• Previous Article
    Structural parameter optimization of linear elastic systems
  • CPAA Home
  • This Issue
  • Next Article
    On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis
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 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.

[1]

Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793

[2]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604

[3]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[4]

Chi Hin Chan, Magdalena Czubak, Luis Silvestre. Eventual regularization of the slightly supercritical fractional Burgers equation. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 847-861. doi: 10.3934/dcds.2010.27.847

[5]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045

[6]

Juan Carlos De los Reyes, Estefanía Loayza-Romero. Total generalized variation regularization in data assimilation for Burgers' equation. Inverse Problems and Imaging, 2019, 13 (4) : 755-786. doi: 10.3934/ipi.2019035

[7]

John D. Towers. The Lax-Friedrichs scheme for interaction between the inviscid Burgers equation and multiple particles. Networks and Heterogeneous Media, 2020, 15 (1) : 143-169. doi: 10.3934/nhm.2020007

[8]

María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049

[9]

John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems and Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333

[10]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

[11]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591

[12]

Sang-Yeun Shim, Marcos Capistran, Yu Chen. Rapid perturbational calculations for the Helmholtz equation in two dimensions. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 627-636. doi: 10.3934/dcds.2007.18.627

[13]

András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks and Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43

[14]

Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035

[15]

Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems and Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020

[16]

Susanna V. Haziot. Study of an elliptic partial differential equation modelling the Antarctic Circumpolar Current. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4415-4427. doi: 10.3934/dcds.2019179

[17]

Dimitra Antonopoulou, Georgia Karali. A nonlinear partial differential equation for the volume preserving mean curvature flow. Networks and Heterogeneous Media, 2013, 8 (1) : 9-22. doi: 10.3934/nhm.2013.8.9

[18]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

[19]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 321-337. doi: 10.3934/naco.2021008

[20]

Jong Uhn Kim. On the stochastic Burgers equation with a polynomial nonlinearity in the real line. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 835-866. doi: 10.3934/dcdsb.2006.6.835

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (108)
  • HTML views (0)
  • Cited by (2)

[Back to Top]