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18th and 25th February, 2014 UCD - p. 1
Finite Difference Time Domain (FDTD)methods for solution of Maxwell’s equations
Case Study in Simulation Sciences
Dr. Eugene KashdanApplied and Computational Mathematics Group,
School of Mathematical Sciences, University College Dublin,
ekashdan@maths.ucd.ie
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 2
Maxwell’s Equations
The Maxwell equations in an isotropic medium are:
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 2
Maxwell’s Equations
The Maxwell equations in an isotropic medium are:
∂ ~B
∂t+ ∇× ~E = 0, (Faraday’s Law)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 2
Maxwell’s Equations
The Maxwell equations in an isotropic medium are:
∂ ~B
∂t+ ∇× ~E = 0, (Faraday’s Law)
∂ ~D
∂t−∇× ~H = − ~J, (Ampere’s law)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 2
Maxwell’s Equations
The Maxwell equations in an isotropic medium are:
∂ ~B
∂t+ ∇× ~E = 0, (Faraday’s Law)
∂ ~D
∂t−∇× ~H = − ~J, (Ampere’s law)
coupled with Gauss’ law
∇ · ~B = 0 (magnetic field)
∇ · ~D = ρ (electric field),
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 2
Maxwell’s Equations
The Maxwell equations in an isotropic medium are:
∂ ~B
∂t+ ∇× ~E = 0, (Faraday’s Law)
∂ ~D
∂t−∇× ~H = − ~J, (Ampere’s law)
coupled with Gauss’ law
∇ · ~B = 0 (magnetic field)
∇ · ~D = ρ (electric field),
where ~J = σ ~E is electric current density, ρ is total electriccharge density, and the constitutive relations are given by~B = µ ~H and ~D = ε ~E.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 3
Properties of materials
µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N
A2 is free space permeability.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 3
Properties of materials
µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N
A2 is free space permeability.
ε – dielectric permittivity.ε = ε0 · εr, ε0 = 1
c2µ0is free space permittivity in
[Fm
].
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 3
Properties of materials
µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N
A2 is free space permeability.
ε – dielectric permittivity.ε = ε0 · εr, ε0 = 1
c2µ0is free space permittivity in
[Fm
].
In general, µr and εr are frequency dependent. Materialswithout such dependence are called "the simple materials".
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 3
Properties of materials
µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N
A2 is free space permeability.
ε – dielectric permittivity.ε = ε0 · εr, ε0 = 1
c2µ0is free space permittivity in
[Fm
].
In general, µr and εr are frequency dependent. Materialswithout such dependence are called "the simple materials".
µr =1 for almost all simple materials except metals (perfectelectric conductors – PEC).
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 3
Properties of materials
µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N
A2 is free space permeability.
ε – dielectric permittivity.ε = ε0 · εr, ε0 = 1
c2µ0is free space permittivity in
[Fm
].
In general, µr and εr are frequency dependent. Materialswithout such dependence are called "the simple materials".
µr =1 for almost all simple materials except metals (perfectelectric conductors – PEC).
εr ≥ 1 is discontinuous at the interface between materials.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 3
Properties of materials
µ – magnetic permeability.µ = µ0 · µr, where µ0 = 4π · 10−7 N
A2 is free space permeability.
ε – dielectric permittivity.ε = ε0 · εr, ε0 = 1
c2µ0is free space permittivity in
[Fm
].
In general, µr and εr are frequency dependent. Materialswithout such dependence are called "the simple materials".
µr =1 for almost all simple materials except metals (perfectelectric conductors – PEC).
εr ≥ 1 is discontinuous at the interface between materials.
