We present an update on the General-relativistic multigrid numerical (Gmunu) code, a parallelised, multi-dimensional curvilinear, general relativistic magnetohydrodynamics code with an efficient non-linear cell-centred multigrid elliptic solver, which is fully coupled with an efficient block-based adaptive mesh refinement module. To date, as described in this paper, Gmunu is able to solve the elliptic metric equations in the conformally flat condition approximation with the multigrid approach and the equations of ideal general-relativistic magnetohydrodynamics by means of high-resolution shock-capturing finite-volume method with reference metric formularised multi-dimensionally in Cartesian, cylindrical or spherical geometries. To guarantee the absence of magnetic monopoles during the evolution, we have developed an elliptical divergence cleaning method by using the multigrid solver. In this paper, we present the methodology, full evolution equations and implementation details of Gmunu and its properties and performance in some benchmarking and challenging relativistic magnetohydrodynamics problems.

We present the implementation of a two-moment-based general-relativistic multigroup radiation transport module in the General-relativistic multigrid numerical (Gmunu) code. On top of solving the general-relativistic magnetohydrodynamics and the Einstein equations with conformally flat approximations, the code solves the evolution equations of the zeroth- and first-order moments of the radiations in the Eulerian-frame. An analytic closure relation is used to obtain the higher order moments and close the system. The finite-volume discretization has been adopted for the radiation moments. The advection in spatial space and frequency-space are handled explicitly. In addition, the radiation–matter interaction terms, which are very stiff in the optically thick region, are solved implicitly. The implicit–explicit Runge–Kutta schemes are adopted for time integration. We test the implementation with a number of numerical benchmarks from frequency-integrated to frequency-dependent cases. Furthermore, we also illustrate the astrophysical applications in hot neutron star and core-collapse supernovae modelings, and compare with other neutrino transport codes.

Gravitational-wave signals from coalescing binary neutron stars (BNSs) can yield important information about the properties of nuclear-matter equation of state. We investigate a direct link between the frequency of the quadrupolar 2 f-mode oscillation (f 2f ) of nonrotating and rotating neutron stars calculated by a nonlinear hydrodynamics code in the conformally flat approximation and the gravitational-wave frequency associated with the peak amplitude ( f max ) of binary neutron stars from a set of publicly available simulations. We find that f max and f 2f differ by about 1%, on average, across 45 equal-mass systems with different total mass and equations of state. Interestingly, assuming that the gravitational-wave frequency is still approximately equal to twice the orbital frequency f orb near the merger, the result indicates that the condition for tidal resonance ( ∣ m ∣ f orb ) m = 2 = f 2 f is satisfied to high accuracy near the merger. Moreover, the well-established universal relation between f max and the tidal deformability of equal-mass binary systems can now be explained by a similar relation between f 2f and the tidal deformability of isolated neutron stars. For unequal-mass binaries, f max increasingly deviates from f 2f of the two stars as the mass ratio decreases from unity. Therefore, it is possible to relate the gravitational-wave signal at the merger of a BNS system directly to the fundamental oscillation modes and the mass ratio. This work potentially brings gravitational-wave asteroseismology to the late-inspiral and merger phases of BNSs.

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