98 | | \vec{F}_{i+1/2}= \frac{1}{2}(\vec{F}_L + \vec{F}_R) - \frac{1}{2}\Sigma_{i=1,m} \alpha_i \lambda_i \vec{K^{(i)}} |

| 98 | \vec{F}_{i+1/2}= \frac{1}{2}(\vec{F}_L + \vec{F}_R) - \frac{1}{2}\Sigma_{i=1,m} \tilde{\alpha}_i \tilde{\lambda}_i \tilde{\vec{K^{(i)}}} |

| 99 | }}} |

| 100 | |

| 101 | where the tilde's are the solution of the Roe approach, and are given by: |

| 102 | |

| 103 | {{{#!Latex |

| 104 | \tilde{\alpha}_i = \hat{\alpha}_i(\tilde{\vec{W}}), ~\tilde{\lambda}_i = \lambda_i(\tilde{\vec{W}}), ~\tilde{\vec{K}^{(i)}} = \vec{K}^{(i)}(\tilde{\vec{W}}) |