Changes between Version 20 and Version 21 of u/erica/RoeSolver


Ignore:
Timestamp:
05/16/13 14:34:49 (11 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/RoeSolver

    v20 v21  
    1414
    1515{{{#!Latex
    16 \vec{U}_t + \frac{\partial F} {\partial U} \vec{U}_x = 0
     16\vec{U}_t + \frac{\partial \vec{F}} {\partial \vec{U}} \vec{U}_x = 0
    1717}}}
    1818
    1919If we assume that
    2020
    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:
    2625
    2726{{{#!Latex
    28 F_{i+1/2}= 1/2(F_L + F_R) - 1/2*SUM[\alpha_i \lambda_i K]
     27F_{i+1/2}= 1/2(F_L + F_R) - 1/2*\sum[\alpha_i \lambda_i K] \mid ^i=1 _m
    2928}}}
    3029