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Boundary Equations

The characteristic equations (13) are

displaymath1389

Write the z-derivatives in the characteristic directions as

  equation1483

so   tex2html_wrap2348

For incoming characteristics these are the terms that must be determined from the physical boundary conditions. For outgoing characteristics these terms are evaluated using one-sided derivatives.

The equations at the boundary can be derived by either (i) solving (43) for tex2html_wrap_inline2316 in terms of tex2html_wrap_inline2318 and substituting in primitive Eqns. 21-24, or (ii) transforming the characteristic equations (12),

displaymath1511

into the primitive equations by multiplying them by tex2html_wrap_inline2320 to obtain

displaymath1520

where the z-derivatives tex2html_wrap_inline2324 can be found as the solution of the system of equations

  equation1533

since we have the matrix tex2html_wrap_inline2326 whose rows are the left eigenvectors, but not the matrix tex2html_wrap_inline2320 .

The primitive z-derivatives tex2html_wrap_inline2316 in terms of the characteristic derivatives tex2html_wrap_inline2318 are:

eqnarray1542

The boundary equations for the primitive variables are obtained by substituting these expressions for the z-derivatives of the primitive variables in the primitive equations (21-24),   tex2html_wrap2350

where tex2html_wrap_inline2338 . Hence,

eqnarray1608

These could also be obtained by solving the system of equations

displaymath1613

or

displaymath1617

with an algebra program.

The equations for the quasi-conserved variables are obtained by multiplying Eqns. 49-52 by tex2html_wrap_inline2340 , which gives

displaymath1621

where tex2html_wrap_inline2342 stands for Eqn. (i). Thus the quasi-conserved boundary equations are   tex2html_wrap2352


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