On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Katharina Schratz; Xiaofei Zhao

doi:10.3934/dcdsb.2020043

Article Contents

2020, Volume 25, Issue 8: 2841-2865. Doi: 10.3934/dcdsb.2020043

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On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Katharina Schratz^1, and
Xiaofei Zhao^2,3,

1.
Fakultät für Mathematik, Karlsruhe Institute of Technology, Germany
2.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
3.
Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China

Received: May 2018

Revised: January 2019

Early access: February 2020

Published: August 2020

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Abstract

This work concerns the time averaging techniques for the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime which have recently gained a lot of attention in numerical analysis. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. To overcome this numerical burden various novel numerical methods with excellent efficiency were derived in recent years. The construction of each method thereby requests essentially the averaged model of the problem. However, the averaged model of each approach is found by different kinds of asymptotic approximation techniques reaching from the modulated Fourier expansion over the multiscale expansion by frequency up to the Chapman-Enskog expansion. In this work we give a first comparison of these recently introduced asymptotic series, reviewing their approximation validity to the KG in the asymptotic limit, their smoothness assumptions as well as their geometric properties, e.g., energy conservation and long-time behaviour of the remainder.

Keywords:

Mathematics Subject Classification: 35Q40, 34C29, 34E05, 81Q05.

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Figure 1. Behavior of the remainder $ \|R\|_{H^1}/ \varepsilon^2 $ of the two-term expansion and the one-term expansion with respect to time

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Figure 2. Behavior of the remainder terms $ \|R_{MFo}\|_{H^1}/ \varepsilon^2 $, $ \|R_{MFe}\|_{H^1}/ \varepsilon^2 $ and $ \|R_{CE}\|_{H^1}/ \varepsilon^2 $ with respect to time under two cases: initial smooth localized wave (2.18) in $ {\mathbb R} $ (left) and smooth planewave (2.19) on torus $ {\mathbb T} $ (right)

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Figure 3. Profiles of the expansions $ u_{MFo}(x, 3) $, $ u_{MFe}(x, 3) $, $ u_{CE}(x, 3) $ and exact solution $ u(x, 3) $ at $ t = 3 $ under $ \varepsilon = 0.1 $: the whole space example (2.18) (up) and the torus example (2.19) (down)

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Figure 4. Energy error $ |\mathcal{H}(t)-H(0)|/|H(0)| $ of the three expansions to the leading order under whole space case (up) and torus case (down)

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Figure 5. Behavior of the MFo remainder under lower regularity initial data (2.20): $ \|R_{MFo}\|_{H^1}/ \varepsilon^2 $ (left) under $ H^5 $-data, $ \|R_{MFo}\|_{H^1}/ \varepsilon^2 $ (middle) and $ \|R_{MFo}\|_{L^2}/ \varepsilon^2 $ (right) under $ H^4 $-data

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Figure 6. Behavior of the MFe remainder under lower regularity initial data (2.20): $ \|R_{MFe}\|_{H^1}/ \varepsilon^2 $ (left) under $ H^3 $-data, $ \|R_{MFe}\|_{H^1}/ \varepsilon^2 $ (middle) and $ \|R_{MFe}\|_{L^2}/ \varepsilon^2 $ (right) under $ H^2 $-data

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Figure 7. Behavior of the CE remainder under lower regularity initial data (2.20): $ \|R_{CE}\|_{H^1}/ \varepsilon^2 $ (left) under $ H^3 $-data, $ \|R_{CE}\|_{H^1}/ \varepsilon^2 $ (middle) and $ \|R_{CE}\|_{L^2}/ \varepsilon^2 $ (right) under $ H^2 $-data

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Figure 8. Behavior of the remainder terms $ \|R_{MFo}\|_{H^1}/ \varepsilon^4 $, $ \|R_{MFe}\|_{H^1}/ \varepsilon^4 $ and $ \|R_{CE}\|_{H^1}/ \varepsilon^4 $ with respect to time under two cases: initial smooth localized wave (2.18) in $ {\mathbb R} $ (left) and smooth planewave (2.19) on torus $ {\mathbb T} $ (right)

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Figure 9. Energy error $ |\mathcal{H}(t)-H(0)|/|H(0)| $ of the three expansions to the next order under whole space case (up) and torus case (down)

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Figure 10. Behavior of the remainders under lower regularity initial data (2.20)

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Table 1. Remainders (in $ H^1 $-norm) of the classical two-term expansion and the one-term with perturbation

$t=0.5$	$\varepsilon =0.05$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
two-term	9.00E-3	2.40E-3	5.98E-4	1.51E-4
one-term	9.60E-3	2.50E-3	6.45E-4	1.64E-4
$t=1$	$\varepsilon =0.05$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
two-term	1.79E-2	4.80E-3	1.20E-3	3.05E-4
one-term	1.84E-2	4.90E-3	1.30E-3	3.15E-4

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Table 2. Remainders (in $ H^1 $-norm) of the leading order modulated Fourier (MFo) expansion, the multiscale expansion by frequency (MFe) and the Chapman-Enskog (CE) expansion

$t=0.5$	$\varepsilon =0.1$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
MFo	3.46E-2	9.90E-3	2.60E-3	6.50E-4
MFe	1.00E-2	2.10E-3	5.35E-4	1.31E-4
CE	1.20E-3	2.58E-4	6.59E-5	1.79E-5
$t=1$	$\varepsilon =0.1$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
MFo	6.15E-2	1.88E-2	5.00E-3	1.30E-3
MFe	9.70E-3	2.40E-3	5.94E-4	1.51E-4
CE	9.29E-4	2.87E-4	7.23E-5	1.74E-5

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Table 3. Time derivative of the remainders: $ \|\partial_tR(\cdot, t)\|_{H^1} $ of the leading order MFo, MFe and CE

$t=0.5$	$\varepsilon =0.1$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
MFo	4.65	4.72	4.71	4.72
MFe	8.65E-1	1.09	1.10	1.12
CE	7.86E-2	6.68E-2	6.70E-2	6.94E-2

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Table 4. Remainders (in $ H^1 $-norm) of the next order modulated Fourier (MFo) expansion, the multiscale frequency (MFe) expansion and the Chapman-Enskog (CE) expansion

$t=0.5$	$\varepsilon =0.1$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
MFo	2.30E-2	2.30E-3	1.82E-4	1.20E-5
MFe	4.99E-6	1.61E-7	1.06E-8	6.69E-10
CE	3.64E-5	1.87E-6	1.20E-7	7.29E-9
$t=1$	$\varepsilon =0.1$	$\varepsilon /2$	$\varepsilon /2^2$	$\varepsilon /2^3$
MFo	5.68E-2	7.60E-3	6.84E-4	4.61E-5
MFe	3.03E-6	2.42E-7	1.56E-8	1.23E-9
CE	2.20E-5	1.58E-6	1.03E-7	6.48E-9

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