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

Abstract

We initiate a systematic implementation of the spectral domain decomposition technique with the Galerkin-Collocation method in situations of interest such as the spherical collapse of a scalar field in the characteristic formulation. We discuss the transmission conditions at the interface of contiguous subdomains that are crucial for the domain decomposition technique for hyperbolic problems. We implemented codes with an arbitrary number of subdomains, and after validating them, we applied to the problem of critical collapse. With a modest resolution, we obtain the Choptuik’s scaling law and its oscillatory component due to the discrete self-similarity of the critical solution.

Appendix: Basis functions

We present the basis functions used in the spectral approximations of $\varPhi , \beta$, and V established in the first and last subdomains. The basis functions for the scalar field $\varPhi$ (cf. Eq. (17)) is

$$\begin{aligned} \psi _k^{(1)}(r)= & {} \frac{1}{2}\left( TL_{k+1}^{(1)}(r) + TL_{k}^{(1)}(r)\right) , \end{aligned}$$

(57)

where $TL^{(1)}_k(r)$ with $k=0,1,..,N_1$ are the rational Chebyshev polynomials defined in the first subdomain. The basis for the metric function $\beta$ is

$$\begin{aligned} \chi _k^{(1)}(r) = \frac{1}{8}\bigg [\frac{(1+2k)}{3+2k}TL_{k+2}^{(1)}(r) +\frac{4(1+k)}{3+2k}TL_{k+1}^{(1)}(r) + TL_{k}^{(1)}(r)\bigg ]. \end{aligned}$$

(58)

Each of the above basis functions satisfies the condition (11). For the last subdomain, we have chosen the corresponding rational Chebyshev function as the basis for all spectral approximations of $\varPhi , \beta$ and V.