σ is electrical conductivity in[
Sm
], which represents conducting
properties of material.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 4
Integral form
∂
∂t
x
S
~B · dS = −∮
C
~E · dl (Faraday’s Law)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 4
Integral form
∂
∂t
x
S
~B · dS = −∮
C
~E · dl (Faraday’s Law)
∂
∂t
x
S
~D · dS =
∮
C
~H · dl −x
S
~J · dS (Ampere’s Law)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 4
Integral form
∂
∂t
x
S
~B · dS = −∮
C
~E · dl (Faraday’s Law)
∂
∂t
x
S
~D · dS =
∮
C
~H · dl −x
S
~J · dS (Ampere’s Law)
Gauss’ law:{
S
~D · dS =y
V
ρdV (electric field)
{
S
~B · dS = 0 (magnetic field)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 5
Equations in scalar form
In Cartesian coordinates, in 3D, Maxwell’s equations are
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 5
Equations in scalar form
In Cartesian coordinates, in 3D, Maxwell’s equations are
ε∂Ex
∂t=
∂Hz
∂y− ∂Hy
∂zµ
∂Hx
∂t= −∂Ez
∂y+
∂Ey
∂z
ε∂Ey
∂t=
∂Hx
∂z− ∂Hz
∂xµ
∂Hy
∂t= −∂Ex
∂z+
∂Ez
∂x
ε∂Ez
∂t=
∂Hy
∂x− ∂Hx
∂yµ
∂Hz
∂t= −∂Ey
∂x+
∂Ex
∂y
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 5
Equations in scalar form
In Cartesian coordinates, in 3D, Maxwell’s equations are
ε∂Ex
∂t=
∂Hz
∂y− ∂Hy
∂zµ
∂Hx
∂t= −∂Ez
∂y+
∂Ey
∂z
ε∂Ey
∂t=
∂Hx
∂z− ∂Hz
∂xµ
∂Hy
∂t= −∂Ex
∂z+
∂Ez
∂x
ε∂Ez
∂t=
∂Hy
∂x− ∂Hx
∂yµ
∂Hz
∂t= −∂Ey
∂x+
∂Ex
∂y
We assume that the medium is loss-free (J = 0) and ε and µ arenot time-dependent)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 6
Transverse mode
■ Transverse electric (TE) modes: no electric field in thedirection of propagation.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 6
Transverse mode
■ Transverse electric (TE) modes: no electric field in thedirection of propagation.
■ Transverse magnetic (TM) modes: no magnetic field in thedirection of propagation.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 6
Transverse mode
■ Transverse electric (TE) modes: no electric field in thedirection of propagation.
■ Transverse magnetic (TM) modes: no magnetic field in thedirection of propagation.
■ Transverse electromagnetic (TEM) modes: neither electricnor magnetic field in the direction of propagation.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 6
Transverse mode
■ Transverse electric (TE) modes: no electric field in thedirection of propagation.
■ Transverse magnetic (TM) modes: no magnetic field in thedirection of propagation.
■ Transverse electromagnetic (TEM) modes: neither electricnor magnetic field in the direction of propagation.
■ Hybrid modes: non-zero electric and magnetic fields in thedirection of propagation.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 6
Transverse mode
■ Transverse electric (TE) modes: no electric field in thedirection of propagation.
■ Transverse magnetic (TM) modes: no magnetic field in thedirection of propagation.
■ Transverse electromagnetic (TEM) modes: neither electricnor magnetic field in the direction of propagation.
■ Hybrid modes: non-zero electric and magnetic fields in thedirection of propagation.
In rectangular waveguides, modes are marked as TEmn,where m is the number of half-wavelengths across the width ofthe waveguide and n – across the height of the waveguide.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 6
Transverse mode
■ Transverse electric (TE) modes: no electric field in thedirection of propagation.
■ Transverse magnetic (TM) modes: no magnetic field in thedirection of propagation.
■ Transverse electromagnetic (TEM) modes: neither electricnor magnetic field in the direction of propagation.
■ Hybrid modes: non-zero electric and magnetic fields in thedirection of propagation.
In rectangular waveguides, modes are marked as TEmn,where m is the number of half-wavelengths across the width ofthe waveguide and n – across the height of the waveguide.
Example: a radio wave in a hollow metal waveguide must havezero tangential electric field amplitude at the walls of thewaveguide, so the transverse pattern of the electric field ofwaves is restricted to those that fit between the walls
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 7
Maxwell’s equations in 2D
We choose z as transverse direction:
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 7
Maxwell’s equations in 2D
We choose z as transverse direction:
■ TE mode
ε∂Ex
∂t=
∂Hz
∂y, ε
∂Ey
∂t= −∂Hz
∂x,
µ∂Hz
∂t= −∂Ey
∂x+
∂Ex
∂y.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 7
Maxwell’s equations in 2D
We choose z as transverse direction:
■ TE mode
ε∂Ex
∂t=
∂Hz
∂y, ε
∂Ey
∂t= −∂Hz
∂x,
µ∂Hz
∂t= −∂Ey
∂x+
∂Ex
∂y.
