INTRNATIONAL JOURNAL OF NUMRICAL ANALYSIS AND MODLING Volume XX Number 0 ages 45 c 03 Institute for Scientific Computing and Information CONVRGNC ANALYSIS OF Y SCHMS FOR MAXWLL S QUATIONS IN DBY AND LORNTZ DISRSIV MDIA V. A. BOKIL AND N. L. GIBSON Abstract. We present discrete energy decay results for te Yee sceme applied to Maxwell s equations in Debye and Lorentz dispersive media. Tese estimates provide stability conditions for te Yee sceme in te corresponding media. In particular we sow tat te stability conditions are te same as tose for te Yee sceme in a nondispersive dielectric. However energy decay for te Maxwell-Debye and Maxwell-Lorentz models indicate tat te Yee scemes are dissipative. Te energy decay results are ten used to prove te convergence of te Yee scemes for te dispersive models. We also sow tat te Yee scemes preserve te Gauss divergence laws on its discrete mes. Numerical simulations are provided to justify te teoretical results. Key words. Maxwell s equations Debye Lorentz dispersive materials Yee FDTD metod energy decay convergence analysis.. Introduction Te Yee sceme is a finite difference time domain (FDTD numerical tecnique for te discretization of Maxwell s equations in a non-dispersive medium suc as free space. It was first presented in [35]. Te Yee sceme was extended to discretize Maxwell s equations in linear dispersive media and analyzed in a series of papers [4 9 3 8 9 0 3] involving dispersive media models suc as te Debye [4 0] Lorentz [8 30] cold plasma [3 36] and cole-cole [9 ] models among oters. Fourier analysis of te Yee sceme in suc dispersive media (see for e.g. [4 3] indicate tat te Yee sceme is stable under te same stability condition as tat in a corresponding (aving te same relative permittivity non-dispersive dielectric. However te Yee sceme in dispersive media is dissipative unlike its counterpart in a non-dispersive non-conductive medium and in addition is more dispersive [5 3]. Te time step in te Yee sceme needs to be cosen to resolve all te time scales associated wit a particular dispersive medium suc as relaxation times resonance times and incident wave periods [3]. Maxwell s equations in suc media ave been sown to constitute a stiff problem and te time step needed to resolve waves in te numerical grid can be extremely small [3]. Researc on te construction and analysis of Yee type finite difference time domain metods for Maxwell s equations in dispersive media is an area of active interest. We refer te reader to te book [33] and te numerous references terein for an introduction to te Yee sceme and its properties. Received by te editors May 7 03. 000 Matematics Subject Classification. 65M06 65M 65Z05.
V. A. BOKIL AND N. L. GIBSON In tis paper we present for te first time an analysis of te Yee sceme in Debye (Maxwell-Debye and Lorentz (Maxwell-Lorentz media by deriving energy decay results tat indicate te conditional stability and dissipative nature of te scemes. We also present a full convergence analysis of te Yee scemes for te Maxwell-Debye and Maxwell-Lorentz models using te derived energy decay results. nergy metods based on variational tecniques for analyzing stability and convergence properties of te Yee sceme in a lossy non-dispersive medium and operator splitting FDTD tecniques ave recently been publised in te literature see for example [7 0 5]. Finite element metods (FM and discontinuous Galerkin (DG metods for Maxwell s equations in various dispersive media ave also recently been publised for example see [6 3 4 5 6 34] and references terein. We construct exact solutions based on numerical dispersion relations for te Maxwell-Debye and Maxwell-Lorentz models wic are useful in understanding te decay of discrete energies in numerical metods for tese models. We use tese exact solutions to justify our stability and convergence analyses in our numerical simulations of te Yee scemes. Te outline of te paper is as follows. In Section we present two dispersive media models and construct te Maxwell-Debye model and Maxwell-Lorentz model in two dimensions. We recall energy decay results for tese models from te literature [3]. In Section 3 we outline te discrete meses and spaces tat te electric magnetic and polarization fields are discretized on and establisg discrete curl operators and teir properties. In Sections 4 and 5 we recall te Yee scemes for te Maxwell-Debye and te Maxwell-Lorentz models respectively. For bot models we sow tat te corresponding Yee scemes are second-order accurate in time establis discrete energy decay results and prove te conditional convergence of te corresponding Yee scemes. In addition we sow tat tese scemes satisfy te Gauss divergence laws on te discrete Yee mes. Numerical simulations based on exact solutions are presented in Sections 6 and 7 tat justify te stability and convergence analyses. Finally conclusions are made in Section 8.. Maxwell s quations in Dispersive Dielectrics We consider Maxwell s equations wic govern te electric field and te magnetic field H in a domain Ω R 3 from time 0 to T given as (.a D t µ 0 B = 0 in Ω (0 T (.b (.c (.d (.e B t + = 0 in Ω (0 T. D = 0 = B in Ω (0 T n = 0 on Ω (0 T (0 x = 0 ; H(0 x = H 0 in Ω. Te fields D B are te electric and magnetic flux densities respectively. On te boundary Ω we impose a perfect conducting (C boundary condition (.d were te vector n is te outward unit normal vector to Ω. Lastly we add initial conditions (.e to te system.
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 3 Witin te dielectric medium we ave constitutive relations tat relate te flux densities D B to te electric and magnetic fields respectively as (.a (.b D = ɛ 0 ɛ + B = µ 0 H were te constants ɛ 0 and µ 0 are te permittivity and permeability of free space and are connected to te speed of ligt in vacuum c 0 by c 0 = / ɛ 0 µ 0. Te quantity is called te electric (macroscopic relaxation polarization and te coefficient ɛ is called te infinite frequency relative permittivity. Te constitutive law (.a incorporates te effects of electric polarization wic is defined as te electric field induced disturbance of te carge distribution in a region []. Tis polarization may ave an instantaneous component as well as ones tat do not occur instantaneously. Te relaxation polarization is te noninstantaneous part of te electric polarization and usually as associated time constants []. Te presence of instantaneous polarization is accounted for by te coefficient ɛ in te constitutive relation (.a. We neglect any additional magnetic effects and assume tat te magnetic constitutive relation (.b for free space is also valid in te dispersive medium. To describe te beavior of te media s macroscopic electric polarization a general integral equation model is employed in wic te polarization explicitly depends on te past istory of te electric field []. Te resulting constitutive law can be given in terms of a convolution involving a displacement susceptibility kernel g as (.3 (t x = t 0 g(t s x(s xds inside te dielectric. Here we consider polarization mecanisms for wic in te time domain te convolution (.3 describing te polarization can be converted to an ordinary differential equation (OD or