Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation. (English) Zbl 1422.65450
Summary: We introduce an accurate and efficient method for the numerical evaluation of nonlocal potentials, including the 3D/2D Coulomb, 2D Poisson and 3D dipole-dipole potentials. Our method is based on a Gaussian-sum approximation of the singular convolution kernel combined with a Taylor expansion of the density. Starting from the convolution formulation of the nonlocal potential, for smooth and fast decaying densities, we make a full use of the Fourier pseudospectral (plane wave) approximation of the density and a separable Gaussian-sum approximation of the kernel in an interval where the singularity (the origin) is excluded. The potential is separated into a regular integral and a near-field singular correction integral. The first is computed with the Fourier pseudospectral method, while the latter is well resolved utilizing a low-order Taylor expansion of the density. Both parts are accelerated by fast Fourier transforms (FFT). The method is accurate (14–16 digits), efficient (\(O(N \log N)\) complexity), low in storage, easily adaptable to other different kernels, applicable for anisotropic densities and highly parallelizable.
MSC:
| 65R10 | Numerical methods for integral transforms |
Keywords:
nonlocal potential solver; convolution integral; separable Gaussian-sum approximation; Coulomb/Poisson/dipole-dipole potential; singular correction integralReferences:
| [1] | Abbamowitz, M.; Stegun, I., Handbook of Mathematical Functions (1965), Dover |
| [2] | Antoine, X.; Tang, Q.; Zhang, Y., On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys., 325, 74-97 (2016) · Zbl 1380.65296 |
| [3] | Bao, W.; Cai, Y., Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6, 1-135 (2013) · Zbl 1266.82009 |
| [4] | Bao, W.; Cai, Y.; Wang, H., Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229, 7874-7892 (2010) · Zbl 1198.82036 |
| [5] | Bao, W.; Jian, H.; Mauser, N.; Zhang, Y., Dimension reduction of the Schrödinger equation with Coulomb and anisotropic confining potentials, SIAM J. Appl. Math., 73, 6, 2100-2123 (2013) · Zbl 1294.35130 |
| [6] | Bao, W.; Jiang, S.; Tang, Q.; Zhang, Y., Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys., 296, 72-89 (2015) · Zbl 1354.65200 |
| [7] | Bao, W.; Tang, Q.; Zhang, Y., Accurate and efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates via the nonuniform FFT, Commun. Comput. Phys., 19, 5, 1141-1166 (2016) · Zbl 1388.65115 |
| [8] | Beylkin, G.; Kurzc, C.; Monzón, L., Fast convolution with the free space Helmholtz Green’s function, J. Comput. Phys., 228, 2770-2791 (2009) · Zbl 1165.65082 |
| [9] | Beylkin, G.; Monzón, L., Approximation by exponential sums, Appl. Comput. Harmon. Anal., 19, 17-48 (2005) · Zbl 1075.65022 |
| [10] | Beylkin, G.; Monzón, L., Approximation by exponential sums revisited, Appl. Comput. Harmon. Anal., 28, 131-149 (2010) · Zbl 1189.65035 |
| [11] | Braess, D.; Hackbusch, W., On the Efficient Computation of High-Dimensional Integrals and the Approximation by Exponential Sums, Multiscale, Nonlinear and Adaptive Approximations (2009), Springer · Zbl 1190.65036 |
| [12] | Cerioni, A.; Genovese, L.; Mirone, A.; Sole, V., Efficient and accurate solver of the three-dimensional screened and unscreened Poisson’s equation with generic boundary conditions, J. Chem. Phys., 137, Article 134108 pp. (2012) |
| [13] | Dutt, A.; Rokhlin, V., Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14, 1368-1393 (1993) · Zbl 0791.65108 |
