Analytical and numerical approaches to asymptotic problems by O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)

By O. Axelsson, L.S. Frank and A. Van Der Sluis (Eds.)

A global convention on Analytical and Numerical ways to Asymptotic difficulties used to be held within the college of technology, college of Nijmegen, The Netherlands from June ninth via June thirteenth, 1980.

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Extra resources for Analytical and numerical approaches to asymptotic problems in analysis: proceedings of the Conference on Analytical and Numerical approaches to Asymptotic Problems, University of Nijmegen, the Netherlands, June 9-13, 1980

Example text

C, (50) , we can obtain p, u, Using the isentropic law we obtain p = Using equation = Ap^ p,p7p^ ^^') • we obtain, by solving for c ('50) c = c^ + ^^ (u^- u) (52) . By substitution of (52) into {^9) and solving for u we obtain o u By substitution of of ( 52) 2P Ax / 1 \ 7" ( 53) into ( into the definition of 52) c c is obtained; by substitution and solving for p we obtain 27 P = If Ij^Ax lies to the right of the left rarefaction wave (3) we obtain p = p^, u = u^, II. The sample point (I^Ax > n p and p = p^.

The sample point (I^Ax > n p and p = p^. Ax lies to the right of the slip line ^ u^At/2). (a) i (53) • (f^)'^"'"' If the right wave is a shock wave (p^ > p and (l) if ) dy Ax lies to the left of the shockline defined by -^ = U we have dt r = p#j u = u^, and p = p^, where p^ is obtained from (15) , -M ^ '* (2) ^= U^, (b) If ^ r_ u^ '* - U (5M r Ax lies to the right of the shockline defined by we have p = p^, u = u^, and p = p^, If the right wave is a rarefaction wave "^ (p^ P The ) . rarefaction wave is bounded on the left by the line defined by dx -TT- = u^ + c^, i^P* where c^ = j and p^ can be obtained from the isentropic law P„p"^ = P*p;^ = A Then we obtain from (55 ; tz> p i = p^, u = u^, dx = u + ) (56 ) /^Pr "'dtrr'rVp and on the right by the line defined by J If (55 ) P* = (-^) (1) .

Rarefaction wave is bounded on the left by the line defined by dx -TT- = u^ + c^, i^P* where c^ = j and p^ can be obtained from the isentropic law P„p"^ = P*p;^ = A Then we obtain from (55 ; tz> p i = p^, u = u^, dx = u + ) (56 ) /^Pr "'dtrr'rVp and on the right by the line defined by J If (55 ) P* = (-^) (1) . -rrr c , c = / , Ax lies to the left of the rarefaction wave, then and p = p^ o 28 (2) If i Ax lies inside the right rarefaction wave, we equate the slope of the characteristic dx = ^ + ^ to the slope of -g^ the line through the origin and (I^^Ax, At/2 ), obtaining Ax 2i u+c=-^.

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