Abstract : During the past several years, significant progress has been made in spectral methods and applications, especially in challenge problems such as numerical approximations for non-local and high-dimensional partial differential equations, where spectral and high-order methods are often preferred to low-order methods due to high-accuracy and lower memory request. This minisymposium aims to bring together active researchers in related areas to present and discuss their newest advances in both mathematical theory and numerical algorithms of efficient spectral numerical approximations for challenge scientific and engineering applications.
Organizer(s) : Hui-Yuan Li (Institute of Software Chinese Academy of Sciences), Li-Lian Wang (Nanyang Technological University), Haijun Yu
[04186] A positive and moment-preserving Fourier spectral method
Format : Talk at Waseda University
Author(s) :
Zhenning Cai (National University of Singapore)
Bo Lin (National University of Singapore)
Abstract : We present a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization problem, we propose an efficient Newton solver that has quadratic convergence rate. Applications to the Boltzmann equation are considered in our numerical tests.
[03237] Efficient structure-preserving spectral methods for plasma simulations
Format : Talk at Waseda University
Author(s) :
Zhiguo Yang (Shanghai Jiao Tong University)
Abstract : In this talk, we present H^1-, H(div) and H(curl)-conforming spectral method with exact preservation of the curl/divergence-free constraints for two typical PDEs arising from plasma simulations. One is the incompressible visco-resistive MHD system and the other one is the Vlasov-Ampere system. Two key ingredients, i.e. exact de Rham complexes and their commuting diagram, and the derivative property of the generalized Jacobi polynomials, are essential for the derivation of the desired basis functions. Besides, we propose a novel efficient solution algorithm based on simultaneous multiple-matrices diagonalisation technique. Ample 2D and 3D numerical examples illustrate both the accuracy and efficiency of the proposed methods.
[04790] A deep adaptive sampling method for the approximation of PDEs
Format : Talk at Waseda University
Author(s) :
Kejun Tang (Changsha Institute for Computing and Digital Economy, Peking University)
Jiayu Zhai (ShanghaiTech University)
Xiaoliang Wan (Louisiana State University)
Abstract : In this work, we develop an adaptive sampling strategy when approximating the PDE solution with a neural network. Two neural networks will be trained simultaneously through a min-max optimization problem, which is formulated by coupling PINN and the optimal transport. One neural network is used as a surrogate model of the true solution and the other neural network is used to optimize the collocation points in the training set. Numerical experiments will be presented.
[04098] A variable time-step scheme for Navier-Stokes equations
Format : Talk at Waseda University
Author(s) :
Yana DI (Beijing Normal University)
Abstract : In the talk, the implicit-explicit (IMEX) second-order backward difference (BDF2) scalar auxiliary variable (SAV) scheme for Navier-Stokes equation with periodic boundary conditions (${Huang\; and\; Shen, SIAM\; J. Numer. Anal., 2021}$) has been generalized to a variable time-step IMEX-BDF2 SAV scheme. We derive global and local optimal $H^1$ error estimates in 2D and 3D, respectively. An adaptive time-stepping strategy has also been designed and numerical examples will confirm the effectiveness and efficiency of our proposed methods.
[03896] Barycentric Interpolation Based on Equilibrium Potential
Format : Talk at Waseda University
Author(s) :
Shuhuang Xiang (Central South University )
Kelong Zhao (Central South University)
Abstract : A novel barycentric interpolation algorithm with specific exponential convergence rate is designed for analytic functions defined on the complex plane, with singularities located near the interpolation region, where the region is compact and can be disconnected or multiconnected. The core of the method is the efficient computation of the interpolation nodes and poles using discrete distributions that approximate the equilibrium logarithmic potential, achieved by solving a Symm's integral equation. It takes different strategies to distribute the poles for isolated singularities and branch points, respectively. In particular, if poles are not considered, it derives a polynomial interpolation with exponential convergence. Numerical experiments illustrate the superior performance of the proposed method.
[04757] Log orthogonal functions in semi-infinite intervals: approximation results and applications
Format : Talk at Waseda University
Author(s) :
Sheng Chen (Beijing Normal University at Zhuhai)
Abstract : We construct two new classes of log orthogonal functions in semi-infinite intervals, log orthogonal functions (LOFs-II) and generalized log orthogonal functions (GLOFs-II), by applying a suitable log mapping to Laguerre polynomials. We develop a basic approximation theory for these new orthogonal functions and show that they can provide uniformly good exponential convergence rates for problems in semi-infinite intervals with slow decay at infinity. We apply them to solve several linear and nonlinear differential equations whose solutions decay algebraically or exponentially with very slow rates and present ample numerical results to show the effectiveness of the approximations by LOFs-II and GLOFs-II.
[05466] A class of efficient spectral methods and error analysis for nonlinear Hamiltonian systems
Format : Online Talk on Zoom
Author(s) :
Waixiang Cao (Beijing Normal University )
Jing An (Guizhou Normal university)
Zhimin Zhimin Zhang (Wayne state University )
Abstract : In this talk, we investigate efficient numerical methods for nonlinear Hamiltonian systems. Three polynomial spectral methods (including spectral Galerkin, Petrov-Galerkin, and collocation methods) coupled with domain decomposition are presented and analyzed. Our main results include the energy and symplectic structure-preserving properties and error estimates. We prove that the spectral Petrov-Galerkin method preserves the energy exactly while both the spectral Gauss collocation and spectral Galerkin methods are energy conserving up to spectral accuracy. While it is well known that collocation at Gauss points preserves symplectic structure, we prove that the Petrov-Galerkin method preserves the symplectic structure up to a Gauss numerical quadrature error and the spectral Galerkin method preserves the symplectic structure up to spectral accuracy error. Finally, we show that all three methods converge exponentially, which makes it possible to simulate the long time behavior of the system. Numerical experiments indicate that our algorithms are efficient.
