42 | | This approach avoids construction of an averaged Jacobian. Instead the method involves solving for the eigen values and vectors for the original Jacobian, deriving an expression for the wave speeds, and then evaluating these expressions using some vector of averaged scalar quantities, typically W, averaged. So from the equation above of the Euler equations in Jacobian form, we can derive the equations for wave speeds: |

| 42 | This approach avoids construction of an averaged Jacobian directly. Instead the method involves 1) assuming the Jacobian consists of reference states for which the left and right states are close to, 2) solving for the eigenvalues and vectors for this 'reference' Jacobian, and the wave speeds, 3) evaluating these expressions using averaged versions of the original scalar quantities. In other words, the Roe Pike method consists of first, deriving expressions for the various quantities needed in the flux, then second, solving for the averaged primitive variables, and third, evaluating the expressions at the reference state equal to the average state. |

| 43 | |

| 44 | From the equation above of the Euler equations in Jacobian form, we can derive the equations for wave speeds: |