Fourier Analysis of Numerical Algorithms for the Maxwell Equations

doi:10.1006/jcph.1996.0068

Journal of Computational Physics

Volume 124, Issue 2, 15 March 1996, Pages 396-416

https://doi.org/10.1006/jcph.1996.0068 Get rights and content

Abstract

The Fourier method is used to analyze the dispersive, dissipative, and isotropy errors of various spatial and time discretizations applied to the Maxwell equations on multi-dimensional grids. Both Cartesian grids and non-Cartesian grids based on hexagons and tetradecahedra are studied and compared. The numerical errors are quantitatively determined in terms of phase speed, wavenumber, propagation direction, gridspacings, and CFL number. The study shows that centered schemes are more efficient and accurate than upwind schemes and the non-Cartesian grids yield superior isotropy than the Cartesian ones. For the centered schemes, the staggered grids produce less errors than the unstaggered ones. A new unstaggered algorithm which has all the best properties is introduced. Using an optimization technique to determine the nodal weights, the new algorithm provides the highest accuracy among all the schemes discussed. The study also demonstrates that a proper choice of time discretization can reduce the overall numerical errors due to the spatial discretization.

References (0)

Cited by (102)

The Crank–Nicolson mixed finite element method for the improved system of time-domain Maxwell's equations
2022, Applied Mathematics and Computation
In this paper, we mainly establish the Crank–Nicolson (CN) mixed finite element (MFE) method for the system of time-domain Maxwell’s equations with a lossy medium. To this end, we first deduce the two-dimensional (2D) improved system of time-domain Maxwell’s equations with the lossy medium, construct the time semi-discretized CN format for the 2D improved system of Maxwell’s equations, and discuss the existence, stability, and error estimates of time semi-discrete solutions. And then, we construct a fully discretized CN MFE (FDCNMFE) format with both symmetry and positive definiteness, and analyze the existence, stability, and errors of FDCNMFE solutions. Furthermore, we make use of some numerical tests to verify the correctness of theoretical results in the 2D case. Finally, as a generalization, we also give the corresponding methods and results for three-dimensional (3D) improved system of time-domain Maxwell’s equations in the appendix A. This paper has advantages in at least the following four aspects. First, the study process herein is to discrete time and then discrete space variables so as to save the semi-discretized MFE discussion for spatial variables, which is a new attempt. Second, the magnetic field strength in the improve system of Maxwell’s equations is independent of the electric intensity such that their MFE subspaces can avoid the constraint of the Babuška-Brezzi (B-B) condition and can be freely chosen, so that the FDCNMFE solutions reach optimal order error estimates. Third, the 2D and 3D FDCNMFE formats reach the second-order accuracy in time and have both symmetry and positive definiteness so as to be solved easily. Fourth, both time semi-discretized CN solutions and the FDCNMFE solutions are unconditionally stable and unconditionally convergent.
Comparative study of grids based on the cubic crystal system for the FDTD solution of the wave equation
2019, Mathematics and Computers in Simulation
In this paper, a scalar wave is solved in the fully explicit finite difference time domain scheme for different stencils based on the cubic crystal system. In particular, we study four systems: the simple cubic, the body-centered cubic, the face-centered cubic and the compact packing cubic. In many papers that are focused on the artificial anisotropy induced by these grids in the propagated wave, one candidate is often better than all the others. In this manuscript, we study the stability, the physical phase velocity error and the anisotropy under two views. Firstly, we consider the same burdens or density of nodes per cubic wavelength and secondly we look at the asymptotic case. We also investigate the computational complexity based on several considerations: burdens, asymptotic time and implementation difficulties. Therefore, we pointed out how each problem or application, due to its different characteristics, has an appropriate grid in order to be treated properly.
Hybrid Newmark-conformal FDTD modeling of thin spoof plasmonic metamaterials
2019, Journal of Computational Physics
