  We present entropy-limited hydrodynamics (ELH): a new approach for the
computation of numerical fluxes arising in the discretization of hyperbolic
equations in conservation form. ELH is based on the hybridisation of an
unfiltered high-order scheme with the first-order Lax-Friedrichs method. The
activation of the low-order part of the scheme is driven by a measure of the
locally generated entropy inspired by the artificial-viscosity method proposed
by Guermond et al. Here, we present ELH in the context of high-order
finite-differencing methods and of the equations of general-relativistic
hydrodynamics. We study the performance of ELH in a series of classical
astrophysical tests in general relativity involving isolated, rotating and
nonrotating neutron stars, and including a case of gravitational collapse to
black hole. We present a detailed comparison of ELH with the fifth-order
monotonicity preserving method MP5, one of the most common high-order schemes
currently employed in numerical-relativity simulations. We find that ELH
achieves comparable and, in many of the cases studied here, better accuracy
than more traditional methods at a fraction of the computational cost (up to 50
% speedup). Given its accuracy and its simplicity of implementation, ELH is a
promising framework for the development of new special- and
general-relativistic hydrodynamics codes well adapted for massively parallel
supercomputers.
