November  2017, 22(9): 3459-3481. doi: 10.3934/dcdsb.2017175

Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group

Department of Mathematics, Indiana State University, Terre Haute, IN 47809, USA

* Corresponding author:vincenzo.isaia@indstate.edu

Received  May 2016 Revised  June 2017 Published  July 2017

A numerical procedure based on the renormalization group (RG) is presented. This procedure will compute the spatial profile and blow up time for self-similar behavior. This will be generated by a family of parabolic IVPs, which includes the semilinear heat equation. This procedure also handles different diffusion structures, finite time extinction problems and exponential absorption with trivial modifications to the power law version. Convergence of the procedure is proved for the semilinear heat equation, which is a marginal perturbation to the heat equation, with a typical class of initial data. Numerical experiments show the accuracy of the method for various related problems. The main feature is simplicity: it will be shown that an explicit numerical method with a fixed mesh size is capable of computing very sharp approximations to these behaviors. This will include a priori detection of the logarithmic correction in the case of the semilinear heat equation.

Citation: Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175
References:
[1]

L. M. AbiaJ. C. López-Marcos and J. Martínez, The Euler method in the numerical integration of reaction-diffusion problems with blow-up, Appl. Num. Math., 38 (2001), 287-313.  doi: 10.1016/S0168-9274(01)00035-6.

[2]

W. K. Abou Salem, On the renormalization group approach to perturbation theory for PDEs, Ann. Henri Poincaré, 11 (2010), 1007-1021.  doi: 10.1007/s00023-010-0046-3.

[3]

D. G. AronsonS. B. Angenent and S. I. Betelú, Renormalization study of two-dimensional convergent solutions of the porous medium equation, Physica D, 138 (2000), 344-359.  doi: 10.1016/S0167-2789(99)00209-2.

[4]

G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781107050242.

[5]

J. Bebernes and S. Bricher, Final time blow-up profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal., 23 (1992), 852-869.  doi: 10.1137/0523045.

[6]

M. Berger and R. V. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), 841-863.  doi: 10.1002/cpa.3160410606.

[7]

G. A. BragaF. Furtado and V. Isaia, Renormalization group calculation of asymptotically self-similar dynamics, Discrete Contin. Dyn. Syst.(suppl.), (2005), 131-141. 

[8]

G. A. Braga and J. M. Moreira, Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients and marginal perturbations, J. Stat. Phys., 148 (2012), 280-295.  doi: 10.1007/s10955-012-0539-1.

[9]

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011.

[10]

J. Bricmont and A. Kupiainen, Renormalizing partial differential equations, Constructive Physics, Berlin: Springer, Lecture Notes in Physics, 446 (1995), 83-115.  doi: 10.1007/3-540-59190-7_23.

[11]

J. BricmontA. Kupiainen and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math., 47 (1994), 893-922.  doi: 10.1002/cpa.3160470606.

[12]

C. J. BuddG. J. Collins and V. A. Galaktionov, An asymptotic and numerical description of self-similar blow-up in quasi-linear parabolic equations, J. Comput. Appl. Math., 97 (1998), 51-80.  doi: 10.1016/S0377-0427(98)00102-2.

[13]

C. J. BuddW. Huang and R. D. Russell, Adaptivity with moving grids, Acta Numer., 18 (2009), 111-241.  doi: 10.1017/S0962492906400015.

[14]

L.-Y. Chen and N. Goldenfeld, Numerical renormalization group calculations for similarity solutions and traveling waves, Physical Review, 51 (1995), 5577–5581. arXiv: chao-dyn/9412005 doi: 10.1103/PhysRevE.51.5577.

[15]

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.

[16]

L.-Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E, 54 (1996), 376–394. arXiv: hep-th/9506161 doi: 10.1103/PhysRevE.54.376.

[17]

R. E. Lee DevilleA. HarkinM. HolzerK. Josić and T. Kaper, Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Phys. D, 237 (2008), 1029-1052.  doi: 10.1016/j.physd.2007.12.009.

[18]

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.

[19]

F. Furtado, Private Communication, (2004).

[20]

V. A. Galaktionov and J. L. Vázquez, A Stability Technique for Evolution Partial Differential Equations -a Dynamical Systems Approach, Birkhauser, Boston, 2004. doi: 10.1007/978-1-4612-2050-3.

[21]

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, 1992.

