Simpler and More Intuitive Interpretation of Stability Conditionfor Finite-Difference Time-Domain MethodHiroyuki ICHIKAWA

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Faculty of Engineering, Ehime University, 3 Bunkyo, Matsuyama 790-8577, Japan

(Received April 28, 2010; Accepted May 21, 2010)

In using a popular numerical electromagnetic solver, finite-different time-domain (FDTD) method, temporal and spatialdiscretization must obey a so-called stability condition. This short paper provides straightforward and easy-to-understand interpretation for the condition than a frequently referred traditional one.# 2010 The Japan Society of Applied Physics

Keywords: FDTD method, stability condition

1. Introduction

Finite-difference time-domain (FDTD) method is a nu-merical solution of Maxwell’s equations and originated inthe field of antenna and microwave technology.1) Nowadaysit would not be exaggeration to say that the FDTD methodis the most widely used numerical electromagnetic fieldanalysis tool even in optics and photonics.

Implementation of the FDTD method begins with deter-mination of the sizes of spatial and temporal discretization,i.e., cell sizes or distances between the closest spatial points�x, �y, �z, and a length of discrete time step �t. However,those values cannot be arbitrarily chosen and must follow awell known stability condition,

�t �1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2 þ ð�zÞ�2

p ; ð1Þ

where v is the fastest phase velocity of the light wave withinthe analyzed region. Equation (1) is also referred toCourant–Friedrichs–Lewy or concisely Courant condition,and its derivation is given, e.g., in the Appendix of ref. 2,in which Maxwell’s equations are treated as eigenvalueproblems.

In this short paper, however, I provide more straight-forward interpretation which is easy to understand howthe stability condition is like by simply referring theoriginal Courant condition in one-dimension as a precon-dition.

2. Concise Summary of a Traditional Approach

Let us summarize a traditional way of deriving eq. (1)based on ref. 2. A set of Maxwell’s curl equations in ahomogeneous medium can be written as

ir � V ¼1

v

@V

@t; ð2Þ

where

V ¼Hffiffiffi�

p þ iEffiffiffiffi�

p ; ð3Þ

and v is a phase velocity in the medium considered. Stabilityof numerical solution of eq. (2) is evaluated by a set of thefollowing eigenvalue equations.

@V

@t¼ v�V; ð4Þ

ir � V ¼ �V; ð5Þ

where � is an eigenvalue. Stable numerical solution foreqs. (4) and (5) is realized, when

j Imf�gj �2

v�t; ð6Þ

j Imf�gj �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2 þ ð�zÞ�2

p ; ð7Þ

respectively. In both cases, Ref�g ¼ 0. As the equality ofeq. (7) is for the most extreme spatial mode, the right handside (RHS) of eq. (6) must be equal to or greater than theRHS of eq. (7). Thus, eq. (1) is obtained. For detailexplanation, refer to ref. 2.

3. In One-Dimension

Courant et al.3) derived a condition under which solutionsof difference equations converged to solutions of partialdifferential equations when discretization size approachednaught. It is a relation between spatial and temporaldiscretization sizes in solving difference equations. Forhyperbolic partial differential equations into which waveequations are categorized, the essence is shown in Fig. 1where discretized spatial and temporal points are plotted.The value at a point S is influenced by the values at pointswithin the shaded area, named domain of determination fordifference equation. The solid and dotted lines are calledlines of determination for difference and differential equa-tions, respectively. As domain of determination for differ-ential equation is within the domain of determination fordifference equation, convergence is obtained. For detailmathematical proof, refer to ref. 3.

Then, the argument above can be summarized as follows:a unit discretized time step must be shorter than the time fortraveling a unit discretized spatial distance. In other words,the longest discretized time step allowed is the distancebetween the closest spatial points divided by the phase�E-mail address: hichikaw@dpc.ehime-u.ac.jp

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velocity in that particular place. This is commonly referredto Courant condition.

A stability condition for one-dimensional FDTD method,

�t ��x

v; ð8Þ

where there is only one spatial component, is an obviousmanifestation of the Counrant condition.

