Yong Zhang, Professor, Center for Applied Mathematics, Tianjin University

Hehu Xie, Professor

Fast convolution-type nonlocal potential solvers in Nonlinear Schrödinger equation and Lightning simulation

14:00-15:00 July 6 (Thursday), Z301

Convolution-type potential are common and important in many science and engineering fields. Efficient and accurate evaluation of such nonlocal potentials are essential in practical simulations. In this talk, I will focus on those arising from quantum physics/chemistry and lightning-shield protection, including Coulomb, dipolar and Yukawa potential that are generated by isotropic and anisotropic smooth and fast-decaying density, as well as convolutions defined on a one-dimensional adaptive finite difference grid. The convolution kernel is usually singular or discontinuous at the origin and/or at the far field, and density might be anisotropic, which together present great challenges for numerics in both accuracy and efficiency. The state-of-art fast algorithms include the kernel truncation method (KTM), NonUniform-FFT based method (NUFFT) and Gaussian-Sum based method (GauSum). Gaussian-sum/exponential-sum approximation and kernel truncation technique, combined with finite Fourier series and Taylor expansion, finally lead to a O(N log N) fast algorithm achieving spectral accuracy. Applications to NLSE, together with a useful recently-developed sum-of- exponential algorithm are reviewed. Tree-algorithm for computing the one-dimensional convolutions in lighting-shield simulation is also covered as the last application.

References:

1.W.Bao,S.Jiang,Q.TangandY.Zhang,Computingthegroundstateanddynamicsofthenonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys., 296 (2015), 72–89.

2.L. Exl, N. Mauser and Y. Zhang, Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation, J. Comput. Phys., 327 (2016), 629–642.

3.X. Antoine, Q. Tang and Y. Zhang, On the ground states and dynamics of space fractional non- linear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys., 325 (2016), pp. 74–97.

4.C. Zhuang, Y. Zhang, X. Zhou, R. Zeng and L. Liu, A fast tree algorithm for electrical field calculation in electrical discharge simulations, IEEE Transactions on Magnetics (2017).

5.L. Greengard, S. Jiang and Y. Zhang, A generic anisotropic kernel truncation method for convolution of free-space Green’s function, SIAM Journal on Scientific Computing, 2018.