[22]

M. A. Herrero and J. J. L. Velázquez, Some results on blow-up for semilinear parabolic problems, IMA Vol. Math. Appl., 47 (1993), 105-125.  doi: 10.1007/978-1-4612-0885-3_7.

[23]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.  doi: 10.1137/1032046.

[24]

Y. Li and Y. W. Qi, The global dynamics of isothermal chemical systems with critical nonlinearity, Nonlinearity, 16 (2003), 1057-1074.  doi: 10.1088/0951-7715/16/3/315.

[25]

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.

[26]

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov (M. Grinfeld translation), Blow-up in Quasilinear Parabolic Equations, De Gruyter Expositions in Mathematics 19, Berlin, 1995. doi: 10.1515/9783110889864.535.

[27]

J. J. L. Velázquez, Local behavior near blow-up points for semilinear parabolic equations, Jour. Diff. Eqns., 106 (1993), 384-415.  doi: 10.1006/jdeq.1993.1113.

[28]

F. B. Weissler, An $L^{∞}$ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math., 38 (1985), 291-295.  doi: 10.1002/cpa.3160380303.

[29]

K. Wilson, Renormalization group and critical phenomena Ⅰ-Ⅱ, Physics Review B, 4 (1971), 3174-3285.  doi: 10.1103/PhysRevB.4.3174.

[30]

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]

L. M. AbiaJ. C. López-Marcos and J. Martínez, The Euler method in the numerical integration of reaction-diffusion problems with blow-up, Appl. Num. Math., 38 (2001), 287-313.  doi: 10.1016/S0168-9274(01)00035-6.

[2]

W. K. Abou Salem, On the renormalization group approach to perturbation theory for PDEs, Ann. Henri Poincaré, 11 (2010), 1007-1021.  doi: 10.1007/s00023-010-0046-3.

[3]

D. G. AronsonS. B. Angenent and S. I. Betelú, Renormalization study of two-dimensional convergent solutions of the porous medium equation, Physica D, 138 (2000), 344-359.  doi: 10.1016/S0167-2789(99)00209-2.

[4]

G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781107050242.

[5]

J. Bebernes and S. Bricher, Final time blow-up profiles for semilinear parabolic equations via center manifold theory, SIAM J. Math. Anal., 23 (1992), 852-869.  doi: 10.1137/0523045.

[6]

M. Berger and R. V. Kohn, A rescaling algorithm for the numerical calculation of blowing-up solutions, Comm. Pure Appl. Math., 41 (1988), 841-863.  doi: 10.1002/cpa.3160410606.

[7]

G. A. BragaF. Furtado and V. Isaia, Renormalization group calculation of asymptotically self-similar dynamics, Discrete Contin. Dyn. Syst.(suppl.), (2005), 131-141. 

[8]

G. A. Braga and J. M. Moreira, Renormalization group analysis of nonlinear diffusion equations with time dependent coefficients and marginal perturbations, J. Stat. Phys., 148 (2012), 280-295.  doi: 10.1007/s10955-012-0539-1.

[9]

J. Bricmont and A. Kupiainen, Universality in blow-up for nonlinear heat equations, Nonlinearity, 7 (1994), 539-575.  doi: 10.1088/0951-7715/7/2/011.

[10]

J. Bricmont and A. Kupiainen, Renormalizing partial differential equations, Constructive Physics, Berlin: Springer, Lecture Notes in Physics, 446 (1995), 83-115.  doi: 10.1007/3-540-59190-7_23.

[11]

J. BricmontA. Kupiainen and G. Lin, Renormalization group and asymptotics of solutions of nonlinear parabolic equations, Comm. Pure Appl. Math., 47 (1994), 893-922.  doi: 10.1002/cpa.3160470606.

[12]

C. J. BuddG. J. Collins and V. A. Galaktionov, An asymptotic and numerical description of self-similar blow-up in quasi-linear parabolic equations, J. Comput. Appl. Math., 97 (1998), 51-80.  doi: 10.1016/S0377-0427(98)00102-2.

[13]

C. J. BuddW. Huang and R. D. Russell, Adaptivity with moving grids, Acta Numer., 18 (2009), 111-241.  doi: 10.1017/S0962492906400015.

[14]

L.-Y. Chen and N. Goldenfeld, Numerical renormalization group calculations for similarity solutions and traveling waves, Physical Review, 51 (1995), 5577–5581. arXiv: chao-dyn/9412005 doi: 10.1103/PhysRevE.51.5577.

[15]

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.

