Changes between Version 43 and Version 44 of u/erica/RoeSolver
 Timestamp:
 05/16/13 16:15:51 (11 years ago)
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u/erica/RoeSolver
v43 v44 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. 49 49 50 = RoePike Approach for Euler Equations = 51 50 52 For the xsplit, 3dimensional Euler equations, we have the following eigen values and vectors: 51 53 … … 54 56 }}} 55 57 56 where a is the local sound speed, given by 58 and 59 60 61 {{{#!Latex 62 \vec{K}^1 = <1, ua, v, w, Hua>, ~\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> 63 }}} 64 65 where u, v, w, are velocities in x, y, z directions, respectively, a is the local sound speed, given by 57 66 58 67 {{{#!Latex … … 60 69 }}} 61 70 62 The corresponding right eigenvectors are: 63 64 {{{#!Latex 65 \vec{K}^1 = <1, ua, v, w, Hua>, ~\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> 66 }}} 67 68 where u, v, w, are velocities in x, y, z directions, respectively, H is the enthalpy given by 71 H is the enthalpy given by 69 72 70 73 {{{#!Latex … … 78 81 }}} 79 82 80 with 83 with 84 81 85 {{{#!Latex 82 86 \vec{V}^2 = u^2 + v^2 + w^2