Entropy-limited hydrodynamics: a novel approach to relativistic hydrodynamics Neutron star mergers: From gravitational waves to nucleosynthesis Hirschegg, Austria — January 18, 2017 Federico Maria Guercilena Institut für Theoretische Physik Goethe-Universität Frankfurt am Main, Germany with David Radice (Princeton) Luciano Rezzolla (Frankfurt)
Besides physical research, there is research on better numerical methods. An ideal numerical scheme should be: • Accurate .
• Parallelizable
... and hopefully easy to implement. The relativistic Euler equations:
where h = (e + ρ )/ρ and P μν = gμν + u ν ν , closed by an EoS ρ = ρ(ρ, e). Cast them into a flux-conservative formulation (the “Valencia formulation"):
Several issues arise when solving these equations, stemming from their non-linearity, most importantly the generation of shocks.
The standard way:  use of HRSC techniques to preserve stability without sacrificing accuracy. HRSC methods feature: – second order of accuracy (or higher) – sharp resolution of discontinuities — no oscillations e.g. PPM, ENO, WENO, MP5... However successful, these methods can potentially suffer from a few shortcomings. . .
Flux-limiter approach
1/2 is a high order, but unfiltered approximation of the flux f LF / 2 is the Lax-Friedrichs flux, which is only first order but stable We want:
where θ guarantees the positivity of the density.
Consider the physical specific entropy s, which for the ideal-gas EoS ρ = (Γ − 1)ρε equals s = log (ֲֲֲֲֲֲֲֲֲThe second principle of thermodynamics can be written:
Therefore one expects the entropy production rate (or entropy residual) R to be a Dirac delta centered at the location of shocks, R = δ(x − x s ). This suggests to define the viscosity as proportional to the entropy residual: Entropy viscosity ν = min[c εΔ | R | , Cmax ] Note the presence of two arbitrary constants, c e and Cmax-
We implemented the ELH scheme in the WhiskyTHC code (Radice et al.), which features: ·  finite differences flux reconstruction... ·  applied to components or characteristics variables. . . • with upwinding; · a positivity preserving limiter.
In the 3+1 decomposition of GR, one can write:
Computed by 2nd order one-sided finite-differences on the past timelevels:
Computed by n th+1 order

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A non-linear, smooth hydrodynamical wave, propagating to the right and tilting in the direction of its motion, until a caustic is produced. Sod's relativistic shock tube test, with initial data: ( ρι, νι, ρι) = (1, 0, 1),    (ρι, νι, ρι) = (0.125, 0, 0.1). TOV star in the Cowling approximation ( i.e. the spacetime is fixed, only the matter is evolved). Figure 5: Viscosity distribution on xy plane at t = 4500 Mo Figure 6: Density distribution on xy plane at t = 4500 Mo
Figure 8:  Continuous lines:  Power spectral density of the central density evolution. Dashed lines: physical oscillation eigenfrequencies
Figure 14: Density distribution on xy plane at t = 61 M ⊙ (i.e. just after collapse) Figure 16:  Black hole mass evolution The entropy limited hydrodynamics scheme is an interesting, robust alternative to the common HRSC schemes. It addresses the issues of – Accuracy – Speed – Ease of implementation and extendability with no tuning of the free parameters. The entropy limited hydrodynamics scheme is an interesting, robust alternative to the common HRSC schemes. It addresses the issues of – Accuracy – Speed – Ease of implementation and extendability with no tuning of the free parameters. Future prospects application to binary neutron stars with nuclear equation of state exploit the simplicity of the scheme to efficiently run on MICs/GPUs • coupling to discontinuous Galerkin scheme • use of truly multidimensional methods The entropy limited hydrodynamics scheme is an interesting, robust alternative to the common HRSC schemes. It addresses the issues of – Accuracy – Speed – Ease of implementation and extendability with no tuning of the free parameters. Future prospects application to binary neutron stars with nuclear equation of state exploit the simplicity of the scheme to efficiently run on MICs/GPUs • coupling to discontinuous Galerkin scheme • use of truly multidimensional methods Thank you!