# An Application of 3-D Kinematical Conservation Laws: Propagation of a 3-D Wavefront

## Abstract

*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

### Keywords

## Get full access to this article

View all available purchase options and get full access to this article.

## References

*Phys. Fluids*, 29 (1986), pp. 2847–2852.

*Int. J. Comput. Fluid Dyn.*, 9 (1997), pp. 1–22.

*-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.

*Eigenvalues of kinematical conservation laws (KCL) based*3

*-D weakly nonlinear ray theory (WNLRT)*, Appl. Math. Comput. to appear.

*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.

*IMA J. Appl. Math.*, 69 (2004), pp. 391–420.

*J. Fluid Mech.*, 523 (2005), pp. 171–198.

*Proc. Indian Acad. Sci. (Math. Sci.)*, 116 (2006), pp. 97–119.

*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.

*SIAM J. Numer. Anal.*, 41 (2003), pp. 135–158.

*SIAM J. Numer. Anal.*, 35 (1998), pp. 2317–2328.

*Arch. Ration. Mech. Anal.*, 130 (1995), pp. 231–276.

*Introduction to Calculus and Analysis*, Vol. II, John Wiley & Sons, New York, 1974.

*J. Differential Equations*, 245 (2008), pp. 3704–3734.

*J. Comput. Appl. Math.*, 74 (1996), pp. 175–192.

*Phys. Fluids*, 7 (1964), pp. 700–706.

*Conservation Form of Equations of Three Dimensional Front Propagation*, Technical report, Department of Mathematics, Indian Institute of Science, Bangalore, 1995.

*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.

*J. Comput. Phys.*, 126 (1996), pp. 202–228.

*SIAM J. Sci. Comput.*, 19 (1998), pp. 1892–1917.

*SIAM J. Numer. Anal.*, 35 (1998), pp. 2147–2168.

*Comm. Pure Appl. Math.*, 7 (1954), pp. 159–193.

*J. Hyperbolic Differ. Equ.*, 1 (2004), pp. 315–327.

*J. Fluid Mech.*, 434 (2001), pp. 119–151.

*Conservation Form of Nonlinear Ray Equations*, Technical report, Department of Mathematics, Indian Institute of Science, Bangalore, 1992.

*J. Comput. Phys.*, 87 (1990), pp. 408–463.

*Nonlinear Hyperbolic Waves in Multi-dimensions*, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 121, Chapman and Hall/CRC, Boca Raton, FL, 2001.

*Indian J. Pure Appl. Math.*, 38 (2007), pp. 467–490.

*J. Fluid Mech.*, 385 (1999), pp. 1–20.

*Multidimensional Delta-Shocks and the Transportation and Concentration Processes*, preprint, 2007; available on http://www.math.ntnu.no/conservation/2007/031. html.

*J. Fluid Mech.*, 73 (1976), pp. 651–671.

*M2AN Math. Model. Numer. Anal.*, 34 (2000), pp. 1203–1231.

*J. Differential Equations*, 112 (1994), pp. 1–32.

*J. Fluid Mech.*, 2 (1957), pp. 145–171.

*Linear and Nonlinear Waves*, John Wiley & Sons, New York, 1974.

## Information & Authors

### Information

#### Published In

#### Copyright

#### History

**Submitted**: 14 August 2008

**Accepted**: 8 May 2010

**Published online**: 15 July 2010

#### MSC codes

#### Keywords

### Authors

## Metrics & Citations

### Metrics

### Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

#### Cited By

- Weakly nonlinear ray theory in inhomogeneous moving media filled with polytropic gasesWave Motion, Vol. 91 | 1 Nov 2019
- Theoretical developments in the study of partial differential equationsIndian Journal of Pure and Applied Mathematics, Vol. 50, No. 3 | 20 August 2019
- Kinematical conservation laws in a space of arbitrary dimensionsIndian Journal of Pure and Applied Mathematics, Vol. 47, No. 4 | 13 October 2016
- A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan ModeSIAM Journal on Scientific Computing, Vol. 34, No. 2 | 10 April 2012
- Numerical Front Propagation Using Kinematical Conservation LawsFinite Volumes for Complex Applications VI Problems & Perspectives | 9 July 2011

## View Options

### Get Access

**Access via your Institution**- Questions about how to access this content? Contact SIAM at
**[email protected]**.