■ TM mode
µ∂Hx
∂t= −∂Ez
∂y, µ
∂Hy
∂t=
∂Ez
∂x,
ε∂Ez
∂t=
∂Hy
∂x− ∂Hx
∂y.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 8
Yee Algorithm
"Classical" FDTD method (Yee, 1966) uses the second ordercentral difference scheme for integration in space and thesecond order Leapfrog scheme for integration in time. This is astaggered non-dissipative scheme both in space and in time.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 8
Yee Algorithm
"Classical" FDTD method (Yee, 1966) uses the second ordercentral difference scheme for integration in space and thesecond order Leapfrog scheme for integration in time. This is astaggered non-dissipative scheme both in space and in time.
A 1D space-time chart of the Yee algorithm
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 8
Yee Algorithm
"Classical" FDTD method (Yee, 1966) uses the second ordercentral difference scheme for integration in space and thesecond order Leapfrog scheme for integration in time. This is astaggered non-dissipative scheme both in space and in time.
A 1D space-time chart of the Yee algorithm
E∆t∆x−E
0∆x
∆t=
H0.5∆t1.5∆x−H
0.5∆t0.5∆x
∆x
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 8
Yee Algorithm
"Classical" FDTD method (Yee, 1966) uses the second ordercentral difference scheme for integration in space and thesecond order Leapfrog scheme for integration in time. This is astaggered non-dissipative scheme both in space and in time.
A 1D space-time chart of the Yee algorithm
E∆t∆x−E
0∆x
∆t=
H0.5∆t1.5∆x−H
0.5∆t0.5∆x
∆x
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 9
Spatial location in 3D
In Cartesian coordinates and three dimensions we have thefollowing spatial distribution of the components:
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 9
Spatial location in 3D
In Cartesian coordinates and three dimensions we have thefollowing spatial distribution of the components:
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 10
Discretization – electric field
Et+∆tx,(i+ 1
2,j,k)
= Etx,(i+ 1
2,j,k) +
∆t
εi+ 12,j,k
[(Ht+∆t
2
z,(i+ 12,j+ 1
2,k)
− Ht+∆t
2
z,(i+ 12,j− 1
2,k)
∆y
)
(Ht+∆t
2
y,(i+ 12,j,k 1
2)− H
t+∆t2
y,(i+ 12,j,k− 1
2)
∆z
)]
Et+∆ty,(i,j+ 1
2,k)
= Ety,(i,j+ 1
2,k) +
∆t
εi,j+ 12,k
[(Ht+∆t
2
x,(i,j+ 12,k+ 1
2)− H
t+∆t2
x,(i,j+ 12,k− 1
2)
∆z
)
(Ht+∆t
2
z,(i+ 12,j+ 1
2,k)
− Ht+∆t
2
z,(i− 12,j+ 1
2,k)
∆x
)
Et+∆tz,(i,j,k+ 1
2)= Et
z,(i,j,k+ 12) +
∆t
εi,j,k+ 12
[(Ht+∆t
2
y,(i+ 12,j,k+ 1
2)− H
t+∆t2
y,(i− 12,j,k+ 1
2)
∆x
)
(Ht+∆t
2
x,(i,j+ 12,k+ 1
2)− H
t+∆t2
x,(i,j− 12,k+ 1
2)
∆y
)]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
● Yee Algorithm
● Spatial location in 3D
● Discretization – electric field
● Discretization – magnetic field
Divergence-free
Numerical stability
18th and 25th February, 2014 UCD - p. 11
Discretization – magnetic field
Ht+ 3∆t
2
x,(i,j+ 12,k+ 1
2)= H
t+∆t2
x,(i,j+ 12,k+ 1
2)+
∆t
µi,j+ 12,k+ 1
2
×
[(Et+∆ty,(i,j+ 1
2,k+1)
− Et+∆ty,(i,j+ 1
