A particle method and numerical study of a quasilinear partial differential equation

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September 2011, 10(5): 1503-1515. doi: 10.3934/cpaa.2011.10.1503

A particle method and numerical study of a quasilinear partial differential equation

Roberto Camassa ^1, , Pao-Hsiung Chiu ^2, , Long Lee ^3, and W.-H. Sheu ^4,

1.	The University of North Carolina at Chapel Hill, Phillips Hall, CB #3250, Chapel Hill, NC 27599-3250, United States
2.	Nuclear Engineering Division, Institute of Nuclear Energy Research, Taoyuan County, 32546, Taiwan
3.	Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036, United States
4.	Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei City 106, Taiwan

Received April 2009 Revised June 2010 Published April 2011

Abstract
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We present a particle method for studying a quasilinear partial differential equation (PDE) in a class proposed for the regularization of the Hopf (inviscid Burger) equation via nonlinear dispersion-like terms. These are obtained in an advection equation by coupling the advecting field to the advected one through a Helmholtz operator. Solutions of this PDE are "regularized" in the sense that the additional terms generated by the coupling prevent solution multivaluedness from occurring. We propose a particle algorithm to solve the quasilinear PDE. "Particles" in this algorithm travel along characteristic curves of the equation, and their positions and momenta determine the solution of the PDE. The algorithm follows the particle trajectories by integrating a pair of integro-differential equations that govern the evolution of particle positions and momenta. We introduce a fast summation algorithm that reduces the computational cost from $O(N^2)$ to $O(N)$, where $N$ is the number of particles, and illustrate the relation between dynamics of the momentum-like characteristic variable and the behavior of the solution of the PDE.

Keywords: Hopf equation., regularization, inviscid Burgers, Quasilinear partial differential equation, particle method, Helmholtz operator.

Mathematics Subject Classification: Primary: 65M25; Secondary: 35G2.

Citation: Roberto Camassa, Pao-Hsiung Chiu, Long Lee, W.-H. Sheu. A particle method and numerical study of a quasilinear partial differential equation. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1503-1515. doi: 10.3934/cpaa.2011.10.1503

References:

[1]	H. S. Bhat and R. C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci., 16 (2006), 615-638. doi: 10.1007/s00332-005-0712-7.
[2]	H. S. Bhat and R. C. Fetecau, The Riemann problem for the Leray-Burgers equation, J. Differential Equations, 246 (2009), 3957-3979. doi: 10.1016/j.jde.2009.01.006.
[3]	R. Camassa, Characteristics and initial value problem of a completely integrable shallow water equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 115-139.
[4]	R. Camassa, J. Huang and L. Lee, On a completely integral numerical scheme for a nonlinear shallow-water wave equation, J. Nonlin. Math. Phys., 12 (2005), 146-162.
[5]	R. Camassa, J. Huang and L. Lee, Integral and integrable algorithm for a nonlinear shallow-water wave equation, J. Comp. Phys., 216 (2006), 547-572. doi: 10.1016/j.jcp.2005.12.013.
[6]	R. Camassa, P. H. Chiu, L. Lee and T. W. H. Sheu, Viscous and inviscid regularizations in a class of evolutionary partial differential equations, J. Comp. Phys., 229 (2010), 6676-6687. doi: 10.1016/j.jcp.2010.06.002.
[7]	A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons, in "Nonlinear Physics: Theory and Experiment, II'' (Gallipoli, 2002), World Sci Publishing, River Edge, NJ, (2003), 37-43.
[8]	H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), 505-528.
[9]	H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44 (2006), 1655-1680. doi: 10.1137/040611975.
[10]	J. Leray, Essai sur le mouvement d'un fluid visqueux emplissant l'space, Acat Math., 63 (1934), 93-258.
[11]	K. Mohseni, H. Zhao and J. Marsden, Shock regularization for the Burgers equation, AIAA Paper 2006-1516, 44th AIAA Aerospace Science Meeting and Exibit Reno, Nevada, Jan, 9-12, 2006.
[12]	G. Norgard and K. Mohseni, A regularization of the Burgers equation using a filtered convective velocity, J. Phys. A: Math. Theor., 41 (2008), 344016, 21pp. doi: 10.1088/1751-8113/41/34/344016.

show all references

References:

[1]	H. S. Bhat and R. C. Fetecau, A Hamiltonian regularization of the Burgers equation, J. Nonlinear Sci., 16 (2006), 615-638. doi: 10.1007/s00332-005-0712-7.
[2]	H. S. Bhat and R. C. Fetecau, The Riemann problem for the Leray-Burgers equation, J. Differential Equations, 246 (2009), 3957-3979. doi: 10.1016/j.jde.2009.01.006.
[3]	R. Camassa, Characteristics and initial value problem of a completely integrable shallow water equation, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 115-139.
[4]	R. Camassa, J. Huang and L. Lee, On a completely integral numerical scheme for a nonlinear shallow-water wave equation, J. Nonlin. Math. Phys., 12 (2005), 146-162.
[5]	R. Camassa, J. Huang and L. Lee, Integral and integrable algorithm for a nonlinear shallow-water wave equation, J. Comp. Phys., 216 (2006), 547-572. doi: 10.1016/j.jcp.2005.12.013.
[6]	R. Camassa, P. H. Chiu, L. Lee and T. W. H. Sheu, Viscous and inviscid regularizations in a class of evolutionary partial differential equations, J. Comp. Phys., 229 (2010), 6676-6687. doi: 10.1016/j.jcp.2010.06.002.
[7]	A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons, in "Nonlinear Physics: Theory and Experiment, II'' (Gallipoli, 2002), World Sci Publishing, River Edge, NJ, (2003), 37-43.
[8]	H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), 505-528.
[9]	H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation, SIAM J. Numer. Anal., 44 (2006), 1655-1680. doi: 10.1137/040611975.
[10]	J. Leray, Essai sur le mouvement d'un fluid visqueux emplissant l'space, Acat Math., 63 (1934), 93-258.
[11]	K. Mohseni, H. Zhao and J. Marsden, Shock regularization for the Burgers equation, AIAA Paper 2006-1516, 44th AIAA Aerospace Science Meeting and Exibit Reno, Nevada, Jan, 9-12, 2006.
[12]	G. Norgard and K. Mohseni, A regularization of the Burgers equation using a filtered convective velocity, J. Phys. A: Math. Theor., 41 (2008), 344016, 21pp. doi: 10.1088/1751-8113/41/34/344016.

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