| [14] | Ethridge, F.; Greengard, L., A new fast-multipole accelerated Poisson solver in two dimensions, SIAM J. Sci. Comput., 23, 3, 741-760 (2001) · Zbl 1002.65131 |
| [15] | Exl, L.; Abert, C.; Mauser, N.; Schrefl, T.; Stimming, H.; Suess, D., FFT-based Kronecker product approximation to micromagnetic long-range interactions, Math. Models Methods Appl. Sci., 24, 1877-1901 (2014) · Zbl 1291.65356 |
| [16] | Exl, L.; Auzinger, W.; Bance, S.; Gusenbauer, M.; Reichel, F.; Schrefl, T., Fast stray field computation on tensor grids, J. Comput. Phys., 231, 2840-2850 (2012) · Zbl 1242.78034 |
| [17] | Exl, L.; Schrefl, T., Non-uniform FFT for the finite element computation of the micromagnetic scalar potential, J. Comput. Phys., 270, 490-505 (2014) · Zbl 1349.78067 |
| [18] | Füsti-Molnar, L.; Pulay, P., Accurate molecular integrals and energies using combined plane wave and Gaussian basis sets in molecular electronic structure theory, J. Chem. Phys., 116, 7795 (2002) |
| [19] | Genovese, L.; Deutsch, T.; Neelov, A.; Goedecker, S.; Beylkin, G., Efficient solution of Poisson equation with free boundary conditions, J. Chem. Phys., 125, Article 074105 pp. (2006) |
| [20] | Genovese, L.; Deutsch, T.; Goedecker, S., Efficient and accurate three-dimensional Poisson solver for surface problems, J. Chem. Phys., 127, Article 054704 pp. (2007) |
| [21] | Greengard, L.; Rokhlin, V., A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numer., 6, 229-269 (1997) · Zbl 0889.65115 |
| [22] | Hackbusch, W.; Khoromskij, B., Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. Part I. Separable approximation of multi-variate functions, Computing, 76, 3, 177-202 (2006) · Zbl 1087.65049 |
| [23] | Jiang, S.; Greengard, L.; Bao, W., Fast and accurate evaluation of nonlocal Coulomb and dipole-dipole interactions via the nonuniform FFT, SIAM J. Sci. Comput., 36, B777-B794 (2014) · Zbl 1307.65184 |
| [24] | Martyna, G.; Tuckerman, M., A reciprocal space based method for treating long range interactions in ab initio and force-field-based calculations in clusters, J. Chem. Phys., 110, 2810-2821 (1999) |
| [25] | Mauser, N.; Zhang, Y., Exact artificial boundary condition for the Poisson equation in the simulation of the 2D Schrödinger-Poisson system, Commun. Comput. Phys., 16, 3, 764-780 (2014) · Zbl 1388.65063 |
| [26] | Piessens, R.; Doncker-Kapenga, E.; Überhuber, C., QUADPACK: A Subroutine Package for Automatic Integration (1987), Springer · Zbl 0508.65005 |
| [27] | Shen, J.; Tang, T., Spectral and High-Order Methods with Applications (2006), Science Press: Science Press Beijing · Zbl 1234.65005 |
| [28] | Stenger, F., Numerical Methods Based on Sinc and Analytic Functions (1993), Springer-Verlag · Zbl 0803.65141 |
| [29] | Mauser, N.; Stimming, H.; Zhang, Y., A novel nonlocal potential solver based on nonuniform FFT for efficient simulation of the Davey-Stewartson equations · Zbl 1375.35387 |
| [30] | Trefethen, L., Spectral Methods in Matlab (2000), Oxford University: Oxford University Oxford · Zbl 0953.68643 |
| [31] | Yi, S.; You, L., Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61, Article 041604(R) pp. (2000) |
| [32] | Yi, S.; You, L., Trapped condensates of atoms with dipole interactions, Phys. Rev. A, 63, Article 053607 pp. (2001) |
| [33] | Zhang, Y.; Dong, X., On the computation of ground states and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230, 2660-2676 (2011) · Zbl 1218.65115 |
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