-volume schemes.

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.

A BGK-Type Flux-Vector Splitting Scheme for the Ultrarelativistic Euler Equations

A gas-kinetic solver is developed for the ultrarelativistic Euler equations. The scheme is based on the direct splitting of the flux function of the Euler equations with inclusion of ``particle'' collisions in the transport process. Consequently, the artificial dissipation in the new scheme is much reduced in comparison with the usual kinetic flux-vector splitting (KFVS) schemes which are based on the free particle transport at the cell interfaces in the gas evolution stage. Although in a usual KFVS scheme the free particle transport gives robust solutions, it gives a smeared solution at the contact discontinuities. The new BGK-type KFVS scheme solves this problem and gives robust and reliable solutions as well as good resolution at the contact discontinuity. In contrast to the classical kinetic theory, we have only a finite domain of dependence due to the structure of the light cone. The scheme is naturally multidimensional and is extended to the two-dimensional case in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction. In the numerical case studies the results obtained from the BGK-type KFVS schemes are compared with the exact Riemann solution, KFVS schemes, upwind schemes, and central schemes.

Second-order accurate kinetic schemes for the ultra-relativistic Euler equations

A second-order accurate kinetic scheme for the numerical solution of the relativistic Euler equations is presented. These equations describe the flow of a perfect fluid in terms of the particle density n, the spatial part of the four-velocity u and the pressure p. The kinetic scheme, is based on the well-known fact that the relativistic Euler equations are the moments of the relativistic Boltzmann equation of the kinetic theory of gases when the distribution function is a relativistic Maxwellian. The kinetic scheme consists of two phases, the convection phase (free-flight) and collision phase. The velocity distribution function at the end of the free-flight is the solution of the collisionless transport equation. The collision phase instantaneously relaxes the distribution to the local Maxwellian distribution. The fluid dynamic variables of density, velocity, and internal energy are obtained as moments of the velocity distribution function at the end of the free-flight phase. The scheme presented here is an explicit method and unconditionally stable. The conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. The scheme also satisfies positivity and L 1-stability. The scheme can be easily made into a total variation diminishing method for the distribution function through a suitable choice of the interpolation strategy. In the numerical case studies the results obtained from the first- and second-order kinetic schemes are compared with the first- and second-order upwind and central schemes. We also calculate the experimental order of convergence and numerical L 1-stability of the scheme for smooth initial data.

Kinetic schemes for the ultra-relativistic Euler equations

We present a kinetic numerical scheme for the relativistic Euler equations, which describe the flow of a perfect fluid in terms of the particle density n, the spatial part of the four-velocity u and the pressure p. The kinetic approach is very simple in the ultra-relativistic limit, but may also be applied to more general cases. The basic ingredients of the kinetic scheme are the phase-density in equilibrium and the free flight. The phase-density generalizes the non-relativistic Maxwellian for a gas in local equilibrium. The free flight is given by solutions of a collision free kinetic transport equation. The scheme presented here is an explicit method and unconditionally stable. We establish that the conservation laws of mass, momentum and energy as well as the entropy inequality are everywhere exactly satisfied by the solution of the kinetic scheme. For that reason we obtain weak admissible Euler solutions including arbitrarily complicated shock interactions. In the numerical case studies the results obtained from the kinetic scheme are compared with the first order upwind and centered schemes.