Plane (FP) relation (Djorgovski & Davis 1987, Dressler et al. 1987).

The paper is organized as follows. The main features of the new N-body code are presented in Section \[seccod\], while Section \[secsim\] describes the set-up and the analysis of the numerical simulations. The results are presented in Section \[secres\] and discussed in Section \[secdis\].

The N-body code {#seccod}
===============

While most N-body codes for simulations in Newtonian gravity are based on the gridless multipole expansion treecode scheme (Barnes & Hut 1986; see also Dehnen 2002), the non-linearity of the MOND field equation (\[eqMOND\]) forces one to resort to other methods, such as the particle-mesh technique (see Hockney & Eastwood 1988). In this approach, particles are moved under the action of a gravitational field which is computed on a grid, with particle-mesh interpolation providing the link between the two representations. In our MOND particle-mesh N-body code, we adopt a spherical grid of coordinates ($r$, $\vartheta$, $\varphi$), made of $\Nr\times\Nth\times\Nph$ points, on which the MOND field equation is solved as in CLN. Particle-mesh interpolations are obtained with a quadratic spline in each coordinate, while time stepping is given by a classical leap-frog scheme (Hockney & Eastwood 1988). The time-step $\Dt$ is 