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.

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We present a new open-source axisymmetric general relativistic hydrodynamics code Gmunu (general-relativistic multigrid numerical solver) which uses a multigrid method to solve the elliptic metric equations in the conformally flat condition (CFC) approximation on a spherical grid. Most of the existing relativistic hydrodynamics codes are based on formulations which rely on a free-evolution approach of numerical relativity, where the metric variables are determined by hyperbolic equations without enforcing the constraint equations in the evolution. On the other hand, although a fully constrained-evolution formulation is theoretical more appealing and should lead to more stable and accurate simulations, such an approach is not widely used because solving the elliptic-type constraint equations during the evolution is in general more computationally expensive than hyperbolic free-evolution schemes. Multigrid methods solve differential equations with a hierarchy of discretizations and its computational cost is generally lower than other methods such as direct methods, relaxation methods, successive over-relaxation. With multigrid acceleration, one can solve the metric equations on a comparable time scale as solving the hydrodynamics equations. This would potentially make a fully constrained-evolution formulation more affordable in numerical relativity simulations. As a first step to assess the performance and robustness of multigrid methods in relativistic simulations, we develop a hydrodynamics code that makes use of standard finite-volume methods coupled with a multigrid metric solver to solve the Einstein equations in the CFC approximation. In this paper, we present the methodology and implementation of our code Gmunu and its properties and performance in some benchmarking relativistic hydrodynamics problems.

We present the implementation of general-relativistic resistive magnetohydrodynamics solvers and three divergence-free handling approaches adopted in the General-relativistic multigrid numerical (Gmunu) code. In particular, implicit–explicit Runge–Kutta schemes are used to deal with the stiff terms in the evolution equations for small resistivity. The three divergence-free handling methods are (i) hyperbolic divergence cleaning (also known as the generalized Lagrange multiplier), (ii) staggered-meshed constrained transport schemes, and (iii) elliptic cleaning through a multigrid solver, which is applicable in both cell-centered and face-centered (stagger grid) magnetic fields. The implementation has been tested with a number of numerical benchmarks from special-relativistic to general-relativistic cases. We demonstrate that our code can robustly recover from the ideal magnetohydrodynamics limit to a highly resistive limit. We also illustrate the applications in modeling magnetized neutron stars, and compare how different divergence-free handling methods affect the evolution of the stars. Furthermore, we show that the preservation of the divergence-free condition of the magnetic field when using staggered-meshed constrained transport schemes can be significantly improved by applying elliptic cleaning.

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.