An Implicit Scheme for Ohmic Dissipation with Adaptive Mesh Refinement

An implicit method for the ohmic dissipation is proposed. The proposed method is based on the Crank-Nicolson method and exhibits second-order accuracy in time and space. The proposed method has been implemented in the SFUMATO adaptive mesh refinement (AMR) code. The multigrid method on the grids of the AMR hierarchy converges the solution. The convergence is fast but depends on the time step, resolution, and resistivity. Test problems demonstrated that decent solutions are obtained even at the interface between fine and coarse grids. Moreover, the solution obtained by the proposed method shows good agreement with that obtained by the explicit method, which required many time steps. The present method reduces the number of time steps, and hence the computational costs, as compared with the explicit method.Comment: Accepted for publication in PASJ. 8 pages, 11 figure

Self-gravitational Magnetohydrodynamics with Adaptive Mesh Refinement for Protostellar Collapse

A new numerical code, called SFUMATO, for solving self-gravitational magnetohydrodynamics (MHD) problems using adaptive mesh refinement (AMR) is presented. A block-structured grid is adopted as the grid of the AMR hierarchy. The total variation diminishing (TVD) cell-centered scheme is adopted as the MHD solver, with hyperbolic cleaning of divergence error of the magnetic field also implemented. The self-gravity is solved by a multigrid method composed of (1) full multigrid (FMG)-cycle on the AMR hierarchical grids, (2) V-cycle on these grids, and (3) FMG-cycle on the base grid. The multigrid method exhibits spatial second-order accuracy, fast convergence, and scalability. The numerical fluxes are conserved by using a refluxing procedure in both the MHD solver and the multigrid method. The several tests are performed indicating that the solutions are consistent with previously published results.Comment: 23 pages, 15 figures. PASJ in press. Document with high resolution figures is available in http://redmagic.i.hosei.ac.jp/~matsu/AMR06/matsumotoAMR.pd

Stability of Dynamically Collapsing Gas Sphere

We discuss stability of dynamically collapsing gas spheres. We use a similarity solution for a dynamically collapsing sphere as the unperturbed state. In the similarity solution the gas pressure is approximated by a polytrope of $P = K \rho ^\gamma$. We examine three types of perturbations: bar ($\ell = 2$) mode, spin-up mode, and Ori-Piran mode. When $\gamma < 1.097$, it is unstable against bar-mode. It is unstable against spin-up mode for any $\gamma$. When $\gamma < 0.961$, the similarity solution is unstable against Ori-Piran mode. The unstable mode grows in proportion to $| t - t_0 | ^{-\sigma}$ while the central density increases in proportion to $\rho_c \propto (t - t_0) ^{-2}$ in the similarity solution. The growth rate, $\sigma$ is obtained numerically as a function of $\gamma$ for bar mode and Ori-Piran mode. The growth rate of the bar mode is larger for a smaller $\gamma$. The spin-up mode has the growth rate of $\sigma = 1/3$ for any $\gamma$.Comment: submitted to PASJ. 7 pages including 6 figures. This paper is also available at http://www.a.phys.nagoya-u.ac.jp/~hanawa/dpnu9922/dpnu9922.htm