systems of ODs governing te evolution of te relaxation polarization driven by te electric field []. In particular we consider two popular models: te Debye model [4] for orientational polarization and te Lorentz model [30] for electronic polarization... Debye Media: Model and nergy estimates. To model wave propagation in polar materials like water we use te single-pole Debye model in wic te susceptibility kernel in (.3 is (.4 g(t x = ɛ 0(ɛ s ɛ e t/τ. τ Tis gives a model for orientational polarization [0] and can be re-written as an OD in time forced by te electric field (.5 τ t + = ɛ 0ɛ (ɛ q in Ω (0 T. In equation (.5 te parameter ɛ s is called te static relative permittivity. Te ratio of static to infinite permittivities is denoted as ɛ q := ɛs ɛ. In general τ ɛ and ɛ s can be functions of space but we assume ere tat all parameters are constant witin te medium ɛ s > ɛ i.e. ɛ q > and τ > 0. To construct a model for electromagnetic wave propagation in a polar material in two dimensions we make te assumption tat no fields exibit variation in te z direction i.e. all partial derivatives wit respect to z are zero. Te electric field and polarization ten ave two components eac = ( x y T = ( x y T and te
4 V. A. BOKIL AND N. L. GIBSON magnetic field as one component H z = H. Combining (.5 wit te constitutive relations (.a and (.b and substituting in te Maxwell curl equations (.a and (.b we get te following system of partial differential equations wic we call te D T Maxwell-Debye model : (.6a (.6b (.6c H t = curl µ 0 t = curl H (ɛ q + ɛ 0 ɛ τ ɛ 0 ɛ τ t = ɛ 0ɛ (ɛ q τ τ were for a vector field U = (U x U y T te scalar curl operator is curl U := Uy x ( T U x y and for a scalar field V te vector curl operator is curl V := V y V x [8]. All te fields in (.6 are functions of position x = (x y T and time t. We first sow tat system (.6 along wit te C boundary conditions (.d and initial conditions (x 0 = 0 (x (x 0 = 0 (x and H(x 0 = H 0 (x for x Ω R is well-posed. To tis end we define te following two function spaces: (.7 (.8 H(curl Ω = {u ( L (Ω curl u L (Ω} H 0 (curl Ω = {u H(curl Ω n u = 0}. Let ( denote te L inner product and te corresponding norm. Multiplying (.6a by µ 0 v L (Ω (.6b by ɛ 0 ɛ u H 0 (curl Ω and (.6c by ɛ 0 ɛ (ɛ q w ( L (Ω integrating over te domain Ω R and applying Green s formula for te curl operator (.9 (curl H u = (H curl u u H 0 (curl Ω we obtain te weak formulation for te D Maxwell-Debye system of equations (.6 (.0a (.0b (.0c ( ɛ 0 ɛ (ɛ q ( µ 0 H t v ( ɛ 0 ɛ t u = ( curl v ( ɛ0 ɛ (ɛ q = (H curl u τ ( ( t w = τ w ɛ 0 ɛ (ɛ q τ w ( u + τ u Te following teorem sows te stability of te D Maxwell-Debye model (.6 by sowing tat te model exibits energy decay.. Teorem. (Maxwell-Debye nergy Decay. Let Ω R and suppose tat te solutions of te weak formulation (.0 for te D Maxwell-Debye system of equations (.6 satisfy te regularity conditions C(0 T ; H 0 (curl Ω C (0 T ; (L (Ω
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 5 C (0 T ; ( L (Ω and H(t C (0 T ; L (Ω along wit C boundary conditions (.d. Ten te system exibits energy decay (. D (t D (0 t 0 were te energy D (t is defined as (. ( D (t = µ0 H(t + ɛ 0 ɛ (t + ɛ0 ɛ (ɛ q (t roof. See [6 3]. In [3] it is also sown tat te Gauss laws are satisfied by te Maxwell-Debye system if te initial fields are divergence free. Te D Maxwell-Debye T scalar equations derived from (.6 are given as H t = ( x µ 0 y y (.3a x (.3b (.3c (.3d (.3e x t y t x t y t = ɛ 0 ɛ H = ɛ 0 ɛ H y (ɛ q τ x (ɛ q τ = ɛ 0ɛ (ɛ q x τ τ x = ɛ 0ɛ (ɛ q y τ τ y. x + ɛ 0 ɛ τ x y + ɛ 0 ɛ τ y.. Lorentz Media: Model and nergy estimates. For Lorentz Media te coice of te kernel function is (.4 g(t x = ɛ 0ωp e t/τ sin(ν 0 t ν 0 were ω p := ω 0 ɛs ɛ is te plasma frequency ω 0 is te resonant frequency of te medium λ := τ is a damping constant and ν 0 = ω0 λ. Te Lorentz model for electronic polarization in differential form is represented wit te second order OD forced by te electric field given as (.5 t + τ t + ω 0 = ɛ 0 ωp. Rewriting te above second order OD as a system of two first order OD s by introducing a new variable J = t te D T Maxwell-Lorentz model is H (.6a t = curl µ 0 (.6b (.6c (.6d t = curl H J ɛ 0 ɛ ɛ 0 ɛ J = t τ J ω0 + ɛ 0 ωp t = J..
6 V. A. BOKIL AND N. L. GIBSON All te fields in (.6 are functions of position x = (x y T and time t. For te Maxwell-Lorentz system (.6 we obtain te weak formulation (.7a ( H µ 0 t v = (curl v v L (Ω (.7b ( ɛ 0 ɛ t u = (H curl u (J u u H 0 (curl Ω (.7c ( ɛ 0 ωp (.7d ( ( ( J p t w = ɛ 0 ωpτ J p w ɛ 0 ɛ (ɛ q ( t q = ɛ 0 ɛ (ɛ q J p q ɛ 0 ɛ (ɛ q w q ( L (Ω. + ( w w ( L (Ω Te following teorem sows te stability of te D Maxwell-Lorentz model (.6 by sowing tat te model exibits energy decay (also see [3]. Teorem. (Maxwell-Lorentz nergy Decay. Let Ω R and suppose tat te solutions of te weak formulation (.7 for te Maxwell-Lorentz system of equations (.6 satisfy te regularity conditions C(0 T ; H 0 (curl Ω C (0 T ; (L (Ω J C (0 T ; ( L (Ω and H(t C (0 T ; L (Ω along wit C boundary conditions (.d. Ten te system exibits energy decay (.8 L (t L (0 t 0 were te energy L (t is defined as (.9 ( L (t = µ0 H(t + ɛ 0 ɛ (t + ɛ0 ɛ (ɛ q (t + ɛ0 ω p J (t. roof. See [6 3]. In [3] it is also sown tat te Gauss laws are satisfied by te Maxwell-Lorentz system if te initial fields are divergence free.
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 7 Te D Maxwell-Lorentz T scalar equations derived from system (.6 on wic te Yee sceme is based are: H t = ( x µ 0 y y (.0a x x = ( H (.0b t ɛ 0 ɛ y J x y = ( H (.0c t ɛ 0 ɛ x + J y (.0d (.0e (.0f (.0g J x t J y t x t y t = ɛ 0 ωp x τ J x ω0 x = ɛ 0 ωp y τ J y ω0 y = J x = J y. 3. Te Yee sceme: Discretization in Space and Time In tis section we consider te finite difference time domain (FDTD Yee sceme for discretizing te D Maxwell-Debye (.6 and D Maxwell-Lorentz models (.6. Consider te spatial domain Ω = [0 a] [0 b] R and time interval [0 T ] wit a b T > 0 and spatial step sizes x > 0 and y > 0 and time step t > 0. Te discretization of te intervals [0 a] [0 b] and [0 T ] is performed as follows [7]. Define L = a/ x J = b/ y and N = T/ t. For l j n N we consider te discretizations (3. (3. (3.3 0 = x 0 x x l x L = a 0 = y 0 y y j y J = b 0 = t 0 t t n t N = T were x l = l x y j = j y and t n = n t for 0 l L 0 j J and 0 n N. Define (x α y β t γ = (α x β y γ t were α is eiter l or l + β is eiter j or j + and γ is eiter n or n + wit l j n N. Te Yee sceme staggers te electric and magnetic fields in space and time. Fields x y and H are staggered in te x and y directions. We define te discrete meses {( } (3.4 τ x := x l+ y j 0 l L 0 j J {( } (3.5 τ y := x l y j+ 0 l L 0 j J (3.6 τ H := {( x l+ y j+ 0 l L 0 j J } to be te sets of spatial grid points on wic te x y and H fields respectively will be discretized. Te components x and J x are discretized at te same spatial locations as te field x wile te components y and J y are discretized at te same spatial locations as te field y. For te time discretization te components x y x y J x and J y are all discretized at integer time steps t n for 0 n N. In te Yee sceme te magnetic field H is staggered in time wit respect to x and y and discretized at time t n+ for 0 n N.
