Fourth order method for Maxwell equations on a staggered mesh

Abstract

To improve the accuracy of the scheme we replace (3) by an implicit scheme to produce a more w e consider fourth order accurate compact accurate scheme. We approximate the spatial schemes for numerical solutions to the Maxwell derivatives to fourth order, using the following im.equations. We use the same mesh stencil as Used plicit method for $f: in the standard Yee scheme. In particular extra ~( 2 , 4 ) : information over a wider stencil is not required. Hence, it is relatively easy to modify an existing code based on the Yee algorithm to make it (4) fourth order accurate. Also, a staggered mesh AY makes the boundary treatment easier. Finally, a staggered grid system gives a lower error than a non-staggered system. Finite Differences SchemeThe normalized TM wave equations are: aEz aHy a H , at ax ay aH, aE, at ay aff, -aEz at a x that will enable us to generalize it easily. -(1) -----We present Yee's difference equation in a form +i[(E) 24 i,j+3/2 +(") ay i,j-l/2 ] We refer to (3) as the Yee scheme and (4) as the Ty scheme. We shall use. second and fourth order accurate time discretizations for the Ty scheme. Equation (4) requires the inversion of a tridiagonal matrix for ( $ f ) , , J .We stress that the only difference between the Yee scheme and the (2,4) scheme is the replacement of (3) by (4). Thus, the scheme is fourth order in space but only second order in time. This is reasonable since we can improve the temporal accuracy by choosing a smaller time step. This increases the work only linearly and does not increase the storage. A fourth order scheme in space enables us to choose a coarser mesh. This decreases the work in each space dimension and also decreases the storage. Hence, the accuracy in space is of greater importance then the accuracy in time.where Yee:At the first and last points we use a fourth order accurate one sided appropximation to the derivative. We note that this is used only in order to globally approximate the derivative. No physical boundary conditions are included at this stage. w,, = Ut+1/2,1 -u,-1/2,3

Comparison of the dispersion properties of several low-dispersion finite-difference time-domain algorithms

“…In the M(2,4) scheme, the global dispersion error is minimized over all propagation angles. Recently, Turkel and Yefet presented an implicit second-order in time, fourth-order in space Ty(2,4) FDTD algorithm [9], [10]. Their algorithm uses implicit spatial derivatives, while maintaining the standard Yee leapfrog timestepping.…”

High-order accurate modeling of electromagnetic wave propagation across media – Grid conforming bodies

“…The compact implicit scheme is a good compromise between an unconditionally stable implicit scheme that requires the inversion of large matrices and an explicit scheme that has a stability condition imposed on the time-step. The scheme in one dimension is derived using a Taylor expansion (see also [26]):…”

“…A fourth order compact implicit scheme for the approximation of the spatial derivatives is then derived [26] as:…”

Numerical Solution of the Time-Domain Maxwell Equations Using High-Accuracy Finite-Difference Methods

“…More efficient methods, which are effective for propagation distances on the order of 10 or 20 wavelengths, include the finite-difference methods of Yee [28] and Shang [20] and the finite-volume method of Mohammadian, Shankar, and Hall [16]. However, for propagation distances greater than 20 wavelengths, these methods, which are second-order accurate, typically require excessive grid densities with correspondingly large computational requirements.The limitations of second-order methods in simulating wave phenomena have led to the development and application of higher-order and optimized finite-difference methods in several fields, including acoustics [13,24] and seismology [10], as well as electromagnetics [12,21,25,27,29,31,34]. Higher-order methods offer increased accuracy for a given node density at the expense of increased cost per node.…”

“…The limitations of second-order methods in simulating wave phenomena have led to the development and application of higher-order and optimized finite-difference methods in several fields, including acoustics [13,24] and seismology [10], as well as electromagnetics [12,21,25,27,29,31,34]. Higher-order methods offer increased accuracy for a given node density at the expense of increased cost per node.…”