2,k)
∆z
)
−(Et+∆t
z,(i,j,k+ 12)− Et+∆t
z,(i,j+1,k+ 12)
∆y
)]
Ht+ 3∆t
2
y,(i+ 12,j,k+ 1
2)= H
t+∆t2
y,(i+ 12,j,k+ 1
2)+
∆t
µi+ 12,j,k+ 1
2
×
[(Et+∆tz,(i+1,j,k+ 1
2)− Et+∆t
z,(i,j,k+ 12)
∆x
)
−(Et+∆t
x,(i+ 12,j,k+1)
− Et+∆tx,(i+ 1
2,j,k)
∆z
)]
Ht+ 3∆t
2
z,(i+ 12,j+ 1
2,k)
= Ht+∆t
2
z,(i+ 12,j+ 1
2,k)
+∆t
µi+ 12,j+ 1
2,k
×
[(Et+∆tx,(i+ 1
2,j+1,k)
− Et+∆tx,(i+ 1
2,j,k)
∆y
)
−(Et+∆t
y,(i+1,j+ 12,k)
− Et+∆ty,(i,j+ 1
2,k)
∆x
)]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
● No charges, no current:
∂ρ∂t
+ ∇ · J = 0
● Applying Yee algorithm
● Concept of order of accuracy
Numerical stability
18th and 25th February, 2014 UCD - p. 12
No charges, no current: ∂ρ∂t
+ ∇ · J = 0
{
Yee cell
~D · dS =ε0
∂t
(
Ex,(i,j+ 12,k+ 1
2) − Ex,(i−1,j+ 1
2,k+ 1
2)
)
∆y∆z
︸ ︷︷ ︸
Term 1
+
ε0
∂t
(
Ey,(i− 12,j+1,k+ 1
2) − Ey,(i− 1
2,j,k+ 1
2)
)
∆x∆z
︸ ︷︷ ︸
Term 2
+
ε0
∂t
(
Ez,(i− 12,j+ 1
2,k+1) − Ez,(i− 1
2,j+ 1
2,k)
)
∆x∆y
︸ ︷︷ ︸
Term 3
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
● No charges, no current:
∂ρ∂t
+ ∇ · J = 0
● Applying Yee algorithm
● Concept of order of accuracy
Numerical stability
18th and 25th February, 2014 UCD - p. 13
Applying Yee algorithm
T1 =
(Hz,(i,j+1,k+ 1
2) − Hz,(i,j,k+ 1
2)
∆y−
Hy,(i,j+ 12,j,k+1) − Hy,(i,j+ 1
2,j,k)
∆z
)
−(
Hz,(i−1,j+1,k+ 12) − Hz,(i−1,j,k+ 1
2)
∆y
−Hy,(i−1,j+ 1
2,j,k+1) − Hy,(i−1,j+ 1
2,j,k)
∆z
)
. . .
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
● No charges, no current:
∂ρ∂t
+ ∇ · J = 0
● Applying Yee algorithm
● Concept of order of accuracy
Numerical stability
18th and 25th February, 2014 UCD - p. 13
Applying Yee algorithm
T1 =
(Hz,(i,j+1,k+ 1
2) − Hz,(i,j,k+ 1
2)
∆y−
Hy,(i,j+ 12,j,k+1) − Hy,(i,j+ 1
2,j,k)
∆z
)
−(
Hz,(i−1,j+1,k+ 12) − Hz,(i−1,j,k+ 1
2)
∆y
−Hy,(i−1,j+ 1
2,j,k+1) − Hy,(i−1,j+ 1
2,j,k)
∆z
)
. . .After substitution:
{
Yee cell
~D · dS = (T1)∆y∆z + (T2)∆x∆z + (T3)∆x∆y = 0
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
● No charges, no current:
∂ρ∂t
+ ∇ · J = 0
● Applying Yee algorithm
● Concept of order of accuracy
Numerical stability
18th and 25th February, 2014 UCD - p. 14
Concept of order of accuracy
According to the Lax-Richtmyer Equivalence Theorem , if a finitedifference scheme has a truncation error of order (p, q) and thescheme is stable, then the difference between the analyticsolution and the numerical solution in appropriate norm is oforder (∆t)p + hq for all finite time.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
● No charges, no current:
∂ρ∂t
+ ∇ · J = 0
● Applying Yee algorithm
● Concept of order of accuracy
Numerical stability
18th and 25th February, 2014 UCD - p. 14
Concept of order of accuracy
According to the Lax-Richtmyer Equivalence Theorem , if a finitedifference scheme has a truncation error of order (p, q) and thescheme is stable, then the difference between the analyticsolution and the numerical solution in appropriate norm is oforder (∆t)p + hq for all finite time.