8 V. A. BOKIL AND N. L. GIBSON Let U be one of te field variables H x y x y J x or J y let (x α y β τ H τ x or τ y and γ be eiter n or n+ wit n N. We define te grid functions or te numerical approximations U γ αβ U(tγ x α y β. We will also use te notation U(t γ to denote te continuous solution on te domain Ω at time t γ and te notation U γ to denote te corresponding grid function on its discrete spatial mes at time t γ. We define in a standard way (see for e.g. [3 0] te centered temporal difference operator and a discrete time averaging operation as δ t U γ αβ := U γ+ αβ U γ αβ (3.7 t U γ αβ := U γ+ αβ + U γ αβ (3.8 and te centered spatial difference operators in te x and y direction respectively as (3.9 U γ U γ δ x U γ αβ := α+ β α β x (3.0 U γ U γ δ y U γ αβ := αβ+ αβ. y Next we define te following staggered discrete l normed spaces (see also [] (3. V H := (3. V := (3.3 V 0 := { } U : τ H R U = (U l+ j+ U H < { } F : τ x τ y R F = (F F xl+ y j lj+ T F < { } F V F = F xl+ x = F 0 l+ y J 0j+ = F xlj+ = 0 0 l L 0 j J were te discrete grid norms are defined as (3.4 (3.5 L J ( F = x y F xl+ + F ylj+ j F V l=0 j=0 L J U H = x y U l+ j+ U V H l=0 j=0 wit corresponding inner products (3.6 L J (F G = x y (F G + F xl+ xl+ ylj+ G ylj+ F G V j j l=0 j=0 (3.7 L J (U V H = x y U l+ j+ V l+ j+ U V V H. l=0 j=0
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 9 Finally we define discrete curl operators on te staggered l normed spaces as (3.8 curl : V 0 V H curl F := δ x F y δ y F x. and (3.9 curl : V H V 0 curl U := (δ y U δ x U T. Te discrete differential operators mimic properties tat are satisfied by teir continuous counterparts. In particular if te C conditions (.d are satisfied on te discrete Yee mes (3.0 F xl+ 0 = F x l+ J = F y 0j+ = F xlj+ = 0 0 l L 0 j J i.e. F V 0 discrete integration by parts (also see [3 0] yields (3. (curl H H = ( curl H. Tus te discrete versions of te curl operators remain adjoint to eac oter wic is essential for obtaining discrete energy estimates [3]. In Sections 4 and 5 we prove discrete energy estimates for te Yee sceme applied to te Maxwell Debye model (.3 and te Maxwell-Lorentz model (.0 respectively. In addition we sow tat te Yee scemes for tese media retain te second order accuracy in space and time tat te sceme enjoys in a non-dispersive medium. However our energy decay results indicate te dissipative nature of te Yee scemes in Debye and Lorentz media as opposed to te non-dissipative nature of te Yee sceme in a non-dispersive dielectric (also see [3 4]. Our energy analysis sows tat te Yee scemes for te Maxwell-Debye and Maxwell-Lorentz models are conditionally stable wit te stability condition. (3. c t < or ν < 0 were te Courant number ν = c t. Ten te stability condition (3. implies tat ν < and we prove convergence of te Yee scemes under tis criteria. 4. Yee Sceme for te Maxwell-Debye System 4.. Discretization. To discretize te D T Maxwell-Debye system (.3 in addition to staggering electric and magnetic components in space and time te lower order terms are discretized using averaging. Using te operators defined in (3.7 (3.8 (3.9 and (3.0 te Yee sceme for te D T Maxwell-Debye system (.3 consists of te following discrete equations: (4.a (4.b (4.c (4.d (4.e δ t H n l+ j+ δ t n+ x l+ j = = ( δ y x n δ µ 0 l+ x j+ y n l+ j+ ɛ q l+ j ɛ 0 ɛ δ y H n+ δ t n+ y lj+ = δ x H n+ ɛ 0 ɛ δ t n+ x = ɛ 0ɛ (ɛ q l+ j τ δ t n+ y lj+ = ɛ 0ɛ (ɛ q τ lj+ τ n+ x + l+ j n+ τɛ 0 ɛ x l+ j ɛ q n+ y + n+ τ lj+ τɛ 0 ɛ n+ x l+ j τ n+ x l+ j n+ y lj+ τ n+ y lj+. y lj+
0 V. A. BOKIL AND N. L. GIBSON We can re-write tis system in vector form as (4.a (4.b δ t H n + µ 0 (curl n = 0 on τ H δ t n+ = (curl H n+ ɛ q n+ + n+ on τ x τ y ɛ 0 ɛ τ τɛ 0 ɛ (4.c δ t n+ ɛ 0 ɛ (ɛ q = n+ τ τ n+ on τ x τ y. 4.. Accuracy: Truncation rror Analysis. Similar to te Yee sceme in free space te Yee sceme for te Maxwell-Debye system is also second-order accurate in bot time and space. Lemma 4. (Yee Sceme Truncation rrors for Maxwell-Debye. Suppose tat te solutions to te two-dimensional Maxwell-Debye equations (.6 or (.3 satisfy te regularity conditions C 3 ([0 T ]; [C 3 (Ω] C 3 ([0 T ]; [C(Ω] and H C 3 ( [0 T ]; [C 3 (Ω]. Let ξ n H ξn+ x ξ n+ y ξ n+ x ξ n+ y te Yee sceme for te Maxwell-Debye model (4.. Ten be te truncation errors for (4.3 { max ξ n H } ξ n+ ξ n+ x ξ n+ y ξ n+ x ( y C D x + y + t were C D = C D (ɛ 0 µ 0 ɛ ɛ q τ does not depend on te mes sizes x y and t.
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA roof. We perform Taylor expansions and substitute te exact solution to obtain te truncation errors for (4.a - (4.e. We ave (4.4 (4.5 (ξ H n l+ j+ = t 4 3 H t 3 (x l+ y j+ t + y 4µ 0 3 x y 3 (x l+ y t n x 4µ 0 3 y x 3 (x y j+ tn (ξ x n+ t 3 x = l+ j 4 t 3 (x l+ y j t + t 3 x 4ɛ 0 ɛ t 3 (x l+ y j t y 3 H 4ɛ 0 ɛ y 3 (x l+ y t n+ (4.6 ( n+ ξy = t 3 y lj+ 4 t 3 (x l y j+ t 3 + t 3 y 4ɛ 0 ɛ t 3 (x l y j+ t 3 + x 3 H 4ɛ 0 ɛ x 3 (x 3 y j+ tn+ (4.7 (4.8 (ξ x n+ t = l+ j 4 + t 8τ ( n+ ξy = t lj+ 4 + t 8τ 3 x t 3 (x l+ y j t 4 t ɛ 0 ɛ (ɛ q 8τ x t (x l+ y j t 43 3 y t 3 (x l y j+ t 5 t ɛ 0 ɛ (ɛ q 8τ y t (x l y j+ t 53 x t (x l+ y j t 4 y t (x l y j+ t 5 were x l x x l+ y j y y j+ t n t t n+ x l x 3 x l+ y j y y j+ tn t i t 3i t n+ for i = and t n t 4i t 5i t n+ for i = 3. 4.3. Discrete nergy stimates for Debye media. In tis section we prove a discrete version of te energy decay property given in Teorem. for te D Maxwell-Debye model (.6. We assume a uniform mes i.e. x = y = > 0. Teorem 4. proves te conditional stability of te D Yee sceme for discretizing te Maxwell-Debye model by sowing te decay of a discrete energy in time.
