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Characteristic Equations

Given a system of n conservation equations for the n conserved variables tex2html_wrap_inline2210 in m-dimensional space:

  equation1072

Note: the choice of conserved variables is unique. The conservation equations can be transformed into ``primitive'', wave-like equations for a corresponding set of n non-unique primitive variables tex2html_wrap_inline2216 ,

  equation1080

where the tex2html_wrap_inline2218 are tex2html_wrap_inline2220 matrices. The choice of primitive variables tex2html_wrap_inline2216 is not unique, and could be taken to be the conserved variables. Assume the two systems are related by

  equation1093

Then the conservation equations can be transformed as follows:

eqnarray1100

where

  eqnarray1130

The eigenvalues of tex2html_wrap_inline2218 are the solution of

  equation1148

The system is hyperbolic if all the eigenvalues of all the tex2html_wrap_inline2218 are real. The left (row), tex2html_wrap_inline2228 , and right (column), tex2html_wrap_inline2230 , eigenvectors of tex2html_wrap_inline2218 , satisfy

  eqnarray1153

The left and right eigenvectors are orthogonal,

displaymath1156

The matrices tex2html_wrap_inline2218 can be individually, but not simultaneously, diagonalized by a similarity transformation

displaymath1158

where the columns of tex2html_wrap_inline2236 are the right eigenvectors tex2html_wrap_inline2230 and rows of its inverse tex2html_wrap_inline2240 are the left eigenvectors tex2html_wrap_inline2228 , and tex2html_wrap_inline2244 is the diagonal matrix, tex2html_wrap_inline2246 .

The Characteristic Equations for a single direction can be found as follows: Combine all the divergence terms except that for the desired direction, say 1, with the source term, so the conservation equations become

displaymath1162

and the primitive equations become

displaymath1168

The 1D Characteristic Equations are obtained by multiplying this last equation by tex2html_wrap_inline2248

  equation1185

or in component form

  equation1198

which gives the variation of tex2html_wrap_inline2216 along the characteristic curve tex2html_wrap_inline2254 in the tex2html_wrap_inline2256 -plane defined by

  equation1209

tangent to the eigenvector tex2html_wrap_inline2258 with eigenvalue tex2html_wrap_inline2260 .

Sometimes it is possible to define a CHARACTERISTIC VARIABLE or Riemann Invariant tex2html_wrap_inline2262 by

  equation1212

if the expression is a perfect differential (the coefficients satisfy Pfaff's condition for the integrability of differential forms or no more than two differentials appear on the right hand side). Then the characteristic equation (13) becomes

  equation1219

These are wave equations with characteristic speeds tex2html_wrap_inline2264 and with Riemann Invariant tex2html_wrap_inline2266 constant along the characteristic tex2html_wrap_inline2268 .


next up previous
Next: Boundary Conditions Up: No Title Previous: No Title

Viggo Hansteen
Wed Sep 19 18:16:24 MET DST 2001