The aim of this paper is to present a new efficient and accurate semi-implicit finite-difference time-domain (FDTD) method for modeling thin plasmonic metamaterials supporting spoof surface plasmons. The presented method is formulated by combining Yu–Mittra's conformal FDTD technique for dielectrics, the simplified conformal finite integration technique for perfectly electric conductors (PECs) and the Newmark-Beta method. The resultant scheme is magnetically implicit, has a more relaxed stability condition than that of Yee's FDTD scheme, and allows for accurate modeling of both PEC and dielectric curved surfaces. Energy conservation and stability of the presented scheme is theoretically proved with the energy inequality method. Its formulation can be generalized to a class of magnetically implicit FDTD schemes with weakly conditional and unconditional stability in a unified manner. The presented scheme is used to simulate thin structures supporting spoof surface plasmon polaritons and spoof localized surface plasmons, and also applied to recent plasmonic devices such as planar frequency splitter, ultrathin rainbow trapping device and compact planar metadisk with split-ring resonators. The accuracy and efficiency of the presented scheme are assessed in comparison with the conventional explicit and implicit FDTD methods, and their conformal variants.
Discrete exterior calculus approach for discretizing Maxwell's equations on face-centered cubic grids for FDTD
2018, Journal of Computational Physics
Maxwell's equations are discretized on a Face-Centered Cubic (FCC) lattice instead of a simple cubic as an alternative to the standard Yee method for improvements in numerical dispersion characteristics and grid isotropy of the method. Explicit update equations and numerical dispersion expressions, and the stability criteria are derived. Also, several tools available to the standard Yee method such as PEC/PMC boundary conditions, absorbing boundary conditions, and scattered field formulation are extended to this method as well. A comparison between the FCC and the Yee formulations is made, showing that the FCC method exhibits better dispersion compared to its Yee counterpart. Simulations are provided to demonstrate both the accuracy and grid isotropy improvement of the method.
Seismic modeling with radial basis function-generated finite differences (RBF-FD) – a simplified treatment of interfaces
2017, Journal of Computational Physics
Citation Excerpt :
The rest of the nodes in the domain are placed via a simulated static repulsion routine, resulting in a hex-like, quasi-uniform arrangement as seen in Fig. 5. Based on the analysis in [13] and tests in [14], we argue that such a structure is at least as inherently resistant to dispersion error as a Cartesian grid of nodes with the same spatial resolution. In 1-D, we determined each support monomial for an FD stencil that crosses an interface by examining every row of (37) associated with time derivatives of constant expansion terms after k applications of the differential operator D. For (both) continuous data fields, all time derivatives of their constant expansion terms had to be equal across the interface for continuity of traction and motion to hold as time passes.
In a previous study of seismic modeling with radial basis function-generated finite differences (RBF-FD), we outlined a numerical method for solving 2-D wave equations in domains with material interfaces between different regions. The method was applicable on a mesh-free set of data nodes. It included all information about interfaces within the weights of the stencils (allowing the use of traditional time integrators), and was shown to solve problems of the 2-D elastic wave equation to 3rd-order accuracy. In the present paper, we discuss a refinement of that method that makes it simpler to implement. It can also improve accuracy for the case of smoothly-variable model parameter values near interfaces. We give several test cases that demonstrate the method solving 2-D elastic wave equation problems to 4th-order accuracy, even in the presence of smoothly-curved interfaces with jump discontinuities in the model parameters.
A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium
2016, Journal of Mathematical Analysis and Applications
This paper is concerned with establishing a reduced-order finite difference time-domain (FDTD) extrapolating scheme based on proper orthogonal decomposition (POD) method for the two-dimensional (2D) Maxwell equations in a lossy medium. For this purpose, we first review the classical FDTD scheme and associated results for the 2D Maxwell equations in a lossy medium. And then, we formulate three sets of POD bases by means of the solutions obtained from the classical FDTD scheme on the very short time interval, establish the POD reduced-order FDTD extrapolating scheme, and furnish the error estimates among the POD reduced-order FDTD solutions and the classical FDTD solutions as well as the accuracy solution for the 2D Maxwell equations in a lossy medium. Finally, we enumerate two numerical examples to confirm that the POD reduced-order FDTD extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium.

View all citing articles on Scopus

^☆: This paper was originally presented at the AIAA 31st Aerospace Sciences Meeting held in Reno, Nevada on Jan. 11–14, 1993.

View full text

Regular ArticleFourier Analysis of Numerical Algorithms for the Maxwell Equations☆

Abstract

Regular Article
Fourier Analysis of Numerical Algorithms for the Maxwell Equations☆