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
DOI:
https://doi.org/10.1090/S0033-569X-05-00976-6
Published electronically:
September 27, 2005
MathSciNet review:
2187929
Full-text PDF Free Access
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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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AnRuII A.M. Anile and G. Russo, Generalized wavefront expansion II: The propagation of step shocks, Wave Motion 10 (1988), 3-18.
BaCh P.B. Bailey and P.J. Chen, Evolutionary behavior of induced discontinuities behind one dimensional shock waves in nonlinear elastic materials, J. Elasticity 15 (1985), 257-269.
Cart H. Cartan, Théorie élémentaire des fonctions analytiques d’une ou plusiers variables complexes, Hermann, Paris, 1961.
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ChGu1 P.J. Chen and M.E. Gurtin, Growth of one dimensional shock waves in materials with memory, Arch. Rat. Mech. Anal. 36 (1970), 33-46.
ChGu2 P.J. Chen and M.E. Gurtin, The growth of one-dimensional shock waves in elastic nonconductors, Int. J. Solids Structures 7 (1971), 5-10.
Du J. Dunwoody, One dimensional shock waves in heat conducting materials with memory , Arch. Rat. Mech. Anal. 50 (1973), 278-289.
FuSc1 Y.B. Fu and N.H. Scott, The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors, Arch. Rat. Mech. Anal. 108 (1989), 11-34.
FuSc2 Y.B. Fu and N.H. Scott, The evolutionary behavior of plane transverse weak nonlinear shock waves in unstrained incompressible isotropic elastic non-conductors, Wave Motion 11 (1989), 351-365.
Gr M.A. Grinfel’d, Ray method of calculating the wave front intensity in nonlinearly elastic material, PMM J. Appl. Math. and Mech. 42 (1978), 883-898.
NKRK N.K.-R. Kevlahan, The propagation of weak shocks in non-uniform flows, J. Fluid Mech. 327 (1996), 161-197.
LPS M.P. Lazarev, P. Prasad and S.K. Singh, An approximate solution of one-dimensional piston problem, ZAMP 46 (1996), 752-771.
Ma V.P. Maslov, Propagation of shock waves in an isentropic nonviscous gas, J. Sov. Math. 13 (1980), 119-163.
NuWa J.W. Nunziato and E.K. Walsh, Propagation and growth of shock waves in inhomogeneous fluids, Phys. Fluids 15 (1972), 1397-1402.
PP P. Prasad, Propagation of a curved shock and nonlinear ray theory, Longman Scientific and Technical (1993).
RaShJe Ch. Radha, V.D. Sharma and A. Jeffrey, An approximate analytical method for describing the kinematics of a bore over a sloping beach, Applicable Anal. 81 (2002), 867-892.
RRPP1 R. Ravindran and P. Prasad, A New Theory of Shock Dynamics, Part I: Analytic Considerations, Appl. Math. Lett. 3(2) (1990), 77-81.
RRPP2 P. Prasad and R. Ravindran, A New Theory of Shock Dynamics, Part II: Numerical Solution, Appl. Math. Lett. 3(3) (1990), 107-109.
RRPP3 R. Ravindran and P. Prasad, On an infinite system of compatibility conditions along a shock ray, Q.J. Mech. Appl. Math. 46 (1993), 131-140.
RR R. Ravindran, Shock Dynamics, J. Indian Inst. Sci. 75 (1976), 517-535.
RPS R. Ravindran, S. Sundar and P. Prasad, Long Term Behaviour of the Solution of a System of Equations from New Theory of Shock Dynamics, Computers Math. Applic. 27(12) (1994), 91-104.
SchuNuWa K.W. Schuler, J.W. Nunziato and E.K. Walsh, Recent results in nonlinear viscoelastic wave propagation, Int. J. Solids Struct. 9 (1973), 1237-1281.
ShaRa1 V.D. Sharma and Ch. Radha, On one-dimensional planar and nonplanar shock waves in a relaxing gas, Phys. Fluids 6(6) (1994), 2177-2190.
ShaRa2 V.D. Sharma and Ch. Radha, Three dimensional shock wave propagation in an ideal gas, Int. J. Non-Linear Mechanics 30(3) (1995), 305-322.
Smol J. Smoller, Shock waves and Reaction-Diffusion Equations, Springer-Verlag, 1967.
SrPr1 R. Srinivasan and P. Prasad, On the propagation of a multidimensional shock of arbitrary strength, Proc. Indian Acad. Sci. (Math. Sci.) 94 (1985), 27-42.
SrPr2 R. Srinivasan and P. Prasad, Corrections to some expressions in “On the propagation of a multidimensional shock of arbitrary strength", Proc. Indian Acad. Sci. (Math. Sci.) 100 (1990), 93-94.
Ti T.C.T. Ting, Further study of one dimensional shock waves in nonlinear elastic media, Q. Appl. Math. 37 (1980), 421-429.
Whit G.B. Whitham, Linear and nonlinear waves, John Wiley and Sons (1974).
Wr T.W. Wright, An intrinsic description of unsteady shock waves, Q. J. Mech. App. Math. 29 (1976), 311-324.
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Additional Information
Jürgen Batt
Affiliation:
Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstr. 39, 80333 München, Germany
Email:
batt@mathematik.uni-muenchen.de
Renuka Ravindran
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560012 India
Email:
renrav@math.iisc.ernet.in
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