4. Extension to Two-Dimension

For two-dimensional FDTD method, it becomes a bitcomplicated. What is critical is definition of a distancebetween the closest spatial points, i.e., the shortest distancebetween spatial points. Let us look at Fig. 2, which shows3�5 spatial points. Here, assuming the dotted lines aswavefronts of a plane wave at various instances, the shortestdistance � between the closest spatial points is defined ashAB, i.e., a distance between two lines A and B connectingdiagonal points. The value is measured

� ¼�x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ2 þ ð�yÞ2

p �y ¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2

p : ð9Þ

Thus, the time step must be shorter than the traveling timebetween the distance�, yielding a stability condition in two-dimension,

�t �1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2

p : ð10Þ

It would be necessary to show why the hAB defined inFig. 2 is the shortest distance. Some readers may argue thatthe distance hCD for example is shorter than hAB. The dottedlines C and D indeed lie on vertically neighbouring points.In the horizontal direction, however, they are not neighbour-ing and thus actual distance between neighbouring points inthe particular direction of propagation should be hCE. Tounderstand this logic, it would suffice to imagine that whatwould happen when the angle � becomes smaller andeventually approaches to naught.

5. Extension to Three-Dimension

Now, extending the argument above, the main interest inthree-dimension is a distance between neighbouring planes,which are defined as a set of planes ABC and DEF in Fig. 3.Black solid circles denote spatial points and a rectangularparallelepiped consisted of dotted lines is a unit spatialstructure. As is easily found, a triangle DEF is in a planeA0B0C0. An equation of a plane ABC is,

x

�xþ

y

�yþ

z

�z¼ 1; ð11Þ

while that of a plane DEF is,

x

2�xþ

y

2�yþ

z

2�z¼ 1: ð12Þ

As planes ABC and DEF are both defined by diagonallines in xy, yz, and zx planes in a unit structure, they are theneighbouring parallel planes in three dimension. Thus, thedistance between them is simply PQ, where P and Q are theends of perpendicular lines drawn from the origin to theplanes ABC and DEF. Acknowledging that

OP ¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2 þ ð�zÞ�2

p ; ð13Þ

Fig. 2. Definition of the shortest distance between spatial points.

Fig. 3. Definition of a distance between two neigboring planes.OP and OQ are perpendicular lines drawn from the origin to theplanes ABC and DEF, respectively. OA ¼ �x, OB ¼ �y,OC ¼ �z. OA0 ¼ 2�x, OB0 ¼ 2�y, OC0 ¼ 2�z.

Fig. 1. In difference equations, the value at S is influenced by thevalues at points within the shaded area. The solid lines are lines ofdetermination for difference equations and the dotted lines are thosefor differential equations.

436 OPTICAL REVIEW Vol. 17, No. 4 (2010) H. ICHIKAWA

OQ ¼2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2 þ ð�zÞ�2

p ; ð14Þ

the distance PQ is,

PQ ¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�xÞ�2 þ ð�yÞ�2 þ ð�zÞ�2

p : ð15Þ

Therefore, Courant condition in three-dimension,

�t � PQ=v; ð16Þ

finally provides the form of frequently referred stabilitycondition in three-dimension as represented by eq. (1).

6. Conclusion

Stability condition for the FDTD method is derivedaccepting Courant condition in one-dimension as a precon-dition. The interpretation provided here is much simpler andmore intuitive than frequently referred approach such as inref. 2. In addition, applying a concept of reciprocal lattice insolid state physics, the shortest distance defined above alsocorresponds to the highest spatial frequency represented asthe distance between diagonal ponits in the reciprocal latticespace. For this extra explanation, it would suffice to statethat G in Fig. 4 corresponds to � in eq. (9).

It is not always easy to correctly understand a condition forstable computation in the FDTD method. One such example

is that the stability condition introduced in Yee’s originalpaper1) was incorrect as indicated in ref. 4. This short paperis intended in particular for beginners of the FDTD methodto understand what the stability condition is like.

References

1) K. S. Yee: IEEE Trans. Antennas Propag. 14 (1966) 302.2) A. Taflove and M. E. Brodwin: IEEE Trans. Microwave Theory

Tech. 23 (1975) 623.3) R. Courant, K. Friedrichs, and H. Lewy: Math. Ann. 100 (1928)

32 [IBM J. Res. Dev. 11 (1967) 215].4) A. Taflove and K. R. Umashankar: Proc. IEEE 77 (1989) 682.

Fig. 4. Reciprocal lattice of spatial points in two-dimensionshown in Fig. 2. The distance G represents the highest spatialfrequency and thus the shortest distance in a real space.

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