Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation
Abstract
:1. Introduction
2. The GSAV Approach for the NLS Equation
3. Construction of the Energy-Preserving Schemes
3.1. Fourier Pseudo-Spectral Approximation of Spatial Derivatives
3.2. Fully Discrete Energy-Preserving Schemes
- The tanh SAV scheme: In this scheme, we select , where is a positive constant to make not too close to numerically since . Thus, we set
- The exponential SAV scheme: The exponential function is a special function that can keep the range constant positive. Thus, we define an exponential scalar auxiliary variable
3.3. Fast Solvers for the Proposed Schemes
4. Numerical Experiments
- LI-EI-i (i = 3 or 4): The paper constructs third and fourth order energy-preserving schemes by using the Runge-Kutta methods shown in Table 1.
- LI-4: A fourth order linearly implicit conservative RK method is based on the GSAV approach [40].
- LI-EI-2: A second order linearly-implicit exponential time differencing conservative scheme is developed in Ref. [33].
- FI-EI-4: A fourth order fully-implicit conservative exponential time differencing method is presented in Ref. [32].
4.1. Two Dimension Case
4.2. Three Dimension Case
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Scheme | |||||
---|---|---|---|---|---|
LI-EI-2 [33] | 5.7897 × 10 | 1.4276 × 10 | 3.5439 × 10 | 8.8282 × 10 | |
Rate | * | 2.0198 | 2.0101 | 2.0051 | |
LI-EI-3 | 1.1578 × 10 | 1.3051 × 10 | 1.6205 × 10 | 2.04488 × 10 | |
Rate | * | 3.1492 | 3.0096 | 2.9863 | |
LI-EI-4 | 9.0072 × 10 | 7.1453 × 10 | 4.9858 × 10 | 3.2867 × 10 | |
Rate | * | 3.6560 | 3.8411 | 3.9231 | |
LI-4 [40] | NaN | 3.2550 × 10 | 2.0345 × 10 | 1.2529 × 10 | |
Rate | * | * | 3.9999 | 4.0213 | |
FI-EI-4 [32] | 8.2450 × 10 | 5.1562 × 10 | 3.2231 × 10 | 2.0145 × 10 | |
Rate | * | 3.9991 | 3.9997 | 3.9999 |
Scheme | ||||||
---|---|---|---|---|---|---|
LI-EI-3 | 5.2663 × 10 | 6.5894 × 10 | 8.2372 × 10 | 1.0295 × 10 | ||
Rate | * | 2.9985 | 2.9999 | 3.0001 | ||
LI-EI-4 | 1.0439 × 10 | 7.1192 × 10 | 4.6417 × 10 | 2.9909 × 10 | ||
Rate | * | 3.8742 | 3.9389 | 3.9559 | ||
LI-EI-3 | 3.26618 × 10 | 4.0989 × 10 | 5.1349 × 10 | 6.4260 × 10 | ||
Rate | * | 2.9942 | 2.9968 | 2.9983 | ||
LI-EI-4 | 6.5017 × 10 | 4.6648 × 10 | 3.1119 × 10 | 2.0090 × 10 | ||
Rate | * | 3.8009 | 3.9059 | 3.9532 | ||
2 | LI-EI-3 | 5.2780 × 10−3 | 6.5852 × 10 | 8.2299 × 10 | 1.0289 × 10 | |
Rate | * | 3.0027 | 3.0002 | 2.9997 | ||
LI-EI-4 | 1.1272 × 10 | 6.6838 × 10 | 4.0734 × 10 | 2.5485 × 10 | ||
Rate | * | 4.0759 | 4.0363 | 3.9985 |
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Fu, Y.; Zheng, Q.; Zhao, Y.; Xu, Z. Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation. Fractal Fract. 2022, 6, 243. https://doi.org/10.3390/fractalfract6050243
Fu Y, Zheng Q, Zhao Y, Xu Z. Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation. Fractal and Fractional. 2022; 6(5):243. https://doi.org/10.3390/fractalfract6050243
Chicago/Turabian StyleFu, Yayun, Qianqian Zheng, Yanmin Zhao, and Zhuangzhi Xu. 2022. "Linearly Implicit High-Order Exponential Integrators Conservative Runge–Kutta Schemes for the Fractional Schrödinger Equation" Fractal and Fractional 6, no. 5: 243. https://doi.org/10.3390/fractalfract6050243