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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

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

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.
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 & 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.  doi: 10.1007/s00332-005-0712-7.  Google Scholar

[2]

H. S. Bhat and R. C. Fetecau, The Riemann problem for the Leray-Burgers equation,, J. Differential Equations, 246 (2009), 3957.  doi: 10.1016/j.jde.2009.01.006.  Google Scholar

[3]

R. Camassa, Characteristics and initial value problem of a completely integrable shallow water equation,, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 115.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jcp.2005.12.013.  Google Scholar

[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.  doi: 10.1016/j.jcp.2010.06.002.  Google Scholar

[7]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons,, in, (2003), 37.   Google Scholar

[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.   Google Scholar

[9]

H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation,, SIAM J. Numer. Anal., 44 (2006), 1655.  doi: 10.1137/040611975.  Google Scholar

[10]

J. Leray, Essai sur le mouvement d'un fluid visqueux emplissant l'space,, Acat Math., 63 (1934), 93.   Google Scholar

[11]

K. Mohseni, H. Zhao and J. Marsden, Shock regularization for the Burgers equation,, AIAA Paper 2006-1516, (2006), 2006.   Google Scholar

[12]

G. Norgard and K. Mohseni, A regularization of the Burgers equation using a filtered convective velocity,, J. Phys. A: Math. Theor., 41 (2008).  doi: 10.1088/1751-8113/41/34/344016.  Google Scholar

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.  doi: 10.1007/s00332-005-0712-7.  Google Scholar

[2]

H. S. Bhat and R. C. Fetecau, The Riemann problem for the Leray-Burgers equation,, J. Differential Equations, 246 (2009), 3957.  doi: 10.1016/j.jde.2009.01.006.  Google Scholar

[3]

R. Camassa, Characteristics and initial value problem of a completely integrable shallow water equation,, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 115.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jcp.2005.12.013.  Google Scholar

[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.  doi: 10.1016/j.jcp.2010.06.002.  Google Scholar

[7]

A. Degasperis, D. D. Holm and A. N. W. Hone, Integrable and non-integrable equations with peakons,, in, (2003), 37.   Google Scholar

[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.   Google Scholar

[9]

H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation,, SIAM J. Numer. Anal., 44 (2006), 1655.  doi: 10.1137/040611975.  Google Scholar

[10]

J. Leray, Essai sur le mouvement d'un fluid visqueux emplissant l'space,, Acat Math., 63 (1934), 93.   Google Scholar

[11]

K. Mohseni, H. Zhao and J. Marsden, Shock regularization for the Burgers equation,, AIAA Paper 2006-1516, (2006), 2006.   Google Scholar

[12]

G. Norgard and K. Mohseni, A regularization of the Burgers equation using a filtered convective velocity,, J. Phys. A: Math. Theor., 41 (2008).  doi: 10.1088/1751-8113/41/34/344016.  Google Scholar

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