FDTD：introduction to the finite-difference time-domain method

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1、Chapter3IntroductiontotheFinite-DifferenceTime-DomainMethod:FDTDin1D3.1IntroductionThefinite-differencetime-domain(FDTD)methodisarguablythesimplest,bothconceptuallyandintermsofimplementation,ofthefull-wavetechniquesusedtosolveproblemsinelectromagnet-ics.Itcanaccuratelytackleawiderangeofp

2、roblems.However,aswithallnumericalmethods,itdoeshaveitsshareofartifactsandtheaccuracyiscontingentupontheimplementation.TheFDTDmethodcansolvecomplicatedproblems,butitisgenerallycomputationallyexpensive.Solutionsmayrequirealargeamountofmemoryandcomputationtime.TheFDTDmethodlooselyfitsintot

3、hecategoryofresonanceregiontechniques,i.e.,onesinwhichthecharacteristicdimensionsofthedomainofinterestaresomewhereontheorderofawavelengthinsize.Ifanobjectisverysmallcomparedtoawavelength,quasi-staticapproximationsgenerallyprovidemoreefficientso-lutions.Alternatively,ifthewavelengthisexce

4、edinglysmallcomparedtothephysicalfeaturesofinterest,ray-basedmethodsorothertechniquesmayprovideamuchmoreefficientwaytosolvetheproblem.TheFDTDmethodemploysfinitedifferencesasapproximationstoboththespatialandtem-poralderivativesthatappearinMaxwell’sequations(specificallyAmpere’sandFaraday’sl

5、aws).ConsidertheTaylorseriesexpansionsofthefunctionf(x)expandedaboutthepointx0withanoffsetof±δ/2:23δδ′1δ′′1δ′′′fx0+=f(x0)+f(x0)+f(x0)+f(x0)+...,(3.1)222!23!223δδ′1δ′′1δ′′′fx0−=f(x0)−f(x0)+f(x0)−f(x0)+...(3.2)222!23!2wheretheprimesindicatedifferentiation.Subtractingtheseconde

6、quationfromthefirstyields3δδ′2δ′′′fx0+−fx0−=δf(x0)+f(x0)+...(3.3)223!2LecturenotesbyJohnSchneider.fdtd-intro.tex3334CHAPTER3.INTRODUCTIONTOTHEFDTDMETHODDividingbyδproduces

fx+δ−fx−δ1δ20202=f′(x)+f′′′(x)+...(3.4)δ03!220Thusthetermontheleftisequaltothederivativeofthefunctionatthep

7、ointx0plusatermwhichdependsonδ2plusaninfinitenumberofothertermswhicharenotshown.Forthetermswhicharenotshown,thenextwoulddependonδ4andallsubsequenttermswoulddependonevenhigherpowersofδ.Rearrangingslightly,thisrelationshipisoftenstatedas

df(x)fx+δ−fx−δ02022=+O(δ).(3.5)dxδx=x0Th