We describe the implementation of a multi-dimensional numerical code to solve the equations for idea! magnetohydrodynamics (MHD) in cylindrical geometry. It is based on an explicit finite difference scheme on an Eulerian grid, called the Total Variation Diminishing (TVD) scheme, which is a second-order-accurate extension of the Roe-type upwind scheme. Multiple spatial dimensions are treated through a Strang-type operator splitting. Curvature and source terms are included in a way to insure the formal accuracy of the code to be second order. The constraint of a divergence-free magnetic field is enforced exactly by adding a correction, which involves solving a Poisson equation. The Fourier Analysis and Cyclic Reduction (FACR) method is employed to solve it. Results from a set of tests show that the code handles flows in cylindrical geometry successfully and resolves strong shocks within two to four computational cells. The advantages and limitations of the code are discussed.

Recent redshift surveys suggest that most galaxies may be distributed on the surfaces of bubbles surrounding large voids. To investigate the quantitative consistency of this qualitative picture of large-scale structure, we study analytically the clustering properties of galaxies in a universe filled with spherical shells. In this paper, we report the results of the calculations for the spatial and angular two-point correlation functions of galaxies. With ∼20 ∼20 of galaxies in clusters and a power law distribution of shell sizes, nsh(R)∼R−α nsh(R)∼R−α , α≃4 α≃4 , the observed slope and amplitude of the spatial two-point correlation function ξgg(r) ξgg(r) can be reproduced. (It has been shown that the same model parameters reproduce the enhanced cluster two-point correlation function, ξcc(r) ξcc(r) ). The corresponding angular two-point correlation function w(θ) w(θ) is calculated using the relativistic form of Limber's equation and the Schecter-type luminosity function. The calculated w(θ θ ) agrees with the observed one quite well on small separations (θ≲2deg θ≲2deg ).