SPH split

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the splitting procedure will be described below.

fuse particles[edit | edit source]

We want all the physical quantities (such as the entropy density, energy density and momentum density) to remain unchanged. Therefore, these quantities are taken to be the average over the particles that will be fused

<math>

s' = \frac{1}{N}\sum_a s_a \\ (q^i)' = (\omega u^i)' = \frac{1}{N}\sum_a (\omega u^i)_a </math>

other constraint is that we want the total entropy and total momentum to be conserved so that

<math>

S = \sum_a \frac{\nu_a s_a}{\sigma_a} = \frac{\nu' s'}{\sigma'} \\ Q^i = \sum_a \frac{\nu_a \omega u^i}{\sigma_a} = \frac{\nu' (\omega u^i)'}{\sigma'} </math> now we can compute

<math>

\frac{\nu'}{\sigma'} = \frac{S}{s'} \\ \frac{\nu' (u^i)'}{\sigma'} = \frac{Q^i}{\omega'} </math> which leads to where omega is completely determined by s.

<math>

(u^i)' = \frac{s'}{S} \frac{Q^i}{\omega'} </math> One more step is needed to compute the weights. First we express the reference density of the old particles

<math>

\rho_a = (\tau\gamma\sigma)_a = \sum_b \nu_b W_{ab} = \sum_b \nu_b W_{ab} + \sum_{c} \nu_c W_{ac} </math> where the index b is for the particles which are fused and the index c is for the particles which are not part of the fusing. We do not want to change the reference density of the particles that will not fuse, so for a particle a which is not part of the fusing procedure

<math>

\rho_a = \sum_b \nu_b W_{ab} + \sum_{c} \nu_c W_{ac} = \nu' W_{ab} + \sum_{c} \nu_c W_{ac} \\ \nu' = \sum_b \frac{ \nu_b W_{ab} }{ W_{ab'} } </math>


the new reference density is

<math>

\rho' = \nu' W_{0} + \sum_{c} \nu_c W_{ac} </math>

In fact we approximate the new weight to be

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\nu' = \sum_b \frac{ \nu_b W_{a'b} }{ W_{0} } </math> where b are the old particle

we also want the derivatives not to change much

<math>

\partial \rho_a = \sum_{b} \nu_b \hat{r}_{ab} W'_{ab} + \sum_{c} \nu_c \hat{r}_{ac} W'_{ac} = \nu' \hat{r}_{ab}W'_{ab} + \sum_{c} \nu_c \hat{r}_{ac} W'_{ac} </math> therefore

<math>

\nu' \hat{r}_{ab'}W'_{ab'} = \sum_{b} \nu_b \hat{r}_{ab} W'_{ab} </math>