  The construction of high-resolution shock-capturing schemes is vital in
producing highly accurate gravitational waveforms from neutron star binaries.
The entropy based flux limiting (EFL) scheme is able to perform fast converging
binary neutron star merger simulations reaching up to fourth-order convergence
in the gravitational waveform phase. Here, we extend the applicability of the
EFL method beyond special/general relativistic hydrodynamics to scalar
conservation laws and show how to treat systems without a thermodynamic
entropy. This is an indication that the method has universal applicability to
any system of partial differential equations that can be written in
conservation form. We also present some further very challenging
special/general relativistic hydrodynamics applications of the EFL method and
use it to construct eccentricity reduced initial data for a specific neutron
star binary and show up to optimal fifth-order convergence in the gravitational
waveform phase for this simulation.
