June  2018, 23(4): 1819-1833. doi: 10.3934/dcdsb.2018089

Singular perturbed renormalization group theory and its application to highly oscillatory problems

1. 

College of Mathematics, Jilin University, Changchun 130012, China

2. 

Beijing Computational Science Research Center, ZPark Ⅱ, No. 10 Dongbeiwang, West Road, Haidian District, Beijing 100094, China

3. 

College of Mathematics State key laboratory of automotive, simulation and control, Jilin University, Changchun 130012, China

Corresponding author: Shaoyun Shi (shisy@jlu.edu.cn)

Received  October 2016 Revised  October 2017 Published  June 2018 Early access  March 2018

Fund Project: This work is supported by NSFC grant (No. 11771177,11301210), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), NSF grant (No. 20140520053JH) and ESF grant (No. JJKH20170776KJ) of Jilin, China.

Renormalization group method in the singular perturbation theory, originally introduced by Chen et al, has been proven to be very practicable in a large number of singular perturbed problems. In this paper, we will firstly reconsider the Renormalization group method under some general conditions to get several newly rigorous approximate results. Then we will apply the obtained results to investigate a class of second order differential equations with the highly oscillatory phenomenon of highly oscillatory properties, which occurs in many multiscale models from applied mathematics, physics and material science, etc. Our strategy, in fact, can be also used to analyze the same problem for related evolution equations with multiple scales, such as nonlinear Klein-Gordon equations in the nonrelativistic limit regime.

Citation: Wenlei Li, Shaoyun Shi. Singular perturbed renormalization group theory and its application to highly oscillatory problems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1819-1833. doi: 10.3934/dcdsb.2018089
References:
[1]

W. Z. BaoX. C. Dong and X. F. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study., 47 (2014), 111-150. 

[2]

W. Z. Bao and X. C. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.

[3]

A. D. Bryuno, Analytic form of differential equations Ⅰ, Ⅱ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262; ibid. 

[4]

L. Y. ChenN. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett., 73 (1994), 1311-1315.  doi: 10.1103/PhysRevLett.73.1311.

[5]

L. Y. ChenN. Goldenfeld and Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E., 54 (1996), 376-394.  doi: 10.1103/PhysRevE.54.376.

[6]

H. Chiba, $C^1$ Approximation of vector fields based on the renormalization group method, SIAM J. Appl. Dyn. Syst., 7 (2008), 895-932.  doi: 10.1137/070694892.

[7]

H. Chiba, Extension and unification of singular perturbation methods for ODEs based on the renormalization group method, SIAM J. Appl. Dyn. Syst., 8 (2009), 1066-1115.  doi: 10.1137/090745957.

[8]

S. I. EiK. Fujii and T. Kunihiro, Renormalization-group method for reduction of evolution equations: invariant manifolds and envelopes, Ann. Physics., 280 (2000), 236-298.  doi: 10.1006/aphy.1999.5989.

[9]

N. GoldenfeldB. P. Athreya and J. A. Dantzig, Renormalization group approach to Multiscale modelling in materials science, J. Stat. Phys., 125 (2006), 1019-1027.  doi: 10.1007/s10955-005-9013-7.

[10]

S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext. Springer-Verlag, Berlin, 2003.

[11]

M. H. Holmes, Introduction to Perturbation Methods, Springer-Verlag, Texts in Applied Mathematics, 20. New York, 2013.

[12]

E. Kirkinis, The Renormalization Group: A perturbation method for the graduate curriculum, SIAM Rev., 54 (2012), 374-388.  doi: 10.1137/080731967.

[13]

I. Moise and R. Temam, Renormalization group method. Applications to Navier-Stokes equation, Discrete Contin. Dyn. Syst., 6 (2000), 191-210. 

[14]

I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations, J. Dynam. Differential Equations, 13 (2001), 275-321.  doi: 10.1023/A:1016680007953.

[15]

R. E. O'Malley and E. Kirkinis, Examples illustrating the use of renormalization techniques on singularly perturbed differential equations, Stud. Appl. Math., 122 (2009), 105-122.  doi: 10.1111/j.1467-9590.2008.00425.x.

[16]

R. E. O'Malley and E. Kirkinis, A combined renormalization group-multiple scale method for singularly perturbed problems, Stud. Appl. Math., 124 (2010), 383-410.  doi: 10.1111/j.1467-9590.2009.00475.x.

[17]

M. PetcuR. Temam and D. Wirosoetisno, Renormalization group method applied to the primitive equations, J. Differential Equations, 208 (2005), 215-257.  doi: 10.1016/j.jde.2003.10.011.