∂u
∂x(i∆x, j∆y, k∆z, n∆t) =
uni+1/2,j,k − un
i−1/2,j,k
∆x+ O[(∆x)2] ,
∂u
∂t(i∆x, j∆y, k∆z, n∆t) =
un+1/2i,j,k − u
n−1/2i,j,k
∆t+ O[(∆t)2]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
● No charges, no current:
∂ρ∂t
+ ∇ · J = 0
● Applying Yee algorithm
● Concept of order of accuracy
Numerical stability
18th and 25th February, 2014 UCD - p. 14
Concept of order of accuracy
According to the Lax-Richtmyer Equivalence Theorem , if a finitedifference scheme has a truncation error of order (p, q) and thescheme is stable, then the difference between the analyticsolution and the numerical solution in appropriate norm is oforder (∆t)p + hq for all finite time.
∂u
∂x(i∆x, j∆y, k∆z, n∆t) =
uni+1/2,j,k − un
i−1/2,j,k
∆x+ O[(∆x)2] ,
∂u
∂t(i∆x, j∆y, k∆z, n∆t) =
un+1/2i,j,k − u
n−1/2i,j,k
∆t+ O[(∆t)2]
However, there is no equivalent of the Lax-Richtmyer Theore m thatextends these results to approximation of equations with va riablecoefficients and includes boundary conditions and forcing f unctions ∗.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 15
TM mode
Hn+1/2x,(i,j) − H
n−1/2x,(i,j)
∆t= − 1
µ
(Enz,(i,j+ 1
2)− En
z,(i,j− 12)
∆y
)
Hn+1/2y,(i,j) − H
n−1/2y,(i,j)
∆t=
1
µ
(Enz,(i+ 1
2,j)
− Enz,(i− 1
2,j)
∆x
)
En+1z,(i,j) − En
z,(i,j)
∆t=
1
ε
[(Hn+1/2
y,(i+ 12,j)
− Hn+1/2
y,(i− 12)
∆x
)
−(H
n+1/2
x,(i,j+ 12)− H
n+1/2
x,(i,j− 12)
∆y
)]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 16
Time eigenvalue problem
Introduce the generic vector component
Vn+1/2(i,j) − V
n−1/2(i,j)
∆t= ΛV n
(i,j)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 16
Time eigenvalue problem
Introduce the generic vector component
Vn+1/2(i,j) − V
n−1/2(i,j)
∆t= ΛV n
(i,j)
and define a solution growth factor
qi,j =V
n+1/2(i,j)
V n(i,j)
=V n
(i,j)
Vn−1/2(i,j)
for all n.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 16
Time eigenvalue problem
Introduce the generic vector component
Vn+1/2(i,j) − V
n−1/2(i,j)
∆t= ΛV n
(i,j)
and define a solution growth factor
qi,j =V
n+1/2(i,j)
V n(i,j)
=V n
(i,j)
Vn−1/2(i,j)
for all n.
The goal: |qi,j | ≤ 1 to avoid uncontrolled growth and blow-upfor all points (i, j).
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 17
Condition of temporal stability
After substitution and excluding V n(i,j):
q2i,j − Λ∆tqi,j − 1 = 0
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 17
Condition of temporal stability
After substitution and excluding V n(i,j):
q2i,j − Λ∆tqi,j − 1 = 0
The solution:
qi,j =Λ∆t
2+
√(
Λ∆t
2
)2
+ 1 = α +√
α1 + 1
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 17
Condition of temporal stability
After substitution and excluding V n(i,j):
q2i,j − Λ∆tqi,j − 1 = 0
The solution:
qi,j =Λ∆t
2+
√(
Λ∆t
2
)2
+ 1 = α +√
α1 + 1
|qi,j | = 1 always if Re[α] = 0 and −1 ≤ Im[α] ≤ 1; henceα = i · Im[α] and Λ = i · Im[Λ].
