Frequency shift in the gravitational potential

The action of the gravitational potential on the propagation of the particles, , in Eq. (19), is finite differenced in full analogy with Eq. (),

$$D_{E}=-\frac{\mu _{j'}G^{E}_{i+1}}{E^{2}_{k'}dE_{k'}}\left[ ... ...frac{dE_{k'}}{E_{k'+dk}-E_{k'}}E_{k'}^{3}F_{i',j',k'}\right] .$$ (83)

This is a valid finite difference representation of . We apply the corrections from Eqs. () and () with (blueshift) if the two following conditions are met: (i) is positive, and (ii), its absolute value is larger than . Otherwise, we choose for a frequency redshift. With this, the chosen sign of in Eqs. () and () implements upwind differencing for the angle-integrated distribution function. Note, that the direction of the finite differencing in energy space must not depend on the individual angle bins. An angle dependence would destroy the important cancellation of redshift and blueshift in the limit of an isotropic distribution function.

In Eq. (), we have introduced the placeholder , whose finite differencing we will now define. Eq. () describes how the terms should combine to the expression . The latter is used in the equation for the total energy evolution. A matching of the energy changes in , , , and to this expression is the last constraint of the list in Eq. () that we have not yet implemented. The term is determined by in the hydrodynamics equations. The finite differencing of the terms and has also been determined in the second line of Eq. () and in the first term of Eq. () respectively. We simply resolve for and obtain

$$G_{i+1}^{E} = \frac{1}{\Gamma _{i+1}}\left( \frac{\partial u}{\alpha _{i'}\partial t}+4\pi r_{i+1}\rho _{i'}\left( 1+e_{i'}+p_{i'+1}/\rho _{i'}\right) \right)$$

$$+ \frac{1}{H_{i'}}\sum _{j,k}\left( A_{i',k'}-\Lambda _{i',k'}\right) F_{i',j',k'}\mu _{j'}w_{j'}E_{k'}^{3}dE_{k'}.$$ (84)

The evaluation of the Lagrangian time derivative on the adaptive grid requires the appropriate corrections. The application of Eq. () to the velocity leads to

$$\frac{\partial u}{\partial t}=\frac{1}{da_{i}}\left\{ \frac{... ...el}u_{i'}^{*}-u_{i'-1}^{\rm rel}u_{i'-1}^{*}\right] \right\} ,$$ (85)

where we have once more used the grid velocity with respect to mass, . The advected velocity is evaluated by upwind differencing:

$$u^{*}_{i'}=\left\{ \begin{array}{ll} u_{i} & \mbox {if}\quad... ... rel}\geq 0\\ u_{i+1} & \mbox {otherwise}. \end{array}\right.$$