V. A. BOKIL AND N. L. GIBSON Teorem 4. (nergy Decay for Maxwell-Debye. If te stability condition (3. is satisfied ten te Yee sceme for te Maxwell-Debye System (4. or (4. satisfies te discrete identity (4.9 δ t n+ D = ɛ n+ 0 ɛ (ɛ q n+ n+ D τɛ 0ɛ (ɛ q for all n were (4.0 D n = µ 0(H n+ H n H + ɛ 0 ɛ n + ɛ0 ɛ (ɛ q n defines a discrete energy. roof. We consider te average of (4.a at n and n+ multiply wit x yh n+ and sum over all spatial nodes on τ H to get (4. µ 0 (δ t H n+ H n+ H + (curl n+ H n+ H = 0. We can rewrite (4. as (4. µ 0 t {(Hn+ 3 H n+ H (H n+ H n H } + (curl n+ H n+ H = 0. We multiply equation (4.b wit x y n+ τ x τ y to get and sum over all spatial nodes on (4.3 ɛ 0 ɛ (δ t n+ n+ + ɛ 0ɛ (ɛ q ( n+ n+ τ τ (n+ n+ = (curl H n+ n+ wic can be re-written as (4.4 ɛ 0 ɛ t { n+ n ɛ 0 ɛ (ɛ q } + n+ τ τ (n+ n+ = (curl H n+ n+. Finally we multiply equation (4.c by x y ɛ 0ɛ (ɛ q n+ spatial nodes on τ x τ y to get and sum over all (4.5 ɛ 0 ɛ (ɛ q (δ t n+ n+ = τ (n+ n+ ɛ 0 ɛ (ɛ q τ n+
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 3 wic can be re-written as (4.6 { n+ n ɛ 0 ɛ (ɛ q } = τ (n+ n+ ɛ 0 ɛ (ɛ q τ n+. ave Adding equations (4. (4.4 and (4.6 and using te definition (4.0 we (4.7 { ( n+ t D (D n } = ɛ 0 ɛ (ɛ q τ }. + (ɛ 0 ɛ (ɛ q n+ We can rewrite tis equation in te form (4.8 n+ D n D t ( = n+ D + D n { n+ ɛ 0 ɛ (ɛ q ( n+ n+ ɛ 0 ɛ (ɛ q τ ɛ 0ɛ (ɛ q n+ n+ wic on utilizing te definitions of te time differencing and averaging operators in (3.7 and (3.8 respectively gives us te discrete identity (4.9 for Debye media. Wat is left to prove is tat te quantity defined in (4.0 is a discrete energy i.e. a positive definite function of te solution to te system (4.. Using te parallelogram law [3] we ave (4.9 (H n+ H n H = H n+ + H n 4 H n+ H n H 4. H Using (4.a and te definitions of te time differencing operator in (3.7 we can rewrite te above as (4.0 4 H n+ H n H = t 4 δ th n H = t 4 curl n H. Substituting equations (4.9 and (4.0 into te definition (4.0 and using te definition of te time averaging operator in (3.8 we can re-write te discrete energy as (4. D n = µ 0 H n H + ɛ 0 ɛ ( n A n + ɛ0 ɛ (ɛ q n were te operator A : V 0 V 0 is defined as (4. A F = (I c t curl curl F F V 0. 4
4 V. A. BOKIL AND N. L. GIBSON In D wit a uniform space mes step size x = y = we ave for all F V 0 te inequality ([3] (4.3 curl F H 8 F from wic we ave (4.4 (A F F = (F F c t (curl F curl F H = F c t curl F H 4 4 F 8c t 4 F = ( ν F. Tus if te stability condition (3. is satisfied ten ν > 0 and te operator A is positive definite and D n defines a discrete energy. We note tat tis stability condition is te same for te Yee sceme applied to a non-dispersive dielectric [33]. Remark: For a nonuniform mes te stability condition is again te same as for te non-dispersive case i.e. ν < wit te Courant number (4.5 ν = c t x + y. 4.4. Convergence Analysis of te Yee sceme for te Maxwell-Debye Model. Te tecnique to prove convergence of te Yee scemes is a classical one (see for e.g. [7] and references terein and employs te energy approac. To prove te convergence of te Yee sceme for te D Maxwell-Debye system for 0 n N we define te error quantities (4.6a (4.6b (4.6c H n = H n H(t n n = n (t n n = n (t n. As was done for te discrete energy estimate in te proof of Teorem 4. we obtain te following identities for te Yee sceme for te Maxwell-Debye model in (4.. (4.7a (4.7b (4.7c µ 0 δ t H n + curl n = ξ n H ɛ 0 ɛ δ t n+ curl H n+ (ɛ q + ɛ0 ɛ n+ τ τ n+ = ξ n+ ɛ 0 ɛ (ɛ q δ t n+ + ɛ 0 ɛ (ɛ q τ n+ τ n+ = ξ n+ were ξh n and ξn+ F = (ξ n+ F x ξ n+ F y T for F = are te local truncation errors for te Maxwell-Debye system as discussed in Lemma 4.. We ave te following result: Teorem 4. (Convergence Analysis of Yee Sceme for Maxwell-Debye. Suppose tat te solutions to te two-dimensional Maxwell-Debye equations (.6 or (.3
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 5 satisfy te regularity conditions C 3 ([0 T ]; [C 3 (Ω] C 3 ([0 T ]; [C(Ω] and H C 3 ( [0 T ]; [C 3 (Ω]. For n 0 let H n+ n and n be te solution to te Yee sceme for te Maxwell-Debye system (4. or (4.. let ξ n H ξn+ x ξ n+ y ξ n+ x ξ n+ y Also be te truncation errors for te Yee sceme for Maxwell-Debye (4. or (4. satisfying te conditions of Lemma 4.. Assume tat te stability condition (3. is satisfied ten for any fixed T > 0 a positive constant C D = C D (ɛ 0 µ 0 ɛ ɛ q ν depending on te medium parameters and te Courant number ν but independent of te mes parameters t x y suc tat { } (4.8 max R n D R 0 D + T C D (ɛ 0 µ 0 ɛ ɛ q ν ( x + y + t 0 n N were te energy of te error at time t n = n t R n D is defined as (4.9 R n D = µ 0(H n+ H n H + ɛ 0 ɛ n + ɛ0 ɛ (ɛ q n roof. We first note based on te proof of Teorem 4. tat te energy of te error (4.9 can be equivalently written in te form (4.30 R n D = µ 0 H n H + ɛ 0 ɛ ( n A n + ɛ0 ɛ (ɛ q n wit te operator A as defined in equation (4.. Next we follow a similar procedure to tat done in te proof of Teorem 4.. Multiplying te average of (4.7a at n and n + by x yh n+ over all spatial nodes on τ H multiplying (4.7b by x y n+ all spatial nodes on τ x. and summing and summing over τ y multiplying (4.7c by x yn+ on τ x τ y and summing over all spatial nodes and finally adding all te results we obtain (4.3 δ t R n+ D R n+ D = τɛ 0 ɛ (ɛ q ɛ 0ɛ (ɛ q n+ n+ + (ξ n+ H H n+ H Neglecting te first term we ave (4.3 + (ξ n+ n+ + (ξ n+ n+. δ t R n+ D R n+ D ξ n+ H H H n+ H + ( ξn+ n + n+ + ξn+ ( n + n+.
6 V. A. BOKIL AND N. L. GIBSON From (4.30 we ave te following bounds (4.33a (4.33b (4.33c H n H (µ 0 R n D n ( ɛ 0 ɛ ( ν R n D n (ɛ 0 ɛ (ɛ q R n D. Te inequalities in (4.33 give us bounds for te averaged electric and polarization terms as (4.34a (4.34b ( n + n+ ( ɛ0 ɛ ( ν n+ R D ( n + n+ (ɛ0 ɛ (ɛ q n+ R D. For a bound on H n+ H we note tat since H n+ (4.35 H n+ H H n H + t δ th n H. = H n + t δ th n we ave Under te assumption of te stability condition (3. and te form of te (nonnegative energy of te error in (4.9 we ave te inequality (4.36 (H n+ H n H ɛ 0 ɛ n µ + 0 ɛ0 ɛ (ɛ q n. Next since t 4 δ th n H = Hn H (Hn+ H n H we ave from (4.36 (4.37 t 4 δ th n H µ 0 µ 0 H n H + ɛ 0 ɛ n + ɛ0 ɛ (ɛ q n Substituting inequalities (4.33 (4.34 (4.35 and (4.37 in (4.3 we ave (4.38 ( δ t R n+ D { } n+ R D C D (ɛ 0 µ 0 ɛ ɛ q ν max ξ n+ H H ξ n+ ξ n+ n+ R D were C D = C D (ɛ 0 µ 0 ɛ ɛ q ν is a constant tat depends on te medium parameters and te Courant number ν but is independent of te mes parameters n+ t x y. Dividing by R D 0 we get (4.39 { } R n+ D Rn D + t C D (ɛ 0 µ 0 ɛ ɛ q ν max ξ n+ H H ξ n+ ξ n+. Recursively applying te inequality (4.39 from n + to 0 we ave (4.40 R n+ D R0 D+ t C D (ɛ 0 µ 0 ɛ ɛ q ν ( n k=0. { } max ξ k+ H H ξ k+ ξ k+.
