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

Vincenzo Michael Isaia

doi:10.3934/dcdsb.2017175

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2017, Volume 22, Issue 9: 3459-3481. Doi: 10.3934/dcdsb.2017175

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

Vincenzo Michael Isaia^,

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

* Corresponding author:vincenzo.isaia@indstate.edu
* Corresponding author:vincenzo.isaia@indstate.edu

Received: May 2016

Revised: June 2017

Published: October 2017

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Abstract

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.

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Mathematics Subject Classification: Primary:65M12;Secondary:35B40, 35C06.

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Figure 1. Experiment 3 -Quasilinear outer expansion, comparison

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Figure 2. Experiment 3 -Quasilinear inner expansion, spatial

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Figure 3. Experiment 4 -$m=0$, log detection in $M_n$

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Figure 4. Experiment 4 -$m=1$, no log detection in $M_n$, $(\Delta \phi^{\infty})_n$

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Figure 5. Experiment 5 -computed $\alpha$, $p=2$: history plots

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Figure 6. Experiment 6 history plots -Gradient Diffusion with $m=0.1$

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Figure 7. Experiment 7 -finite time extinction, $p=0.5$: history plots

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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-6	1.8e-6	1.7e-5	3.0e-5	< 1e-10	< 1e-10
$\Delta \phi^1_n$	2.4e-5	5.9e-6	3.4e-5	1.7e-5	< 1e-10	< 1e-10
$\Delta^* \phi^{\infty}_n$	3.7e-3	3.9e-3	4.9e-3	2.6e-2	4.9e-4	1.3e-2
$\Delta^* \phi^1_n$	8.5e-3	8.5e-3	9.6e-3	1.5e-2	6.5e-6	9.2e-3
$T^*_n$	1.802831	1.238660	1.127843	31.31502	2.102335	1.433876
$T^*_{d}$	1.802793	1.238642	1.127830	31.56750	2.102333	1.433861

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