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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Calculation of shocks using solutions of systems of ordinary differential equations

Authors: Jürgen Batt and Renuka Ravindran
Journal: Quart. Appl. Math. 63 (2005), 721-746
MSC (2000): Primary 74J40, 35F25
Published electronically: September 27, 2005
MathSciNet review: 2187929
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Abstract: The method of intrinsic characterisation of shock wave propagation avoids the cumbersome task of solving the basic systems of equations before and after the shock, and has been used by various authors for direct calculation of relevant quantities on the shock. It leads to an infinite hierarchy of ordinary differential equations, which, due to the absence of a mathematical theory, is truncated to a finite system. In most practical cases, but not in all, the solutions of the truncated systems approximate the solution of the infinite system satisfactorily. The mathematical question of the error generated is completely open. We precisely define the concept of approximation and rigorously justify the local correctness of the approximation method for positive real analytic initial data for the inviscid Burgers’ equation, which has certain features in common with systems appearing in literature. At the same time we show that the nonuniqueness of the infinite system can lead to wrong results when the initial data are only $C^{\infty }$ and that blow-up of the solutions of the truncated systems are an obstacle for straightforward global approximation. Global approximation is achieved by recomputing the initial conditions for the approximating solutions in finitely many time steps. The results obtained will have to be taken into account in a future theory for more advanced systems.

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Additional Information

Jürgen Batt
Affiliation: Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstr. 39, 80333 München, Germany

Renuka Ravindran
Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560012 India

Keywords: Infinite system of odes, intrinsic description of shock propagation.
Received by editor(s): February 25, 2005
Published electronically: September 27, 2005
Article copyright: © Copyright 2005 Brown University