Maxwell's equations can be solved numerically in space manifold and the time by discrete exterior calculus as a kind of lattice gauge theory.Since the stable conditions of this method is very severe restriction, we combine the implicit scheme of time variable and discrete exterior calculus to derive an unconditional stable scheme. It is an generation of implicit Yee-like scheme, since it can be implemented in space manifold directly. The analysis of its unconditional stability and error is also accomplished.
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…
Mean-square convergence of a semi-discrete scheme for stochastic nonlinear Maxwell equations
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）
We present two strategies for designing passivity preserving higher order discretization methods for Maxwell's equations in nonlinear Kerr-type media. Both approaches are based on variational approximation schemes in space and time. This allows to rigorously prove energy conservation or dissipation, and thus passivity, on the fully discrete level. For linear media, the proposed methods coincide with certain combinations of mixed finite element and implicit Runge-Kutta schemes. The order optimal convergence rates, which can thus be expected for linear problems, are also observed for nonlinear p…
We focus on non-stationary Maxwell equations defined on a regular patch of elements as considered in the isogeometric analysis (IGA). We apply the time-integration scheme following the ideas developed by the finite difference community [M. Hochbruck, T. Jahnke, R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations, Numerishe Mathematik, 2015] to derive a weak formulation resulting in discretization with Kronecker product matrices. We take the tensor product structure of the computational patch of elements from the IGA framework as an advantage, allowing for linear computationa…
The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equ…
Numerical schemes that preserve the structure of the kinetic equations can provide stable simulation results over a long time. An electromagnetic particle-in-cell solver for the Vlasov-Maxwell equations that preserves at the discrete level the non-canonical Hamiltonian structure of the Vlasov-Maxwell equations has been presented in [Kraus et al. 2017]. Whereas the original formulation has been obtained for Cartesian coordinates, we extend the formulation to curvilinear coordinates in this paper. For the discretisation in time, we discuss several (semi-)implicit methods either based on a Hamilt…
This paper proposes a charge-conserving, variational, spatio-temporal discretization for the drift-kinetic Vlasov-Maxwell system, utilizing finite-elements for the electromagnetic fields and the particle-in-cell approach for the Vlasov distribution. The proposed scheme is fully electromagnetic, dealing with fields instead of potentials, and includes the effects of polarization and magnetization in the Gauss and Ampère-Maxwell laws, a consequence of reducing the full particle dynamics to drift-center dynamics. There is, however, no need to invert the Gauss law: it is satisfied automatically at …
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）
Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an anti-symmetric structure of the discretized equations that is inherited from a wea…
The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a …
This paper discusses energy-conserving time-discretizations for finite element particle-in-cell discretizations of the Vlasov--Maxwell system. A geometric spatially discrete system can be obtained using a standard particle-in-cell discretization of the particle distribution and compatible finite element spaces for the fields to discretize the Poisson bracket of the Vlasov--Maxwell model (see Kraus et al., J Plasma Phys 83, 2017). In this paper, we derive energy-conserving time-discretizations based on the discrete gradient method applied to an antisymmetric splitting of the Poisson matrix. Fir…
Stability and convergence analysis for the domain decomposition finite element/finite difference (FE/FD) method is presented. The analysis is designed for semi-discrete finite element scheme for the time-dependent Maxwell's equations. The explicit finite element schemes in different settings of the spatial domain are constructed and domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies.
