Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell's equations. This is accomplished by combining time exponential integrator and Fourier pseudospectral methods. Fast computation is implemented in the scheme by using the Fast Fourier Transform algorithm which is well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and infinite-order accuracy in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.
We propose an exponential integrator Fourier pseudospectral method DEI-FP for solving the "Good" Boussinesq (GB) equation. The numerical scheme is based on a Deuflhard-type exponential integrator and a Fourier pseudospectral method for temporal and spatial discretizations, respectively. The scheme is fully explicit and efficient due to the fast Fourier transform. Rigorous error estimates are established for the method without any CFL-type condition constraint. In more details, the method converges quadratically and spectrally in time and space, respectively. Extensive numerical experiments are…
Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
Beijing Computational Science Research Center（北京计算科学研究中心）；IRMAR, Université de Rennes 1（雷恩第一大学 IRMAR）
Uniform and optimal error estimates of an exponential wave integrator sine pseudospectral method for the nonlinear Schrodinger equation with wave operator
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；University of Maryland（马里兰大学）
Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods are presented and analyzed for the long-time dynamics of the Dirac equation with small potentials characterized by $\varepsilon \in (0, 1]$ a dimensionless parameter. Based on the (symmetric) exponential wave integrator for temporal derivatives in phase space followed by applying the Fourier pseudospectral discretization for spatial derivatives, the EWI-FP methods are explicit and of spectral accuracy in space and second-order accuracy in time for any fixed $\varepsilon = \varepsilon_0$. Uniform error bounds are rigorously…
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Purdue University（普渡大学）；Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（南锡大东部国家研究学院）
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \varepsilon \ll 1$ a dimensionless parameter up to the time at $O(1/\varepsilon^2)$. The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear pa…
In this paper we propose and investigate a general approach to constructing local energy-preserving algorithms which can be of arbitrarily high order in time for solving Hamiltonian PDEs. This approach is based on the temporal discretization using continuous Runge-Kutta-type methods, and the spatial discretization using pseudospectral methods or Gauss--Legendre collocation methods. The local energy conservation law of our new schemes is analyzed in detail. The effectiveness of the novel local energy-preserving integrators is demonstrated by coupled nonlinear Schrödinger equations and 2D nonlin…
The Vlasov--Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov--Maxwell system and propose a dynamic low-rank integrator. The key idea is to approximate the dynamics of the system by constraining it to a low-rank manifold. This is accomplished by a projection onto the tangent space. There, the dynamics…
Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation are proposed and analyzed. To obtain a high spatial accuracy, new regularized Bermúdez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme …
A Uniformly Accurate Multiscale Time Integrator for the Klein-Gordon-Schrödinger Equations in the Nonrelativistic Regime via Simplified Transmission Conditions
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Department of Mathematics, National University of Singapore（新加坡国立大学数学系）
A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions
A uniformly accurate (UA) multiscale time integrator Fourier pseoduspectral method for the Klein-Gordon-Schrodinger equations in the nonrelativistic limit regime
We establish uniform error bounds of an exponential wave integrator Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the nonlinear Klein-Gordon equation (NKGE) with a cubic nonlinearity whose strength is characterized by $\varepsilon^2$ with $\varepsilon \in (0, 1]$ a dimensionless parameter. When $0 < \varepsilon \ll 1$, the problem is equivalent to the long-time dynamics of the NKGE with small initial data (and $O(1)$ cubic nonlinearity), while the amplitude of the initial data (and the solution) is at $O(\varepsilon)$. For the long-time dynamics of the NKGE up to the tim…
We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws during the time evolution. Hence, we analyze the capability…
Numerical analysis of the second-order time-dependent saddle point Maxwell system via a parameter-free discontinuous Galerkin method: The first optimal ${\bf L}^{2}$-norm error estimates
Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）
Metriplectic particle-in-cell integrators for the Landau collision operator
Princeton Plasma Physics Laboratory（普林斯顿等离子体物理实验室）；Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Technische Universität München, Zentrum für Mathematik（慕尼黑工业大学数学中心）；Courant Institute of Mathematical Sciences（纽约大学柯朗数学科学研究所）
Dispersion Analysis of Finite Difference and Discontinuous Galerkin Schemes for Maxwell's Equations in Linear Lorentz Media
University of Science and Technology of China（中国科学技术大学）；King Mongkut’s Institute of Technology Ladkrabang（拉加邦国王科技大学）；Oregon State University（俄勒冈州立大学）；Michigan State University（密歇根州立大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
Max-Planck-Institut für Plasmaphysik（马克斯·普朗克等离子体物理研究所）；Technische Universität München, Zentrum Mathematik（慕尼黑工业大学数学中心）；Department of Physics and Institute for Fusion StudiesThe University of Texas at Austin（德克萨斯大学奥斯汀分校物理系与聚变研究学院）
