Numerical Study of Singularity Formation in Relativistic Euler Flows

Abstract

The formation of singularities in relativistic flows is not well understood.Smooth solutions to the relativistic Euler equations are known to have a finite lifespan; the possible breakdown mechanisms are shock formation, violation of the subluminal conditions and mass concentration. We propose a new hybrid Glimm/centralupwind scheme for relativistic flows. The scheme is used to numerically investigate, for a family of problems, which of the above mechanisms is involved.

A numerical exploration of first-order relativistic hydrodynamics

“…For all three of the systems of PDEs considered herethe relativistic Euler, BDNK, and MIS equations-all of the discrete elements in the algorithm are second-order accurate, with two exceptions: first is that the perfect fluid part of the flux f P F , as a result of the slope limiter, converges at second order only in regions where the solution is smooth, elsewhere it is first order; the other is in the MIS π xx 2 evolution equation (35), which uses a first-order upwind discretization for the advection operator (46). As a result-see Fig.…”

“…A naive scheme of the form (38) should work in principle as long as the solutions are smooth. However, solutions to the relativistic Euler equations ( 1), ( 26)- (28), are not generically smooth, as discontinuities in , v (shockwaves) can form dynamically [45,46]. In these cases the physical solution is given not by direct solution of the PDEs (38)-as derivative terms , v diverge-but instead by solution to the weak formulation of the equations [47].…”

“…As mentioned earlier, a well-known property of the inviscid equations is that flows which are initially smooth and subsonic can evolve to a state with discontinuities. While the formation of these discontinuities is nontrivial and not yet fully understood [45] [46], it is simpler to see why they persist once formed (beyond the intuition that without viscosity there is no mechanism to smooth them out). This comes from considering the characteristics of the PDEs, which for the relativistic Euler equations with conformal fluid equation of state, evaluated in the rest frame v = 0 of the fluid, are…”