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doi: 10.3934/dcdsb.2021149
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Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system
Gengen Zhang ,
South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou 510631, China
* Corresponding author: Gengen Zhang
Received October 2020 Early access May 2021
Fund Project: The author is supported by the National Natural Science Foundation of China (Grant No.11701110), the China Postdoctoral Science Foundation (Grant No.2020M682746)
Figure(7) / Table(3)
In this paper we develop a time splitting combined with exponential wave integrator (EWI) Fourier pseudospectral (FP) method for the quantum Zakharov system (QZS), i.e. using the FP method for spatial derivatives, a time splitting technique and an EWI method for temporal derivatives in the Schrödinger-like equation and wave-type equations, respectively. The scheme is fully explicit and efficient due to fast Fourier transform. Numerical experiments for the QZS are presented to illustrate the accuracy and capability of the method, including accuracy tests, convergence of the QZS to the classical Zakharov system in the semi-classical limit, soliton-soliton collisions and pattern dynamics of the QZS in one-dimension, as well as the blow-up phenomena of QZS in two-dimension.
Keywords: Quantum Zakharov system, time splitting method, exponential wave integrator method, soliton-soliton collisions, pattern dynamics.
Mathematics Subject Classification: Primary: 35Q55, 65M06, 65M70; Secondary: 65M12.
Citation: Gengen Zhang. Time splitting combined with exponential wave integrator Fourier pseudospectral method for quantum Zakharov system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021149
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Figure 1. Convergence of $ E $ (left) and $ N $ (right) between the QZS and classical ZS in Example $ 4.1 $
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Figure 2. Inelastic collision between two solitons in Example $ 4.2 $ under case (i)
Figure 3. Inelastic collision between two solitons in Example $ 4.2 $ under case (ii)
Figure 4. Inelastic collision between two solitons in Example $ 4.2 $ under case (iii)
Figure 5. Pattern dynamics of QZS in Example $ 4.3 $, contours of $ |E| $
Figure 7. Plot of energy density $ |E|^2 $ and ion density fluctuation $ N $ in Example $ 4.5 $, $ \mu = \nu = 20 $, $ \varepsilon = \frac{1}{2^5} $
Table 1. Spatial errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ \tau = 10^{-5} $
$ e_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
$ \varepsilon =\frac{1}{2^1} $ 1.41e-2 6.83e-5 3.78e-9 4.77e-12
$ \varepsilon =\frac{1}{2^{7}} $ 5.06e-2 2.60e-4 7.75e-9 5.25e-12
$ \varepsilon =\frac{1}{2^{11}} $ 5.06e-2 2.61e-4 8.20e-9 5.25e-12
$ n_{\varepsilon} $ $ h = 1 $ $ 1/2 $ $ 1/2^2 $ $ 1/2^3 $
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Table 2. Temporal errors of the scheme at $ T = 1 $ for Example 4.1 with different $ \varepsilon $, $ h = 1/2^4 $
$ e_{\varepsilon} $ $ \tau_0 = 1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
$ \varepsilon =\frac{1}{2^1} $ 1.85e-3 3.80e-4 7.31e-5 1.38e-5 3.55e-6 8.03e-7
rate - 2.28 2.38 2.41 1.96 2.14
$ \varepsilon =\frac{1}{2^{7}} $ 7.63e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
$ \varepsilon =\frac{1}{2^{11}} $ 7.64e-4 1.87e-4 4.66e-5 1.16e-5 2.90e-6 7.20e-7
$ n_{\varepsilon} $ $ \tau_0 =1/20 $ $ \tau_0/2 $ $ \tau_0 /2^2 $ $ \tau_0 /2^3 $ $ \tau_0 /2^4 $ $ \tau_0 /2^5 $
Table 3. Temporal errors of the scheme at $ T = 1 $ for Example $ 4.4 $ with different $ \varepsilon $, $ h = 1/2^4 $
$ \varepsilon =\frac{1}{2^{11}} $ 1.92e-2 4.79e-3 1.20e-3 2.99e-04 7.47e-5 1.85e-5
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