Multidomain Galerkin-Collocation method: spherical collapse of scalar fields II

We follow the strategy initiated in Ref. [1] and proceed with the implementation of the Galerkin-Collocation domain decomposition (GCDD) applied to the dynamics of a spherical self-gravitating scalar field with the field equation in the Cauchy formulation. We have adopted the areal slicing gauge. We have presented a detailed implementation for an arbitrary number of subdomains and adopted the simplest form of the transmission conditions. Further, by an appropriated choice of the basis functions in the inner subdomain, we eliminated exactly the 1/r terms near the origin present in the field equations. The code is validated using two error measures: the conservation of the ADM mass and the Hamiltonian constraint that must be satisfied during the spacetime dynamics. In general, both error measures converge exponentially in all subdomains. As a useful illustration of placing more subdomains near the strong-field region, meaning an efficient concentrating of collocation points near the origin, we exhibited the formation of an apparent horizon even though the numerical integration diverges.