Abstract

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface $\Omega_t$ in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic $7\times7$ system that is highly nonlinear. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features.

MSC codes

  1. 35L60
  2. 35L65
  3. 35L67
  4. 35L80
  5. 58J47
  6. 65M06

Keywords

  1. kinematical conservation laws
  2. ray theory
  3. nonlinear waves
  4. kinks
  5. weakly hyperbolic system
  6. finite difference scheme

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References

1.
A. M. Anile and G. Russo, Corrugation stability for plane relativistic shock waves, Phys. Fluids, 29 (1986), pp. 2847–2852.
2.
P. Arminjon, M. C. Viallon, and A. Madrane, A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured grids, Int. J. Comput. Fluid Dyn., 9 (1997), pp. 1–22.
3.
K. R. Arun and P. Prasad, 3-D kinematical conservation laws (KCL): Evolution of a surface in $\mathbb{R}^3$—in particular propagation of a nonlinear wavefront, Wave Motion, 46 (2009), pp. 293–311.
4.
K. R. Arun and P. Prasad, Eigenvalues of kinematical conservation laws (KCL) based 3-D weakly nonlinear ray theory (WNLRT), Appl. Math. Comput. to appear.
5.
S. Baskar and P. Prasad, Kinematical conservation laws applied to study geometrical shapes of a solitary wave, in Wind over Waves II: Forecasting and Fundamentals, S. Sajjadi and J. Hunt, eds., Horwood, Chichester, UK, 2003, pp. 189–200.
6.
S. Baskar and P. Prasad, Riemann problem for kinematical conservation laws and geometrical features of nonlinear wavefronts, IMA J. Appl. Math., 69 (2004), pp. 391–420.
7.
S. Baskar and P. Prasad, Propagation of curved shock fronts using shock ray theory and comparison with other theories, J. Fluid Mech., 523 (2005), pp. 171–198.
8.
S. Baskar and P. Prasad, Formulation of the problem of sonic boom by a maneuvering aerofoil as a one parameter family of Cauchy problems, Proc. Indian Acad. Sci. (Math. Sci.), 116 (2006), pp. 97–119.
9.
F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci. 22, World Scientific, River Edge, NJ, 1994, pp. 171–190.
10.
F. Bouchut, S. Jin, and X. Li, Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), pp. 135–158.
11.
Y. Brenier and E. Grenier, Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 2317–2328.
12.
G. Q. Chen and P. T. Kan, Hyperbolic conservation laws with umbilic degeneracy (I), Arch. Ration. Mech. Anal., 130 (1995), pp. 231–276.
13.
R. Courant and F. John, Introduction to Calculus and Analysis, Vol. II, John Wiley & Sons, New York, 1974.
14.
V. G. Danilov and D. Mitrovic, Delta shock wave formation in the case of triangular hyperbolic system of conservation laws, J. Differential Equations, 245 (2008), pp. 3704–3734.
15.
B. Enquist and O. Runborg, Multi-phase computations in geometrical optics, J. Comput. Appl. Math., 74 (1996), pp. 175–192.
16.
C. S. Gardner and M. D. Kruskal, Stability of plane magnetohydrodynamic shocks, Phys. Fluids, 7 (1964), pp. 700–706.
17.
M. Giles, P. Prasad, and R. Ravindran, Conservation Form of Equations of Three Dimensional Front Propagation, Technical report, Department of Mathematics, Indian Institute of Science, Bangalore, 1995.
18.
F. Huang, Existence and uniqueness of discontinuous solutions for a class of non-strictly hyperbolic systems, in Advances in Nonlinear Partial Differential Equations and Related Areas (Beijing, 1997), World Scientific, River Edge, NJ, 1998, pp. 187–208.
19.
G. S. Jiang and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), pp. 202–228.
20.
G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19 (1998), pp. 1892–1917.
21.
G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher, and E. Tadmor, High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM J. Numer. Anal., 35 (1998), pp. 2147–2168.
22.
P. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math., 7 (1954), pp. 159–193.
23.
R. J. LeVeque, The dynamics of pressureless dust clouds and delta waves, J. Hyperbolic Differ. Equ., 1 (2004), pp. 315–327.
24.
A. Monica and P. Prasad, Propagation of a curved weak shock, J. Fluid Mech., 434 (2001), pp. 119–151.
25.
K. W. Morton, P. Prasad, and R. Ravindran, Conservation Form of Nonlinear Ray Equations, Technical report, Department of Mathematics, Indian Institute of Science, Bangalore, 1992.
26.
H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408–463.
27.
P. Prasad, Nonlinear Hyperbolic Waves in Multi-dimensions, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 121, Chapman and Hall/CRC, Boca Raton, FL, 2001.
28.
P. Prasad, Ray theories for hyperbolic waves, kinematical conservation laws (KCL) and applications, Indian J. Pure Appl. Math., 38 (2007), pp. 467–490.
29.
P. Prasad and K. Sangeeta, Numerical simulation of converging nonlinear wavefronts, J. Fluid Mech., 385 (1999), pp. 1–20.
30.
V. M. Shelkovich, Multidimensional Delta-Shocks and the Transportation and Concentration Processes, preprint, 2007; available on http://www.math.ntnu.no/conservation/2007/031. html.
31.
B. Sturtevant and V. A. Kulkarny, The focusing of weak shock waves, J. Fluid Mech., 73 (1976), pp. 651–671.
32.
O. Runborg, Some new results in multiphase geometrical optics, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 1203–1231.
33.
D. Tan, T. Zhang, and Y. Zheng, Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations, 112 (1994), pp. 1–32.
34.
G. B. Whitham, A new approach to problems of shock dynamics. I. Two dimensional problems, J. Fluid Mech., 2 (1957), pp. 145–171.
35.
G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, 1974.

Information & Authors

Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 2604 - 2626
ISSN (online): 1095-712X

History

Submitted: 14 August 2008
Accepted: 8 May 2010
Published online: 15 July 2010

MSC codes

  1. 35L60
  2. 35L65
  3. 35L67
  4. 35L80
  5. 58J47
  6. 65M06

Keywords

  1. kinematical conservation laws
  2. ray theory
  3. nonlinear waves
  4. kinks
  5. weakly hyperbolic system
  6. finite difference scheme

Authors

Affiliations

M. Lukáčová-Medviďová

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