21 | | \partial where, the Jacobian matrix A= is a NON-constant coefficient matrix, the Euler equations comprise a NON-linear set of equations. An easier system to solve would be one that is a linear, constant coefficient system of equations. We can transform the Euler equations into this simpler case, if we make transformations of variables in the matrix to be some average function of the left and right data state variables. This results in the set of equations: |

22 | | |

23 | | Where now the Jacobian is an averaged matrix, representing constant coefficients for the Euler equations. |

24 | | |

25 | | In considering an integral form of the conservation laws now that an exact solution to the 'approximate' Riemann problem is feasible, one can solve for the numerical flux in terms of 1) wave strengths, 2) eigenvalues, 3) right eigenvectors of the Jacobian. One possible algebraic form for the flux function is: |

| 21 | {{{#!Latex |

| 22 | \hat{A} = \frac{\partial \vec{F}}{\partial \vec{U}} |

| 23 | }}} |

| 24 | where, A-hat is a Jacobian matrix of averaged/constant values, we can derive an expression for the numerical flux in terms of 1) wave strengths (alpha), and the 2) eigenvalues (lambda) and 3) right eigenvectors (K) of the 'averaged' Jacobian: |