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 7 Using t = T/N and 0 n N and noting tat te sum in te second term on te rigt in (4.40 from Lemma 4. is O( t + x + y we get (4.4 R n D R 0 D + T C D (ɛ 0 µ 0 ɛ ɛ q ν ( t + y + x. Note we use C D as a generic constant absorbing all constants tat arise in te above inequalities. Finally taking te maximum for n from 0 to N- in (4.4 we obtain (4.8. Remark 4.. We note tat te convergence result in Teorem 4. does not old at te stability limit ν = (see also [7]. It is sown in [7] tat te Yee sceme in a non-dispersive dielectric need not be stable at te stability limit. However (3. provides a necessary and sufficient criteria for te stability of te Yee sceme. (Also see [5]. 4.5. Discrete Divergence for te Maxwell-Debye Model. If te initial fields satisfy te Gauss divergence laws (4.4a (4.4b D = ρ B = 0 were ρ is te electric carge density ten one can sow from te Maxwell curl equations tat te Gauss divergence laws are satisfied for all time [9]. Tis is done by applying te divergence operator to te Maxwell curl equations using te fact tat te divergence of te curl is zero and utilizing te continuity equation (assume tere are no sources J s = 0 to get (4.43 ρ t + J c = 0. It is well known tat te Yee sceme for te Maxwell equations in free space (linear media preserves te divergence property of te solution at te discrete level [8]. We now sow tat tis remains true for te Yee sceme applied to te Maxwell-Debye system. 4.5.. Discrete Divergence Operator. To define te discrete divergence operator we define te discrete mes of interior vertices (4.44 τ 0 := {(x l y j l L j J } to be te set of spatial grid points on wic te discrete divergence of an electric field will be defined. On te mes τ 0 we define te staggered l normed space (4.45 V 0 := { V : τ 0 R V = (V lj V 0 < } were te discrete grid norm is defined as L J ( (4.46 V 0 = x y V lj V V 0 l= j=
8 V. A. BOKIL AND N. L. GIBSON wit corresponding inner product L J (4.47 (V W 0 = x y (V lj W lj V W V 0 l= j= We define te discrete divergence operator on V 0 as (4.48 div : V 0 V 0 (div F lj := (δ x F x lj +(δ y F y lj l L j J. Lemma 4. (Discrete Divergence for te Maxwell-Debye System. For te Yee sceme applied to te Maxwell-Debye system given in (4. te discrete divergence of te initial grid functions is preserved for all n 0 i.e. we ave te identity (4.49 div D n = div D 0 on τ 0. roof. We note tat te discrete derivative operators δ t δ x and δ y commute. From te Yee equations (4.b and (4.c we ave on te mes τ x τ y (4.50 δ t D n+ = δt (ɛ 0 ɛ n+ + n+ = curl H. Terefore div curl H = δ x δ y H δ y δ x H = 0. Tus applying te discrete divergence operator div to (4.50 we obtain (4.5 δ t div D n+ = δt (ɛ 0 ɛ div n+ + div n+ = div curl H = 0 on τ. 0 We finally ave (4.5 div D n+ = div D n on τ 0. Applying te identity (4.5 recursively in discrete time we obtain (4.49. Remark 4.. We note tat for te D T Maxwell model te divergence of te magnetic flux density is not defined. Tus te divergence free or solenoidal nature of te magnetic flux density is lost in te two dimensional model [8].
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 9 5. Yee Sceme for te Maxwell-Lorentz System 5.. Discretization. Te discrete approximation of te D Maxwell-Lorentz system (.6 by te Yee sceme is (5.a (5.b (5.c δ t H n l+ j+ = ( δ y x n δ µ 0 l+ x j+ y n l+ j+ δ t n+ x = ( l+ δ j y H n+ ɛ 0 ɛ J n+ l+ j xl+ j δ t n+ y lj+ = ɛ 0 ɛ ( δ x H n+ + J n+ lj+ ylj+ (5.d (5.e (5.f (5.g δ t J n+ xl+ j = ɛ 0 ωp n+ x l+ j τ J n+ xl+ ω 0 n+ x j l+ j δ t J n+ ylj+ = ɛ 0 ω p n+ y lj+ τ J n+ ylj+ ω 0 n+ y lj+ δ t n+ x = J n+ l+ j xl+ j δ t n+ y lj+ = J n+ ylj+. We can re-write tis system in vector form as (5.a (5.b (5.c δ t H n + µ 0 (curl n = 0 on τ H δ t n+ = (curl H n+ n+ J on τ x τ y ɛ 0 ɛ δ t J n+ = ɛ 0 ωp n+ τ Jn+ ω0 n+ on τ x τ y (5.d δ t n+ = J n+ on τ x τ y. 5.. Accuracy: Truncation rror Analysis. Te Yee sceme for te Maxwell- Lorentz system is also second-order accurate in bot time and space. Lemma 5. (Yee Sceme Truncation rrors for Maxwell-Lorentz. Suppose tat te solutions to te two-dimensional Maxwell-Lorentz equations (.6 or (5. satisfy te regularity conditions C 3 ([0 T ]; [C 3 (Ω] J C 3 ([0 T ]; [C(Ω] and H C 3 ( [0 T ]; [C 3 (Ω]. Let ξ n H ξn+ x ξ n+ y ξ n+ J x ξ n+ J y ξ n+ x ξ n+ y truncation errors of te Yee sceme for te Maxwell-Lorentz model (5.. Ten be te (5.3 { max ξ n } ξ n+ ξ n+ x ξ n+ y ξ n+ J x ξ n+ J y ξ n+ x ( y C L x + y + t H were C L = C L (ɛ 0 µ 0 ɛ ɛ q τ ω 0 does not depend on te mes sizes x y and t.
0 V. A. BOKIL AND N. L. GIBSON roof. We perform Taylor expansions and substitute te exact solution to obtain te truncation errors for te equations in system (5.. We ave (5.4 (5.5 (5.6 (5.7 (5.8 (ξ H n i+ j+ = t 4 3 H t 3 (x l+ y j+ t + y 4µ 0 3 x y 3 (x l+ y t n x 4µ 0 3 y x 3 (x y j+ tn (ξ x n+ t 3 x = l+ j 4 t 3 (x l+ y j t + t 3 x 4ɛ 0 ɛ t 3 (x l+ y j t y 3 H 4ɛ 0 ɛ y 3 (x l+ y t n+ ( n+ ξy = t 3 y lj+ 4 t 3 (x l y j+ t 3 + t 3 y 4ɛ 0 ɛ t 3 (x l y j+ t 3 + x 3 H 4ɛ 0 ɛ x 3 (x 3 y j+ tn+ ( n+ t 3 J x ξjx = l+ j 4 t 3 (x l+ y t x j t 4 ɛ 0 ω p 8 t (x l+ y j t 4 + t J x 8τ t (x l+ y j t 43 + ω 0 t x 8 t (x l+ y j t 44 n+ (ξ Jy = t 3 J y 4 t 3 (x l y j+ t 5 ɛ 0 ωp t y 8 t (x l y j+ t 5 lj+ + t 8τ J y t (x l+ y j t 53 + ω 0 t y 8 t (x l+ y j t 54 (5.9 (5.0 (ξ x n+ t = l+ j 4 ( n+ ξy = t lj+ 4 3 x t 3 (x l+ y j t 6 t J x 8 t (x l+ y j t 6 3 y t 3 (x l y j+ t 7 t J y 8 t (x l y j+ t 7 were x l x x l+ y j y y j+ and t n t t n+. Next x l x 3 x l+ y j y y j+ tn t li t n+ for i { 3 4} and l = 3... 7. 5.3. Discrete nergy stimates for Lorentz media. Teorem 5. (nergy Decay for Maxwell-Lorentz. If stability condition (3. is satisfied ten te Yee sceme for te Maxwell-Lorentz System (5. or (5. satisfies te discrete identity (5. δ t n+ L = n+ J n+ L τɛ 0ω0
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA for all n were (5. L n = µ 0(H n+ H n H + ɛ 0 ɛ n + ɛ0 ɛ (ɛ q n + J n ɛ0 ω p defines a discrete energy. roof. Te proof is similar to te proof of Teorem 4. for te Yee sceme for te Maxwell-Debye system. We point out te differences ere. We multiply equation (5.b wit x y n+ and sum over all spatial nodes on τ x τ y to get (5.3 ɛ 0 ɛ (δ t n+ n+ + (J n+ n+ = (curl H n+ n+ wic can be re-written as (5.4 ɛ 0 ɛ t { n+ n } n+ + (J n+ = (curl H n+ n+. Next we multiply equation (5.c by x y n+ ɛ 0ω J p on τ x τ y to get and sum over all spatial nodes (5.5 ɛ 0 ωp (δ t J n+ n+ J = ( n+ n+ J τɛ 0 ωp (J n+ J n+ ɛ 0 ɛ (ɛ q (J n+ n+ wic we re-write as (5.6 ɛ 0 ωp { J n+ J n Finally we multiply equation (5.d by nodes on τ x (5.7 τ y to get } n+ ( n+ J + τɛ 0 ωp J n+ x ɛ 0ɛ (ɛ q n+ + ɛ 0 ɛ (ɛ q (J n+ n+ = 0. and sum over all spatial ɛ 0 ɛ (ɛ q (δ t n+ n+ ɛ 0 ɛ (ɛ q (J n+ n+ = 0 wic can be re-written as (5.8 tɛ 0 ɛ (ɛ q { n+ n } ɛ 0 ɛ (ɛ q (J n+ n+ = 0. Adding equations (4. (5.4 (5.6 and (5.8 and using te definition (5. we ave (5.9 { ( n+ L t (L n } = { } n+ ɛ 0 τωp J.
