Toward high-precision inspiral gravitational waveforms from binary neutron star mergers in numerical relativity

We report the performance of a newly implemented fourth-order accurate finite-volume Harten-Lax-van Leer-contact Riemann solver in the adaptive-mesh-refinement numerical relativity code -. First, we validate our implementation in one-dimensional special relativistic hydrodynamics tests, i.e., a simple wave and shock tube test, which have analytic solutions. We demonstrate that the fourth-order convergence is achieved for the smooth flow, which cannot be achieved in our original second-order accurate finite-volume Riemann solver. We also show that our new solver is robust for the strong shock wave emergence problem. Second, we validate the implementation in a dynamical spacetime by demonstrating that - perfectly preserves the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>π</a:mi></a:math> symmetry without imposing the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi></c:math> symmetry in a short-term (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>∼</e:mo><e:mn>20</e:mn><e:mtext> </e:mtext><e:mtext> </e:mtext><e:mi>ms</e:mi></e:math> in the inspiral and subsequent postmerger phase) nonspinning equal-mass binary neutron star merger simulations. Finally, we quantify the accuracy of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo>≈</g:mo><g:mn>28</g:mn></g:math> cycle inspiral gravitational waveforms from binary neutron star mergers by conducting a resolution study with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo>≈</i:mo><i:mn>78</i:mn></i:math>, 94, 118, and 135 m. We find that the fourth-order accurate Riemann solver achieves the convergence order <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo>≈</k:mo><k:mn>2.1</k:mn><k:mo>±</k:mo><k:mn>0.05</k:mn><k:mi>–</k:mi><k:mn>2.4</k:mn><k:mo>±</k:mo><k:mn>0.27</k:mn></k:math>, i.e., slightly evolving with time, in the inspiral gravitational wave phase, while the second-order accurate Riemann solver achieves the convergence order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mo>≈</m:mo><m:mn>2.0</m:mn><m:mo>±</m:mo><m:mn>0.5</m:mn></m:math>. The residual phase error towards the continuum limit at the merger is <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mn>0.27</o:mn><o:mo>±</o:mo><o:mn>0.07</o:mn><o:mtext> </o:mtext><o:mtext> </o:mtext><o:mi>rad</o:mi></o:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mn>0.58</q:mn><q:mo>±</q:mo><q:mn>0.22</q:mn><q:mtext> </q:mtext><q:mtext> </q:mtext><q:mi>rad</q:mi></q:math> out of a total phase of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mo>≈</s:mo><s:mn>176</s:mn><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>rad</s:mi></s:math>, respectively, for the fourth- and second-order accurate Riemann solver.