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RoeSolver

-              v11
+              v12
 = The ROE Solver =
 The Roe solver is an approximation means for the numerical flux of the Godunov method, which is derived through linearizing a hyperbolic system of equations. For instance, the Euler equations in conservative form are written:
+The Roe solver is an approximation means for the numerical flux of the Godunov method, which is derived through linearizing a hyperbolic system of equations. For instance, the Euler equations in conservative form are written
 {{{#!Latex
 \vec{U}_t + \vec{F}(\vec{U})_x = 0
 }}}
+can be written in matrix form as:
+}}}, which using the chain rule is identical to
 {{{#!Latex
 U_t + F(U)_x = 0
+\vec{U}_t + \frac{\partial F}{\partial U} \vec{U}_x = 0
 }}}
 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: