Simulations of gravitational collapse in null coordinates. I. Formulation and weak-field tests in generalized Bondi gauges

We present a code for numerical simulations of the collapse of regular initial data to a black hole in null coordinates. We restrict to twist-free axisymmetry with scalar field matter. Our coordinates are (u,x,θ,φ), where the retarded time u labels outgoing null cones emerging from a regular central worldline, the angles (θ,φ) label the null generators of each null cone, and the radial coordinate x labels points along these generators. We focus on a class of generalized Bondi radial coordinates x with the twin properties that $x=0$ is the central worldline and that the numerical domain (u≥0,0≤x≤xₘₐₓ) is a subset of the domain of dependence of the initial data on (u=0,0≤x≤xₘₐₓ). In critical collapse, an appropriate choice of these coordinates can be made to zoom in on the accumulation point of scale echoes of the critical solution, without the need for explicit mesh refinement. We introduce a novel numerical scheme that in effect reduces the angular resolution at small radius, such that the time step Δu for an explicit numerical scheme is limited by the radial resolution Δx, rather than Δx(Δθ)². We present convergence tests in the weak-field regime, where we have exact solutions to the linearized scalar and gravitational-wave equations.