Changes between Version 43 and Version 44 of u/erica/RoeSolver

Timestamp:

05/16/13 16:15:51 (12 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/RoeSolver

@@ -48 +48 @@
 Now, we can either assume the independent variables u, rho, etc. of these functions for alpha, lambda, and K are 1) reference states which we set, or 2) some general averaged versions of the variables, which we have to solve for. Toro goes through the algebraic analysis for the general case for the Euler equations. The results are as follows.
 
+= Roe-Pike Approach for Euler Equations = 
+
 For the x-split, 3-dimensional Euler equations, we have the following eigen values and vectors:
 
@@ -54 +56 @@
 }}}
 
-where a is the local sound speed, given by
+and
+
+
+{{{#!Latex
+\vec{K}^1 = <1, u-a, v, w, H-ua>, ~\vec{K}^2 = <1,u,v,w,V^2/2>, ~\vec{K}^3 = <0,0,1,0,v>, ~\vec{K}^4 = <0,0,0,1,w>, ~\vec{K}^5 = <1, u+a, v, w, H+ua> 
+}}}
+
+where u, v, w, are velocities in x, y, z directions, respectively, a is the local sound speed, given by
 
 {{{#!Latex
@@ -60 +69 @@
 }}}
 
-The corresponding right eigenvectors are:
-
-{{{#!Latex
-\vec{K}^1 = <1, u-a, v, w, H-ua>, ~\vec{K}^2 = <1,u,v,w,V^2/2>, ~\vec{K}^3 = <0,0,1,0,v>, ~\vec{K}^4 = <0,0,0,1,w>, ~\vec{K}^5 = <1, u+a, v, w, H+ua> 
-}}}
-
-where u, v, w, are velocities in x, y, z directions, respectively, H is the enthalpy given by
+H is the enthalpy given by
 
 {{{#!Latex
@@ -78 +81 @@
 }}}
 
-with 
+with
+ 
 {{{#!Latex
 \vec{V}^2 = u^2 + v^2 + w^2