V. A. BOKIL AND N. L. GIBSON We can rewrite tis equation in te form n+ L (5.0 ( L n n+ = t n+ D + D n ɛ 0 τωp J wic on utilizing te definitions of te time differencing and averaging operators in (3.7 and (3.8 respectively gives us te discrete identity (5. for Lorentz media. As for te case of Debye media if te stability condition (3. is satisfied te quantity defined in (5. is a discrete energy i.e. a nonnegative function of te solution to te system (5.. Te rest of te proof is similar to te proof of Teorem 4. for te case of Debye media. 5.4. Convergence Analysis of te Yee sceme for te Maxwell-Lorentz Model. Define te error quantities (5.a (5.b (5.c (5.d We obtain te identities (5.a µ 0 δ t H n + (curl n = ξ n H on τ H (5.b H n = H n H(t n n = n (t n J n = J n J (t n n = n (t n. ɛ 0 ɛ δ t n+ (curl H n+ n+ + J = ξ n+ on τ x τ y (5.c δ t J n+ n+ + ɛ 0 ω pτ J n+ ɛ 0 ωp (5.d ɛ 0 ɛ (ɛ q δ t n+ + ɛ 0 ɛ (ɛ q J n+ ɛ 0 ɛ (ɛ q n+ = ξ n+ on τ x = ξ n+ J τ y. For te Maxwell-Lorentz system we ave te following result: on τ x τ y Teorem 5. (Convergence Analysis of Yee Sceme for Maxwell-Lorentz. Suppose tat te solutions to te two-dimensional Maxwell-Lorentz equations (.6 satisfy te regularity conditions C 3 ([0 T ]; [C 3 (Ω] J C 3 ([0 T ]; [C(Ω] and H C 3 ( [0 T ]; [C 3 (Ω]. For n 0 let H n+ n J n and n be te solution to te Yee sceme for te Maxwell-Lorentz system (5. or (5. and let ξ n H ξn+ ξ n+ J ξ n+ be te truncation errors wit ξ k F = (ξk F x ξ k F y T F = J satisfying te conditions of te Lemma 5.. If te stability condition (3. is satisfied ten for any fixed T > 0 a positive constant C L = C L (ɛ 0 µ 0 ɛ ɛ q τ ω 0 ν
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 3 depending on te medium parameters and te Courant number ν but independent of te mes parameters t x y suc tat { } (5.3 max R n L R 0 L + T C L (ɛ 0 µ 0 ɛ ɛ q ω 0 ν ( t + y + x 0 n N were te energy of te error at time n t R n L is defined as (5.4 R n L = µ 0 H n H + ɛ 0 ɛ ( n A n + ɛ0 ɛ (ɛ q n + J n ɛ0 ω p roof. We follow a similar procedure to te convergence analysis for te Yee sceme for te Maxwell- Debye model in Teorem 4. we: multiply te average of (5.a at n and n + by x yh n+ by x y n+ x yj n+ x y n+ obtain (5.5 and sum over all spatial nodes on τ H multiply (5.b and sum over all spatial nodes on τ x and sum over all spatial nodes on τ x and sum over all spatial nodes on τ x τ y multiply (5.c by τ y multiply (5.d by τ y δ t R n+ L R n+ L = τɛ 0 ωp J n+ + (ξ n+ H H n+ H xpanding (5.5 we ave (5.6 and add te results to + (ξ n+ n+ + (ξ n+ J J n+ +(ξ n+ n+ δ t R n+ L R n+ L ξ n+ H H H n+ H + ( ξn+ n + n+ + ( ξn+ J J n + J n+ ( + ξn+ n + n+ Substituting inequalities (4.33 (4.34 (4.35 and (4.37 in (5.6 and using te bound (5.7 we ave (5.8 ( δ t R n+ L ( J n + J n+ ɛ0 ωpr n+ L { } n+ R L C L max ξ n+ H ξ n+ ξ n+ J ξ n+ n+ R L were C L = C L (ɛ 0 µ 0 ɛ ɛ q τ ω 0 ν is a constant tat depends on te medium parameters and te Courant number ν but is independent of te mes parameters n+ t x y. Dividing by R L 0 using recursion in time t = T/N te.
4 V. A. BOKIL AND N. L. GIBSON results of Lemma 5. and summing from n to 0 we finally obtain (5.9 R n L R 0 L + T C L (ɛ 0 µ 0 ɛ ɛ q ω 0 τ ν ( t + y + x. Remark 5.. As noted in Remark 4. for te convergence result in Teorem 4. for te D Yee Maxwell-Debye sceme te convergence result in Teorem 5. for te D Yee Maxwell-Lorentz sceme does not old at te stability limit ν =. 5.5. Discrete Divergence for te Maxwell-Lorentz Model. Lemma 5. (Discrete Divergence for te Maxwell-Lorentz System. For te Yee sceme applied to te Maxwell-Lorentz system given in (5. te discrete divergence of te initial grid functions is preserved for all n 0 i.e. we ave te identity (5.30 div D n = div D 0 on τ. 0 roof. Te proof is te same as te proof of Lemma 4. for Debye media. 6. Numerical Simulations of te Yee Sceme for te Maxwell-Debye Model We perform numerical simulations of system (4. on te domain Ω = [0 ] [0 ]. For our simulation we assume a uniform mes wit x = y =. We use T = parameter values µ 0 = ɛ 0 = (i.e. c 0 = ɛ = ɛ q = (ɛ s = and τ =. 6.. An xact Solution for te Maxwell-Debye Model. We use an exact solution introduced in [7] to te Maxwell-Debye system (.3 wic we use to initialize our simulations. We define te wave vector as k = (k x k y T were k x = π k x k y = π k y and te corresponding wave number is k = kx + ky. We also define te function α D (θ k := θ θ + k. Maxwell-Debye system (.3 is (6.a (6.b (6.c = = ( x ( x Te exact solution to te H = k π e θt cos(k x x cos(k y y θ = π k ye θt cos(k x x sin(k y y y θ π k xe θt sin(k x x cos(k y y k y = π α D(θ k e θt cos(k x x sin(k y y y k x π α D(θ k e θt sin(k x x cos(k y y were te parameter θ is a real number. We note tat for k x and k y integers te exact solution (6. satisfies te perfect conductor conditions (.d on te
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 5 boundary of te domain Ω and te electric and polarization fields are divergence free on Ω. Te wave number k and parameter θ are related by te equation (6. θ 3 θ + k θ k = 0. Te energy defined in (. for te exact solution (6. can be computed to be (6.3 D (t = k e θt ( k + θ π + α(θ k. In our simulations we use various values of te Courant number (3. ν x = ν y = ν and various values of k x = k y = k. Te real root of equation (6. depends on te value of te wave number k = k. In particular for k = θ.053. Te exact dispersion relation for Debye media [5 3] relating te wave number k to te angular frequency ω is (6.4 k = ω ɛs iωτɛ. c 0 iωτ Tus using te cosen values for te parameters c 0 = τ = ɛ = and ɛ s = ɛ q = in (6.4 and squaring bot sides te dispersion relation can be written as (6.5 (iω 3 (iω + k iω k = 0. As noted in [7] comparing te dispersion relation (6.5 to te relation (6. for real θ we note tat te exact solution (6. corresponds to a solution for te Maxwell- Debye system (.3 for a purely imaginary angular frequency ω = iθ. 6.. Relative and nergy rrors. For te discrete solution produced we compute relative errors defined as ( RD (t n = (t n n + H(t n H n H + (t n n (6.6 ( RD (t n (6.7 Relative rror = max 0 n N D (t n were te grid norms and H are defined in (3.4 and (3.5 respectively. We also define te nergy rror for te discrete solutions as d D (6.8 nergy rror = max dt (tn+ δt n+ D 0 n N d D dt (tn+ ( were te discrete energy D n is defined in (4.9 and d D dt t n+ is te time derivative of te exact energy (6.3 computed at te time point t n+. Table presents te Relative errors (6.7 and confirms te second order accuracy of te Yee sceme for various values of t k and ν. Here N N wit N t = T and refers to te number of time iterations performed. We note tat te largest value of t cosen (0.0 is suc tat t/τ (τ = in tis example is O(0 or lower. Tis is in agreement wit results obtained in [3] wic indicate tat to resolve all time scales in te problem we must coose t = O(0 τ for Debye media. Table presents te nergy errors (6.8. Te results in tis table indicate tat te energy error decreases in a second order accurate manner and provides anoter confirmation of te second order accuracy of te Yee sceme. In Figures and we plot te relative errors (6.6 and te energy errors (6.8 respectively for te various values of N ν and k as presented in te corresponding
