By Eduard L. Stiefel

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But discontinuous solutions (including shocks and contact discontinuities) of the new system may not be solutions to the original Euler, as we will see later in this chapter. Interestingly, it can be shown that the isentropic system is still a valid approximation of the original Euler in the case of weak discontinuities. 5) where R is a constant. In an ideal gas, it can be shown that the internal energy e is a function of the temperature T alone. 1. The isentropic Euler system 25 where C v is a constant.

We desire that the two waves do not overlap along the ~ -axis. In fact we trivially have that the exit value of ~ of the first rarefaction wave is less than the entering value of ~ of the second rarefaction wave. 3. 1. 3. Two rarefaction waves For a given (p_, u_), a two-parameter family of global solutions can now be constructed. Construct a backward rarefaction wave, as in the previous paragraph, starting at (Pe, ue) = (p-, u_) and ending at an arbitrary point (pi, Ui) along the backward wave curve.

7. Wave front, wave back, particle path, and characteristics in a two-shock structure present several other entropy conditions with the same effect for the isentropic Euler system. We notice that particles travel with velocity u. Small disturbances travel with velocities A± = u ± c, where c = p' (p) is the sound speed. The A± characteristics are also called forward and backward moving waves because of their relative motion with respect to the particles' motion. The right side of a forward wave or the left side of a backward wave is called the wavefront.

### An Introduction to Numerical Mathematics by Eduard L. Stiefel

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