Calculation of shocks using solutions of systems of ordinary differential equations

Author:

Batt Jürgen,Ravindran Renuka

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 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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics

Reference31 articles.

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Evolutionary behavior of weak shocks in a non-ideal gas;Journal of Theoretical and Applied Physics;2013

2. Evolution of weak shocks in one dimensional planar and non-planar gasdynamic flows;International Journal of Non-Linear Mechanics;2012-10

3. Front Matter;Monographs & Surveys in Pure & Applied Math;2010-04-29

4. Kinematics of a shock wave of arbitrary strength in a non-ideal gas;Quarterly of Applied Mathematics;2009-05-05

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