Abstract
When the projected characteristics of a quasi-linear first-order PDE intersect within the domain of interest, the solution ceases to exist as a well-defined function. In the case of PDEs derived from an integral balance equation, however, it is possible to relax the requirement of continuity and obtain a single-valued solution that is smooth on either side of a shock front of discontinuity and that still satisfies the global balance law. The speed of propagation of this front is obtained as the ratio between the jump of the flux and the jump of the solution across the shock.
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Notes
- 1.
- 2.
For the sake of simplicity, we are assuming that no other shocks develop after the breaking time \(t_b\) that we have calculated.
- 3.
The generalization of this formula to three dimensions is known as Reynolds’ transport theorem, with which you may be familiar from a course in Continuum Mechanics.
- 4.
Naturally, because of the sign convention we used for the flux, the formula found in most books changes the sign of the right-hand side.
- 5.
- 6.
For the possibility of extending the values \(u^-\) and \(u^+\) into the shaded region and producing a legitimate shock, see Box 4.1.
References
Arnold VI (2003) Catastrophe theory. Springer, Heidelberg
Knobel R (2000) An introduction to the mathematical theory of waves. American Mathematical Society, Providence
Lax PD (1973) Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Regional conference series in applied mathematics, vol 11. SIAM, Philadelphia
Poston T, Stewart I (2012) Catastrophe theory and its applications. Dover, New York
Zauderer E (1998) Partial differential equations of applied mathematics, 2nd edn. Wiley-Interscience, New York
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Epstein, M. (2017). Shock Waves. In: Partial Differential Equations. Mathematical Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-55212-5_4
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DOI: https://doi.org/10.1007/978-3-319-55212-5_4
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