Frequency shift from observer motion

We continue our description with the observer correction in Eq. (20). We start with a review of the finite differencing procedure presented in Mezzacappa_Bruenn_93a in order to extend it from local to global energy conservation. Global energy conservation requires the cancellation of observer correction terms in Eq. () with terms from space propagation and angular advection. Therefore, it is convenient to express the observer corrections in terms of the same physical quantities and we have used before,

$$A = \left( \frac{\partial \ln \rho }{\alpha \partial t}+\frac{2u}{r}\right)$$

$$B = \left( 1-\mu ^{2}\right) \frac{u}{r}.$$

$$O_{E} = \left( \mu ^{2}A-B\right) \frac{1}{E^{2}}\frac{\partial }{\partial E}\left[ E^{3}F\right] .$$ (70)

In the following, we solve the equation for the frequency shift correction, , by the method of characteristics. We convert the time derivative at constant energy to a time derivative along the characteristic in the energy dimension of the momentum phase space. This eliminates the partial derivative with respect to energy and facilitates a conservative finite differencing of . Note that the characteristic used in this section implements the frequency shift correction alone, it differs from the free propagation characteristic prescribed by the full Boltzmann equation. As in Bruenn_85,Mezzacappa_Bruenn_93a we write the prefactor of the correction as a time derivative of a quantity ,

$$\frac{\partial \ln R_{f}}{\alpha \partial t}=\mu ^{2}A-B,$$

such that becomes

$$0=E^{3}\left( \frac{\partial F}{\partial t}\right) _{E}+\fra... ...\partial t}E\frac{\partial }{\partial E}\left[ E^{3}F\right] .$$ (71)

It is now possible to transform from the ``Eulerian'' variable, , to a ``Lagrangian'' variable, , along the characteristic by the chain rule (the subscript of the bracket denotes the variable that is kept constant for the differentiation). Eq. () simplifies to

$$\begin{eqnarray*} 0 & = & \left( \frac{\partial }{\partial t}\left[ E^{3}F\right... ...c{\partial }{\partial t}\left[ E^{3}F\right] \right) _{E/R_{f}}. \end{eqnarray*}$$

For a small section of the energy phase space , this relationship leads to

$$\left( \frac{\partial }{\partial t}\left[ E^{2}F\Delta E\right] \right) _{E/R_{f}}=0.$$ (72)

The validity of Eq. () for arbitrary distribution functions leads to the following interpretation: The observer correction shifts the particles that initially reside in the energy interval along the characteristic with constant in the energy phase space. This allows us to determine the evolution of any other particle property in analogy to Eq. (). For example, the specific energy of the particles in this phase space interval, , evolves according to

$$\left( \frac{\partial }{\partial t}\left[ E^{3}F\Delta E\rig... ...t) _{E/R_{f}}=\frac{\partial \ln R_{f}}{\partial t}d\epsilon .$$ (73)

A finite difference representation of Eqs. () and () has been given in Mezzacappa_Bruenn_93a. Consider a particle energy group , with a neighbor group , . Eq. () tells us that the number of particles before the correction, , is equal to the number of particles after the correction. The distribution after the correction is represented by a diminished number of particles in group and an additional number of particles in the neighbor group

$$F_{i',j',k'}E_{k'}^{2}dE_{k'}-\left[ \left( F_{i',j',k'}E_{k... ...E_{k'}-n^{-}_{i',j',k'}\right) +n^{+}_{i',j',k'+dk}\right] =0.$$ (74)

Eq. () now defines a similar relationship for the particle energies

$$F_{i',j',k'}E_{k'}^{3}dE_{k'} - \left[ \left( F_{i',j',k'}E_{k'}^{3}dE_{k'}-E_{k'}n^{-}_{i',j',k'}\right) +E_{k'+dk}n^{+}_{i',j',k'+dk}\right]$$

$$= -\left( \mu ^{2}_{j'}A_{i',k'}-B_{i',j',k'}\right) F_{i',j',k'}E^{3}_{k'}dE_{k'}\alpha _{i'}dt,$$ (75)

where and stand for a finite difference representation of Eq. (). Eqs. () and () uniquely define the solution

$$n^{-}_{i',j',k'} = \left( \mu _{j'}^{2}A_{i',k'}-B_{i',j',k'}\right) \frac{dE_{k'}}{E_{k'+dk}-E_{k'}}E^{3}_{k'}F_{i',j',k'}\alpha _{i'}dt$$

$$n^{+}_{i',j',k'} = n_{i',j',k'-dk}^{-},$$ (76)

which leads, by the update , to the following finite difference representation of the frequency shift term in the Boltzmann equation (15):

$$O_{E} = \frac{1}{E^{2}_{k'}dE_{k'}}$$

$$\times \left[ \left( \mu _{j'}^{2}A_{i',k'-dk}-B_{i',j',k'-dk}\right) \frac{dE_{k'-dk}}{E_{k'}-E_{k'-dk}}E_{k'-dk}^{3}F_{i',j',k'-dk}\right.$$

$$- \left. \left( \mu _{j'}^{2}A_{i',k'}-B_{i',j',k'}\right) \frac{dE_{k'}}{E_{k'+dk}-E_{k'}}E_{k'}^{3}F_{i',j',k'}\right] .$$ (77)

Finally, we calculate the contribution of the frequency shift to energy conservation,

$$\sum _{j,k}\left( \Gamma _{i+1}+u_{i+1}\mu _{j'}\right) O_{E,i'}w_{j'}E_{k'}^{3}dE_{k'}.$$

The summation over in Eq. () with weight and measure of integration can be simplified with a discrete ``integration by parts'' as in Eq. (). If we neglect for the time being the boundary terms because of a small in the lowest energy group and a small in the highest energy group (a correction for these terms will be made later in subsection ), the energy contribution from becomes

$$O^{1-4}_{E}:\qquad -\sum _{j,k}\left( \Gamma _{i+1}+u_{i+1}\... ...',k'}-B_{i',j',k'}\right) F_{i',j',k'}w_{j'}E_{k'}^{3}dE_{k'}.$$ (78)

If we use the finite difference representations () and () for and respectively, we find in the comparison of Eq. () with Eqs. () and () that we have indeed matched the terms and to machine precision Liebendoerfer_00.