6 V. A. BOKIL AND N. L. GIBSON Table. Relative rrors for te D Yee Maxwell-Debye sceme. k = π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50.0 0 3 4.57 0 4.53 0 4 00.99 0 4.0.4 0 4.0 6.30 0 5.00 00 7.46 0 5.00.84 0 5.00.57 0 5.00 400.86 0 5.00 7.0 0 6.00 3.93 0 6.00 800 4.65 0 6.00.77 0 6.00 9.83 0 7.00 k = 5π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50 5.39 0 3.0 0 3.0 0 3 00.39 0 4.96 4.97 0 4.0.54 0 4.0 00 3.44 0 4.0.4 0 4.0 6.34 0 5.00 400 8.57 0 5.00 3.09 0 5.00.58 0 5.00 800.4 0 5.00 7.7 0 6.00 3.95 0 6.00 k = 0π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50.3 0 4.08 0 3.0 0 3 00.79 0 3.4 9.75 0 4.06 4.94 0 4.03 00 6.74 0 4.05.4 0 4.0.3 0 4.0 400.67 0 4.0 6.00 0 5.00 3.06 0 5.00 800 4.6 0 5.00.50 0 5.00 7.66 0 6.00 tables. An O( reference is provided to visually see te second order accuracy of te Yee sceme. 6.3. Convergence Analysis of te Discrete Divergence. We verify te identity in (4.49 by computing te maximum absolute grid error in te discrete divergence as follows (6.9 max 0 n N div D n div D 0 0 were te grid norm 0 is defined in (4.46. Table 3 presents te absolute errors (6.9 of te D Yee Maxwell-Debye sceme for various values of t k and ν.
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 7 Table. nergy rrors for te D Yee Maxwell-Debye sceme. k = π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50.67 0 3 6.44 0 4 3.60 0 4 00 4.6 0 4.0.60 0 4.0 8.97 0 5.0 00.04 0 4.00 4.00 0 5.00.4 0 5.00 400.59 0 5.00 9.99 0 6.00 5.59 0 6.00 800 6.48 0 6.00.50 0 6.00.40 0 6.00 k = 5π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50 6.68 0 3.3 0 3.0 0 3 00.6 0 3.06 5.79 0 4.0.99 0 4.00 00 3.98 0 4.0.44 0 4.0 7.44 0 5.0 400 9.9 0 5.00 3.60 0 5.00.86 0 5.00 800.47 0 5.00 9.89 0 6.00 4.64 0 6.00 k = 0π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50.46 0 5.4 0 3.5 0 3 00 3.53 0 3.05.3 0 3.09 6.30 0 4.00 00 8.55 0 4.04 3.06 0 4.0.56 0 4.0 400. 0 4.0 7.6 0 5.00 3.90 0 5.00 800 5.7 0 5.00.90 0 5.00 9.75 0 6.00 Again N N wit N t = T refers to te number of time steps performed. All errors are sufficiently small to suggest tat tey are due to roundoff. 7. Numerical Simulations of te Yee Sceme for te Maxwell-Lorentz Model We perform numerical simulations of system (5. on te domain Ω = [0 ] [0 ] using exact solutions for wic ɛ 0 = µ 0 = ɛ = ω 0 = τ = 0.4 and ɛ s = ɛ q =. 7.. An xact Solution for te Maxwell-Lorentz Model. We define te functions α L (θ k := θ + θ + k and β L (θ k := θ + k. We consider
8 V. A. BOKIL AND N. L. GIBSON Table 3. Discrete Divergence rrors for te D Yee Maxwell- Debye sceme. k = π N ν = 0.3 ν = 0.5 ν = 0.7 50 9.47 0 4.6 0 3.5 0 3 00.43 0 3 4.05 0 3 5.8 0 3 00 7. 0 3.5 0.6 0 400.04 0 3.4 0 4.70 0 800 5.76 0 9.66 0.34 0 k = 5π N ν = 0.3 ν = 0.5 ν = 0.7 50 4.66 0.68 0.47 0 00.99 0 5.48 0 7.6 0 00 7.9 0.30 0 0.98 0 0 400.3 0 0 3.83 0 0 6.9 0 0 800 6.86 0 0.7 0 9.7 0 9 k = 0π N ν = 0.3 ν = 0.5 ν = 0.7 50 6.99 0.4 0 0.89 0 0 00.5 0 0.48 0 0 5.40 0 0 00 3.9 0 0.09 0 9.3 0 9 400.7 0 9 3.35 0 9 4.9 0 9 800 6.44 0 9 8.73 0 9.5 0 8 te following exact solution to te Maxwell-Lorentz system (.6 ( x θ = = π k ye θt cos(k x x sin(k y y (7.a y θ π k xe θt sin(k x x cos(k y y (7.b (7.c (7.d = J = H = k π e θt cos(k x x cos(k y y k y = π α L(θ k e θt cos(k x x sin(k y y y k x π α L(θ k e θt sin(k x x cos(k y y k y = J π β L(θ k e θt cos(k x x sin(k y y. y k x π β L(θ k e θt sin(k x x cos(k y y ( x ( Jx
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 9 In te above te wave number and parameter θ are related by te equation (7. θ 4 τ θ3 + ( + k θ k τ θ + k = 0. As in te Debye model for k x and k y integers te exact solution (7. satisfies te perfect conductor conditions on te boundary of te domain Ω and te electric and polarization fields are divergence free on Ω. Te energy defined in (.9 for te exact solution (7. can be computed to be (7.3 (t = k e θt βl ( + β L + α L (θ k π. Te real root of equation (7. depends on te value of te wave number k. In particular for k = θ 0.5087. Te exact dispersion relation for Lorentz media [5 3] relating te wave number k to te angular frequency ω for te cosen values of parameters is (7.4 k = ω (ω τ + iω c 0 (ω τ + iω. Squaring bot sides te dispersion relation can be written as (7.5 (iω 4 τ (iω3 + ( + k (iω k τ (iω + k = 0. Comparing te disersion relation (7.5 to te relation (7. for real θ we note tat te exact solution (7. corresponds to a solution for te Maxwell-Lorentz system (.6 for a purely imaginary angular frequency ω = iθ. 7.. Relative and nergy rrors. For te discrete solution produced we compute relative errors defined as (7.6 RL (t n = (7.7 Relative rror = ( (t n n + H(t n H n H + (t n n + J (t n J n max 0 n N ( RL (t n L (t n were te grid norms and H are defined in (3.4 and (3.5 respectively. We also define te nergy rror for te discrete solutions as d L (7.8 nergy rror = max dt (tn+ δt n+ L 0 n N d L dt (tn+ ( were te discrete energy L n is defined in (5. and d L dt t n+ is te time derivative of te exact energy (7.3 computed at te time point t n+. Table 4 presents te Relative errors (7.7 and confirms te second order accuracy of te Yee sceme for various values of t k and ν. Here again N N wit N t = T and refers to te number of time iterations performed. Table 5 presents te nergy errors (7.8. Te results in tis table indicate tat te energy error decreases in a second order accurate manner and provides anoter confirmation of te second order accuracy of te Yee sceme.