[18]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, Texts in Applied Mathematics, 59. New York, 2007.

[19]

A. SarkarJ. K. BhattacharjeeS. Chakraborty and D. B. Banerjee, Center or limit cycle: Renormalization group as a probe, Eur. Phys. J. D, 64 (2011), 479-489.  doi: 10.1140/epjd/e2011-20060-1.

[20]

K. Walid and S. Ablou, On the renormalization group Approach to perturbation theory for PDEs, Ann. Henri Poincaré, 11 (2010), 1007-1021.  doi: 10.1007/s00023-010-0046-3.

[21]

M. Ziane, On a certain renormalization group method, J. Math. Phys., 41 (2000), 3290-3299.  doi: 10.1063/1.533307.

show all references

References:
[1]

W. Z. BaoX. C. Dong and X. F. Zhao, Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations, J. Math. Study., 47 (2014), 111-150. 

[2]

W. Z. Bao and X. C. Dong, Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime, Numer. Math., 120 (2012), 189-229.  doi: 10.1007/s00211-011-0411-2.

[3]

A. D. Bryuno, Analytic form of differential equations Ⅰ, Ⅱ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262; ibid. 

[4]

L. Y. ChenN. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett., 73 (1994), 1311-1315.  doi: 10.1103/PhysRevLett.73.1311.

[5]

L. Y. ChenN. Goldenfeld and Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E., 54 (1996), 376-394.  doi: 10.1103/PhysRevE.54.376.

[6]

H. Chiba, $C^1$ Approximation of vector fields based on the renormalization group method, SIAM J. Appl. Dyn. Syst., 7 (2008), 895-932.  doi: 10.1137/070694892.

[7]

H. Chiba, Extension and unification of singular perturbation methods for ODEs based on the renormalization group method, SIAM J. Appl. Dyn. Syst., 8 (2009), 1066-1115.  doi: 10.1137/090745957.

[8]

S. I. EiK. Fujii and T. Kunihiro, Renormalization-group method for reduction of evolution equations: invariant manifolds and envelopes, Ann. Physics., 280 (2000), 236-298.  doi: 10.1006/aphy.1999.5989.

[9]

N. GoldenfeldB. P. Athreya and J. A. Dantzig, Renormalization group approach to Multiscale modelling in materials science, J. Stat. Phys., 125 (2006), 1019-1027.  doi: 10.1007/s10955-005-9013-7.

[10]

S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Universitext. Springer-Verlag, Berlin, 2003.

[11]

M. H. Holmes, Introduction to Perturbation Methods, Springer-Verlag, Texts in Applied Mathematics, 20. New York, 2013.

[12]

E. Kirkinis, The Renormalization Group: A perturbation method for the graduate curriculum, SIAM Rev., 54 (2012), 374-388.  doi: 10.1137/080731967.

[13]

I. Moise and R. Temam, Renormalization group method. Applications to Navier-Stokes equation, Discrete Contin. Dyn. Syst., 6 (2000), 191-210. 

[14]

I. Moise and M. Ziane, Renormalization group method. Applications to partial differential equations, J. Dynam. Differential Equations, 13 (2001), 275-321.  doi: 10.1023/A:1016680007953.

[15]

R. E. O'Malley and E. Kirkinis, Examples illustrating the use of renormalization techniques on singularly perturbed differential equations, Stud. Appl. Math., 122 (2009), 105-122.  doi: 10.1111/j.1467-9590.2008.00425.x.

[16]

R. E. O'Malley and E. Kirkinis, A combined renormalization group-multiple scale method for singularly perturbed problems, Stud. Appl. Math., 124 (2010), 383-410.  doi: 10.1111/j.1467-9590.2009.00475.x.

[17]

M. PetcuR. Temam and D. Wirosoetisno, Renormalization group method applied to the primitive equations, J. Differential Equations, 208 (2005), 215-257.  doi: 10.1016/j.jde.2003.10.011.

[18]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, Texts in Applied Mathematics, 59. New York, 2007.

[19]

A. SarkarJ. K. BhattacharjeeS. Chakraborty and D. B. Banerjee, Center or limit cycle: Renormalization group as a probe, Eur. Phys. J. D, 64 (2011), 479-489.  doi: 10.1140/epjd/e2011-20060-1.

[20]

K. Walid and S. Ablou, On the renormalization group Approach to perturbation theory for PDEs, Ann. Henri Poincaré, 11 (2010), 1007-1021.  doi: 10.1007/s00023-010-0046-3.

[21]

M. Ziane, On a certain renormalization group method, J. Math. Phys., 41 (2000), 3290-3299.  doi: 10.1063/1.533307.

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