[16]

L.-Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory, Phys. Rev. E, 54 (1996), 376–394. arXiv: hep-th/9506161 doi: 10.1103/PhysRevE.54.376.

[17]

R. E. Lee DevilleA. HarkinM. HolzerK. Josić and T. Kaper, Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations, Phys. D, 237 (2008), 1029-1052.  doi: 10.1016/j.physd.2007.12.009.

[18]

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.

[19]

F. Furtado, Private Communication, (2004).

[20]

V. A. Galaktionov and J. L. Vázquez, A Stability Technique for Evolution Partial Differential Equations -a Dynamical Systems Approach, Birkhauser, Boston, 2004. doi: 10.1007/978-1-4612-2050-3.

[21]

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, 1992.

[22]

M. A. Herrero and J. J. L. Velázquez, Some results on blow-up for semilinear parabolic problems, IMA Vol. Math. Appl., 47 (1993), 105-125.  doi: 10.1007/978-1-4612-0885-3_7.

[23]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.  doi: 10.1137/1032046.

[24]

Y. Li and Y. W. Qi, The global dynamics of isothermal chemical systems with critical nonlinearity, Nonlinearity, 16 (2003), 1057-1074.  doi: 10.1088/0951-7715/16/3/315.

[25]

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.

[26]

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov (M. Grinfeld translation), Blow-up in Quasilinear Parabolic Equations, De Gruyter Expositions in Mathematics 19, Berlin, 1995. doi: 10.1515/9783110889864.535.

[27]

J. J. L. Velázquez, Local behavior near blow-up points for semilinear parabolic equations, Jour. Diff. Eqns., 106 (1993), 384-415.  doi: 10.1006/jdeq.1993.1113.

[28]

F. B. Weissler, An $L^{∞}$ blow-up estimate for a nonlinear heat equation, Comm. Pure Appl. Math., 38 (1985), 291-295.  doi: 10.1002/cpa.3160380303.

[29]

K. Wilson, Renormalization group and critical phenomena Ⅰ-Ⅱ, Physics Review B, 4 (1971), 3174-3285.  doi: 10.1103/PhysRevB.4.3174.

[30]

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

Figure 1.  Experiment 3 -Quasilinear outer expansion, comparison
Figure 2.  Experiment 3 -Quasilinear inner expansion, spatial
Figure 3.  Experiment 4 -$m=0$, log detection in $M_n$
Figure 4.  Experiment 4 -$m=1$, no log detection in $M_n$, $(\Delta \phi^{\infty})_n$
Figure 5.  Experiment 5 -computed $\alpha$, $p=2$: history plots
Figure 6.  Experiment 6 history plots -Gradient Diffusion with $m=0.1$
Figure 7.  Experiment 7 -finite time extinction, $p=0.5$: history plots
Table 1.  Exp.1 and 2 -convergence and accuracy errors
Exp. $m=0$$m=1$
1 and 2$p=2$$p=3$$p=4$$p=1.05$$p=2$$p=3$
$\Delta \phi^{\infty}_n$9.9e-61.8e-61.7e-53.0e-5< 1e-10< 1e-10
$\Delta \phi^1_n$2.4e-55.9e-63.4e-51.7e-5< 1e-10< 1e-10
$\Delta^* \phi^{\infty}_n$3.7e-33.9e-34.9e-32.6e-24.9e-41.3e-2
$\Delta^* \phi^1_n$8.5e-38.5e-39.6e-31.5e-26.5e-69.2e-3
$T^*_n$1.8028311.2386601.12784331.315022.1023351.433876
$T^*_{d}$1.8027931.2386421.12783031.567502.1023331.433861
Exp. $m=0$$m=1$
1 and 2$p=2$$p=3$$p=4$$p=1.05$$p=2$$p=3$
$\Delta \phi^{\infty}_n$9.9e-61.8e-61.7e-53.0e-5< 1e-10< 1e-10
$\Delta \phi^1_n$2.4e-55.9e-63.4e-51.7e-5< 1e-10< 1e-10
$\Delta^* \phi^{\infty}_n$3.7e-33.9e-34.9e-32.6e-24.9e-41.3e-2
$\Delta^* \phi^1_n$8.5e-38.5e-39.6e-31.5e-26.5e-69.2e-3
$T^*_n$1.8028311.2386601.12784331.315022.1023351.433876
$T^*_{d}$1.8027931.2386421.12783031.567502.1023331.433861
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