We present an asymptotic preserving method for the radiative transfer equations in the framework of PN method. An implicit and explicit method is proposed to solve the P N system based on the order analysis of the expansion coefficients of the distribution function. The order of each coefficient expanded in the Knudsen number is found through the process of Maxwellian iteration, and the coefficients of high order are treated explicitly while that of low order are treated implicitly in each equation of P N system. Energy inequality is proved for this numerical scheme. Several numerical examples…
Structure-preserving methods can be derived for the Vlasov-Maxwell system from a discretisation of the Poisson bracket with compatible finite-elements for the fields and a particle representation of the distribution function. These geometric electromagnetic particle-in-cell (GEMPIC) discretisations feature excellent conservation properties and long-time numerical stability. This paper extends the GEMPIC formulation in curvilinear coordinates to realistic boundary conditions. We build a de Rham sequence based on spline functions with clamped boundaries and apply perfect conductor boundary condi…
This paper is the first one of two serial articles, whose goal is to prove convergence of HX Preconditioner (proposed by Hiptmair and Xu 2007) for Maxwell's equations with jump coefficients. In this paper we establish various extensions of the regular Helmholtz decomposition for edge finite element functions defined in three dimensional domains. The functions defined by the regular Helmholtz decompositions can preserve the zero tangential complement on faces and edges of polyhedral domains and some non-Lipchitz domains, and possess stability estimates with only a $logarithm$ factor. These regu…
The time harmonic Maxwell equations are of current interest in computational science and applied mathematics with many applications in modern physics. In this work, we present parallel finite element solver for the time harmonic Maxwell equations and compare different preconditioners. We show numerically that standard preconditioners like incomplete LU and the additive Schwarz method lead to slow convergence for iterative solvers like generalized minimal residuals, especially for high wave numbers. Even more we show that also more specialized methods like the Schur complement method also yield…
A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，拉格朗日分析几何实验室）；University of Strathclyde（思克莱德大学）；University of Bath（巴斯大学）；UPMC Univ Paris 06, LJLL（巴黎第六大学皮埃尔和玛丽·居里大学，拉格朗日分析实验室）
Numerical discretization of the large-scale Maxwell's equations leads to an ill-conditioned linear system that is challenging to solve. The key requirement for successive solutions of this linear system is to choose an efficient solver. In this work we use Perfectly Matched Layers (PML) to increase this efficiency. PML have been widely used to truncate numerical simulations of wave equations due to improving the accuracy of the solution instead of using absorbing boundary conditions (ABCs). Here, we will develop an efficient solver by providing an alternative use of PML as transmission conditi…
In this work, we consider the time-harmonic Maxwell's equations and their numerical solution with a domain decomposition method. As an innovative feature, we propose a feedforward neural network-enhanced approximation of the interface conditions between the subdomains. The advantage is that the interface condition can be updated without recomputing the Maxwell system at each step. The main part consists of a detailed description of the construction of the neural network for domain decomposition and the training process. To substantiate this proof of concept, we investigate a few subdomains in …
Classically transmission conditions between subdomains are optimized for a simplified two subdomain decomposition to obtain optimized Schwarz methods for many subdomains. We investigate here if such a simplified optimization suffices for the magnetotelluric approximation of Maxwell's equation which leads to a complex diffusion problem. We start with a direct analysis for 2 and 3 subdomains, and present asymptotically optimized transmission conditions in each case. We then optimize transmission conditions numerically for 4, 5 and 6 subdomains and observe the same asymptotic behavior of optimize…
Convergence of parallel overlapping domain decomposition methods with impedance boundary conditions for time-harmonic Maxwell equations in heterogeneous media
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen Research Institute of Big Data（深圳工业与应用数学中心，深圳大数据研究院）；University of Bath（巴斯大学）；Shenzhen Loop Area Institute（深圳河套学院）
This work studies the parallelization and empirical convergence of two finite difference acoustic wave propagation methods on 2-D rectangular grids, that use the same alternating direction implicit (ADI) time integration. This ADI integration is based on a second-order implicit Crank-Nicolson temporal discretization that is factored out by a Peaceman-Rachford decomposition of the time and space equation terms. In space, these methods highly diverge and apply different fourth-order accurate differentiation techniques. The first method uses compact finite differences (CFD) on nodal meshes that r…
In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations. Convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement the near-kernel space made of the gradient of scalar functions. This extends the results in [2] to the variable coefficient case and non-convex domains at the expense of a larger coarse space.
A matrix basis formulation is introduced to represent the 3 x 3 dyadic Green's functions in the frequency domain for the Maxwell's equations and the elastic wave equation in layered media. The formulation can be used to decompose the Maxwell's Green's functions into independent TE and TM components, each satisfying a Helmholtz equation, and decompose the elastic wave Green's function into the S-wave and the P-wave components. In addition, a derived vector basis formulation is applied to the case for acoustic wave sources from a non-viscous fluid layer.
We discuss the applicability of a unified hyperbolic model for continuum fluid and solid mechanics to modeling non-Newtonian flows and in particular to modeling the stress-driven solid-fluid transformations in flows of viscoplastic fluids, also called yield-stress fluids. In contrast to the conventional approaches relying on the non-linear viscosity concept of the Navier-Stokes theory and representation of the solid state as an infinitely rigid non-deformable solid, the solid state in our theory is deformable and the fluid state is considered rather as a "melted" solid via a certain procedure …
In collisionless and weakly collisional plasmas, the particle distribution function is a rich tapestry of the underlying physics. However, actually leveraging the particle distribution function to understand the dynamics of a weakly collisional plasma is challenging. The equation system of relevance, the Vlasov-Maxwell-Fokker-Planck (VM-FP) system of equations, is difficult to numerically integrate, and traditional methods such as the particle-in-cell method introduce counting noise into the distribution function. In this thesis, we present a new algorithm for the discretization of VM-FP syste…
An implicit multiscale solver for steady kinetic equations based on wave-particle decomposition
Hong Kong University of Science and Technology（香港科技大学）；Shenzhen Research Institute, Hong Kong University of Science and Technology（香港科技大学深圳研究院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算力学研究所）
