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We provide a road towards obtaining gravitational waveforms from inspiralingmaterial binaries with an accuracy viable for third-generation gravitationalwave detectors, without necessarily advancing computational hardware ormassively-parallel software infrastructure. We demonstrate a proof-of-principle1+1-dimensional numerical implementation that exhibits up to 7th-orderconvergence for highly dynamic barotropic stars in curved spacetime, andnumerical errors up to 6 orders of magnitude smaller than a standard method.Aside from high-order interpolation errors (Runge's phenomenon), there are noobvious fundamental obstacles to obtaining convergence of even higher order.The implementation uses a novel surface-tracking method, where the surface isevolved and high-order accurate boundary conditions are imposed there.Computational memory does not need to be allocated to fluid variables in thevacuum region of spacetime. We anticipate the application of this new method tofull $3\! +\! 1$-dimensional simulations of the inspiral phase of compactbinary systems with at least one material body. The additional challenge of adeformable surface must be addressed in multiple spatial dimensions, but it isalso an opportunity to input more precise surface tension physics.

Extremely High-Order Convergence in Simulations of Relativistic Stars