Calculation of shocks using solutions of systems of ordinary differential equations

Batt, Jürgen; Ravindran, Renuka

doi:10.1090/S0033-569X-05-00976-6

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
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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.

A. M. Anile and G. Russo, Generalized wavefront expansion. I. Higher order corrections for the propagation of weak shock waves, Wave Motion 8 (1986), no. 3, 243–258. MR 841962, DOI https://doi.org/10.1016/S0165-2125%2886%2980047-6
A. M. Anile and G. Russo, Generalized wavefront expansion. II. The propagation of step shocks, Wave Motion 10 (1988), no. 1, 3–18. MR 928792, DOI https://doi.org/10.1016/0165-2125%2888%2990003-0
Paul B. Bailey and Peter J. Chen, Evolutionary behavior of induced discontinuities behind one-dimensional shock waves in nonlinear elastic materials, J. Elasticity 15 (1985), no. 3, 257–269. MR 804498, DOI https://doi.org/10.1007/BF00041424
Henri Cartan, Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Enseignement des Sciences. Hermann, Paris, 1961 (French). Avec le concours de Reiji Takahashi. MR 0147623

Ch P.J. Chen, Selected Topics in wave propagation, Nordhoff, Leyden, 1976.

Peter J. Chen and Morton E. Gurtin, On the growth of one-dimensional shock waves in materials with memory, Arch. Rational Mech. Anal. 36 (1970), 33–46. MR 270599, DOI https://doi.org/10.1007/BF00255745

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.

J. Dunwoody, One-dimensional shock waves in heat conducting materials with memory. I. Thermodynamics, Arch. Rational Mech. Anal. 47 (1972), 117–148. MR 468663, DOI https://doi.org/10.1007/BF00251226
Y. B. Fu and N. H. Scott, The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic nonconductors, Arch. Rational Mech. Anal. 108 (1989), no. 1, 11–34. MR 1006727, DOI https://doi.org/10.1007/BF01055751
Y. B. Fu and N. H. Scott, The evolutionary behaviour of plane transverse weak nonlinear shock waves in unstrained incompressible isotropic elastic nonconductors, Wave Motion 11 (1989), no. 4, 351–365. MR 1007976, DOI https://doi.org/10.1016/0165-2125%2889%2990040-1
M. A. Grinfel′d, Ray method of calculating the wave front intensity in nonlinearly elastic material; Russian transl., J. Appl. Math. Mech. 42 (1978), no. 5, 883–898. MR 620884
N. K.-R. Kevlahan, The propagation of weak shocks in non-uniform flows, J. Fluid Mech. 327 (1996), 161–197. MR 1422761, DOI https://doi.org/10.1017/S0022112096008506
M. P. Lazarev, Phoolan Prasad, and S. K. Sing, An approximate solution of one-dimensional piston problem, Z. Angew. Math. Phys. 46 (1995), no. 5, 752–771. MR 1355537, DOI https://doi.org/10.1007/BF00949078

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.

Phoolan Prasad, Propagation of a curved shock and nonlinear ray theory, Pitman Research Notes in Mathematics Series, vol. 292, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1245728
Ch. Radha, V. D. Sharma, and A. Jeffrey, An approximate analytical method for describing the kinematics of a bore over a sloping beach, Appl. Anal. 81 (2002), no. 4, 867–892. MR 1929550, DOI https://doi.org/10.1080/0003681021000004474
Renuka Ravindran and Phoolan Prasad, A new theory of shock dynamics. I. Analytic considerations, Appl. Math. Lett. 3 (1990), no. 2, 77–81. MR 1052254, DOI https://doi.org/10.1016/0893-9659%2890%2990019-8
Phoolan Prasad and Renuka Ravindran, A new theory of shock dynamics. II. Numerical solution, Appl. Math. Lett. 3 (1990), no. 3, 107–109. MR 1077884, DOI https://doi.org/10.1016/0893-9659%2890%2990150-A
Renuka Ravindran and Phoolan Prasad, On an infinite system of compatibility conditions along a shock ray, Quart. J. Mech. Appl. Math. 46 (1993), no. 1, 131–140. MR 1222383, DOI https://doi.org/10.1093/qjmam/46.1.131
Renuka Ravindran, Shock dynamics, J. Indian Inst. Sci. 75 (1995), no. 5, 517–535. MR 1383042
R. Ravindran, S. Sundar, and P. Prasad, Long time behaviour of the solution of a system of equations from new theory of shock dynamics, Comput. Math. Appl. 27 (1994), no. 12, 91–104. MR 1284133, DOI https://doi.org/10.1016/0898-1221%2894%2990089-2

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.

V. D. Sharma and Ch. Radha, On one-dimensional planar and nonplanar shock waves in a relaxing gas, Phys. Fluids 6 (1994), no. 6, 2177–2190. MR 1275083, DOI https://doi.org/10.1063/1.868220

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.

R. Srinivasan and Phoolan Prasad, On the propagation of a multidimensional shock of arbitrary strength, Proc. Indian Acad. Sci. Math. Sci. 94 (1985), no. 1, 27–42. MR 830891, DOI https://doi.org/10.1007/BF02837254
R. Srinivasan and Phoolan Prasad, Corrections to some expressions in: “On the propagation of a multidimensional shock of arbitrary strength” [Proc. Indian Acad. Sci. Math. Sci. 94 (1985), no. 1, 27–42; MR0830891 (87k:76040)], Proc. Indian Acad. Sci. Math. Sci. 100 (1990), no. 1, 93–94. MR 1051095, DOI https://doi.org/10.1007/BF02881119
T. C. T. Ting, Further study on one-dimensional shock waves in nonlinear elastic media, Quart. Appl. Math. 37 (1979/80), no. 4, 421–429. MR 564733, DOI https://doi.org/10.1090/S0033-569X-1980-0564733-5
G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954

Wr T.W. Wright, An intrinsic description of unsteady shock waves, Q. J. Mech. App. Math. 29 (1976), 311-324.