  We provide a road towards obtaining gravitational waveforms from inspiraling
material binaries with an accuracy viable for third-generation gravitational
wave detectors, without necessarily advancing computational hardware or
massively-parallel software infrastructure. We demonstrate a proof-of-principle
1+1-dimensional numerical implementation that exhibits up to 7th-order
convergence for highly dynamic barotropic stars in curved spacetime, and
numerical errors up to 6 orders of magnitude smaller than a standard method.
Aside from high-order interpolation errors (Runge's phenomenon), there are no
obvious fundamental obstacles to obtaining convergence of even higher order.
The implementation uses a novel surface-tracking method, where the surface is
evolved and high-order accurate boundary conditions are imposed there.
Computational memory does not need to be allocated to fluid variables in the
vacuum region of spacetime. We anticipate the application of this new method to
full $3\! +\! 1$-dimensional simulations of the inspiral phase of compact
binary systems with at least one material body. The additional challenge of a
deformable surface must be addressed in multiple spatial dimensions, but it is
also an opportunity to input more precise surface tension physics.
