تقرير
Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications
| العنوان: | Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications |
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| المؤلفون: | Duo, Siwei, Zhang, Yanzhi |
| المصدر: | Computer Methods in Applied Mechanics and Engineering, 355 (2019), pp. 639-662 |
| سنة النشر: | 2018 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Numerical Analysis |
| الوصف: | In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed finite difference methods provide a fractional analogue of the central difference schemes to the fractional Laplacian, and as $\alpha \to 2^-$, they collapse to the central difference schemes of the classical Laplace operator $-\Delta$. We prove that our methods are consistent if $u \in C^{\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, and the local truncation error is ${\mathcal O}(h^\epsilon)$, with $\epsilon > 0$ a small constant and $\lfloor \cdot \rfloor$ denoting the floor function. If $u \in C^{2+\lfloor\alpha\rfloor, \alpha-\lfloor\alpha\rfloor+\epsilon}({\mathbb R}^d)$, they can achieve the second order of accuracy for any $\alpha \in (0, 2)$. These results hold for any dimension $d \ge 1$ and thus improve the existing error estimates for the finite difference method of the one-dimensional fractional Laplacian. Extensive numerical experiments are provided and confirm our analytical results. We then apply our method to solve the fractional Poisson problems and the fractional Allen-Cahn equations. Numerical simulations suggest that to achieve the second order of accuracy, the solution of the fractional Poisson problem should {\it at most} satisfy $u \in C^{1,1}({\mathbb R}^d)$. One merit of our methods is that they yield a multilevel Toeplitz stiffness matrix, an appealing property for the development of fast algorithms via the fast Fourier transform (FFT). Our studies of the two- and three-dimensional fractional Allen-Cahn equations demonstrate the efficiency of our methods in solving the high-dimensional fractional problems. Comment: 24 pages, 6 figures, and 6 tables |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1016/j.cma.2019.06.016 |
| URL الوصول: | http://arxiv.org/abs/1804.02718 |
| رقم الأكسشن: | edsarx.1804.02718 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1016/j.cma.2019.06.016 |
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