30 V. A. BOKIL AND N. L. GIBSON Table 4. Relative rrors for te D Yee Maxwell-Lorentz sceme. k = π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50 4.43 0 4.6 0 4 8.45 0 5 00. 0 4.00 4.04 0 5.00.times0 5.00 00.76 0 5.00.0 0 5.00 5.7 0 6.00 400 6.90 0 5.00.5 0 6.00.3 0 6.00 800.7 0 6.00 6.30 0 6.00 3.9 0 7.00 k = 5π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50.4 0 3 8.89 0 4 4.49 0 4 00 6.4 0 4.98.9 0 4.0. 0 4.00 00.5 0 4.0 5.47 0 5.00.79 0 5.00 400 3.79 0 5.00.37 0 5.00 6.97 0 6.00 800 9.47 0 6.00 3.4 0 6.00.74 0 6.00 k = 0π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50 5.45 0 3.8 0 3 8.97 0 4 00.4 0 3.4 4.34 0 4.06.0 0 4.03 00 3.00 0 4.04.07 0 4.0 5.47 0 5.0 400 7.45 0 5.0.68 0 5.00.37 0 5.00 800.86 0 5.00 6.68 0 6.00 3.4 0 6.00
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 3 0 k =π 0 k =5π 0 k =0π Reference Reference Reference 0 0 0 0 3 0 3 0 3 Relative rror 0 4 0 5 0 4 0 5 0 4 0 5 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N Figure. Relative errors for different wave numbers (k x = k y = k in te D Yee Maxwell-Debye sceme for Courant numbers ν = 0.3 0.5 and 0.7. 0 k =π 0 k =5π 0 k =0π Reference Reference Reference 0 0 0 0 3 0 3 0 3 nergy rror 0 4 0 4 0 4 0 5 0 5 0 5 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N Figure. nergy errors for different wave numbers (k x = k y = k in te D Yee Maxwell-Debye sceme for Courant numbers ν = 0.3 0.5 and 0.7.
3 V. A. BOKIL AND N. L. GIBSON Table 5. nergy rrors for te D Yee Maxwell-Lorentz sceme. k = π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50.00 0 4 6.59 0 5 3.55 0 5 00 4.97 0 5.0.64 0 5.0 9.4 0 6.96 00.4 0 5.00 4.0 0 6.00.3 0 6.98 400 3.0 0 6.00.0 0 6.00 5.84 0 7.99 800 7.74 0 7.00.56 0 7.00.47 0 7.00 k = 5π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50 3.37 0 4.0 0 4 6.43 0 5 00 7.99 0 5.08.99 0 5.0.63 0 5.98 00.97 0 5.0 7.44 0 6.00 4.07 0 6.00 400 4.9 0 6.00.86 0 6.00.0 0 6.00 800.3 0 6.00 4.65 0 7.00.55 0 7.00 k = 0π N ν = 0.3 ν = 0.5 ν = 0.7 rror Rate rror Rate rror Rate 50 4.9 0 4. 0 4 6.9 0 5 00 8.86 0 5.8 3.3 0 5.87.75 0 5.98 00.3 0 5.99 8. 0 6.0 4.44 0 6.98 400 5.50 0 6.0.06 0 6.00. 0 6.00 800.37 0 6.0 5.3 0 7.00.78 0 7.00
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 33 Table 6. Discrete Divergence rrors for te D Yee Maxwell- Lorentz sceme. k = π N ν = 0.3 ν = 0.5 ν = 0.7 50.7 0 3.96 0 3 3.59 0 3 00 4.0 0 3 6.9 0 3.0 0 00. 0.05 0.83 0 400 3.34 0 5.78 0 8.06 0 800 9.8 0.6 0.30 0 k = 5π N ν = 0.3 ν = 0.5 ν = 0.7 50 9.59 0.35 0 4.37 0 00 3.39 0 7.80 0. 0 0 00.6 0 0.3 0 0 3.6 0 0 400 5.9 0 0 8.36 0 0 9.9 0 0 800.8 0 9.96 0 9.77 0 9 k = 0π N ν = 0.3 ν = 0.5 ν = 0.7 50.8 0 0.5 0 0.66 0 0 00 3.6 0 0 3.83 0 0 8.60 0 0 00.8 0 9.64 0 9.63 0 9 400.85 0 9 5.09 0 9 8.09 0 9 800 9.96 0 9.55 0 8.9 0 8
34 V. A. BOKIL AND N. L. GIBSON 0 k =π 0 k =5π 0 k =0π 0 Reference 0 Reference 0 Reference 0 3 0 3 0 3 Relative rror 0 4 0 5 0 4 0 5 0 4 0 5 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N Figure 3. Relative errors for different wave numbers (k x = k y = k in te D Yee Maxwell-Lorentz sceme for Courant numbers ν = 0.3 0.5 and 0.7. 0 k =π 0 k =5π 0 k =0π 0 Reference 0 Reference 0 Reference 0 3 0 3 0 3 nergy rror 0 4 0 4 0 4 0 5 0 5 0 5 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N 0 6 0 7 ν = 0.3 ν = 0.5 ν = 0.7 0 0 0 3 N Figure 4. nergy errors for different wave numbers (k x = k y = k in te D Yee Maxwell-Lorentz sceme for Courant numbers ν = 0.3 0.5 and 0.7.
CONVRGNC ANALYSIS OF Y SCHMS FOR DISRSIV MDIA 35 In Figures 3 and 4 we plot te relative errors (7.6 and te energy errors (7.8 respectively for te various values of N ν and k as presented in te corresponding tables. An O( reference is provided to visually see te second order accuracy of te Yee sceme. 7.3. Convergence Analysis of Discrete Divergence. Finally we verify te identity in (5.30 by computing te maximum absolute grid error in te discrete divergence as defined in (6.9. Table 6 presents te absolute errors in te discrete divergence of solutions to te D Yee Maxwell-Lorentz sceme for various values of t k and ν. Again N N wit N t = T refers to te number of time steps performed. All errors are sufficiently small to suggest tat tey are due to roundoff. 8. Conclusions In tis paper we ave presented an accuracy stability and convergence analysis of te Yee sceme for Maxwell s equations in Debye and Lorentz dispersive media using energy tecniques. Tis researc fills an important gap in te literature on Yee metods for dispersive media models by explicitly computing energy decay inequalities for tese metods wic aid in a convergence analysis of te numerical scemes. Our analysis assumes dispersive media parameters tat are constant. However te generality of te energy analysis will allow an extension of our results to te case of parameters tat are functions of space and/or time. We construct novel exact solutions for te Maxwell-Debye and Maxwell-Lorentz models tat torougly justify our numerical results. Tese exact solutions will also be elpful to justify analysis of oter numerical tecniques. Acknowledgements Te researc presented ere was partially funded by te NSF grant DMS # 083. References [] H. T. Banks M. W. Buksas and T. Lin lectromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts vol. Society for Industrial Matematics (SIAM 000. [] lectromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts vol. FR of Frontiers in Applied Matematics SIAM iladelpia A 000. [3]. Bécace and. Joly On te analysis of Bérenger s perfectly matced layers for Maxwell s equations Matematical Modelling and Numerical Analysis 36 (00 pp. 87 9. [4] B. Bidégaray-Fesquet Stability of FD-TD scemes for Maxwell-Debye and Maxwell- Lorentz equations SIAM J. Numer. Anal. 46 (008 pp. 55 566. [5] V. A. Bokil and N. L. Gibson Analysis of Spatial Hig-Order Finite Difference Metods for Maxwell s quations in Dispersive Media IMA J. Numer. Anal. 3 (0 pp. 96 956.