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 17
Condition of temporal stability
After substitution and excluding V n(i,j):
q2i,j − Λ∆tqi,j − 1 = 0
The solution:
qi,j =Λ∆t
2+
√(
Λ∆t
2
)2
+ 1 = α +√
α1 + 1
|qi,j | = 1 always if Re[α] = 0 and −1 ≤ Im[α] ≤ 1; henceα = i · Im[α] and Λ = i · Im[Λ].
In this case
−1 ≤ Im[Λ]∆t
2≤ 1 ⇒ − 2
∆t≤ Im[Λ] ≤ 2
∆t
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 18
Space eigenvalue problem
We isolate space differentiation operation:
− 1
µ
(Ez,(i,j+ 1
2) − Ez,(i,j− 1
2)
∆y
)
= ΛHx,(i,j)
1
µ
(Ez,(i+ 1
2,j) − Ez,(i− 1
2,j)
∆x
)
= ΛHy,(i,j)
1
ε
[(Hy,(i+ 1
2,j) − Hy,(i− 1
2)
∆x
)
−(
Hx,(i,j+ 12) − Hx,(i,j− 1
2)
∆y
)]
= ΛEz,(i,j)
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 18
Space eigenvalue problem
We isolate space differentiation operation:
− 1
µ
(Ez,(i,j+ 1
2) − Ez,(i,j− 1
2)
∆y
)
= ΛHx,(i,j)
1
µ
(Ez,(i+ 1
2,j) − Ez,(i− 1
2,j)
∆x
)
= ΛHy,(i,j)
1
ε
[(Hy,(i+ 1
2,j) − Hy,(i− 1
2)
∆x
)
−(
Hx,(i,j+ 12) − Hx,(i,j− 1
2)
∆y
)]
= ΛEz,(i,j)
and look for solution in form of the plane wave with kx and ky arethe components of wave-vector in x and y directions:
Ez,(I,J) = Ez0exp[i(kxI∆x + kyJ∆y)]
Ex,(I,J) = Hx0exp[i(kxI∆x + kyJ∆y)]
Ey,(I,J) = Hy0exp[i(kxI∆x + kyJ∆y)]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 19
Condition of spatial stability
After substitution and applying Euler’s Identity:
Hx0=
2iEz0
λµ∆ysin
(ky∆y
2
)
, Hy0=
2iEz0
λµ∆xsin
(kx∆x
2
)
Ez0=
2i
Λε
[2iHy0
∆xsin
(kx∆x
2
)
− 2iHx0
∆ysin
(ky∆y
2
)]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 19
Condition of spatial stability
After substitution and applying Euler’s Identity:
Hx0=
2iEz0
λµ∆ysin
(ky∆y
2
)
, Hy0=
2iEz0
λµ∆xsin
(kx∆x
2
)
Ez0=
2i
Λε
[2iHy0
∆xsin
(kx∆x
2
)
− 2iHx0
∆ysin
(ky∆y
2
)]
After substituting Hx0and Hy0
and eliminating Ez0we obtain:
Λ2 = − 4
µε
[1
(∆x)2sin2
(kx∆x
2
)
+1
(∆y)2sin2
(ky∆y
2
)]
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 19
Condition of spatial stability
After substitution and applying Euler’s Identity:
Hx0=
2iEz0
λµ∆ysin
(ky∆y
2
)
, Hy0=
2iEz0
λµ∆xsin
(kx∆x
2
)
Ez0=
2i
Λε
[2iHy0
∆xsin
(kx∆x
2
)
− 2iHx0
∆ysin
(ky∆y
2
)]
After substituting Hx0and Hy0
and eliminating Ez0we obtain:
Λ2 = − 4
µε
[1
(∆x)2sin2
(kx∆x
2
)
+1
(∆y)2sin2
(ky∆y
2
)]
Using the fact that Re[Λ] = 0 and the bounds of sin(.) we write:
−2c
√
1
(∆x)2+
1
(∆y)2≤ Im[Λ] ≤ 2c
√
1
(∆x)2+
1
(∆y)2,
where c = 1√
εµ is a speed of light.
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 20
Enforcing stability of Yee algorithm
2c
√
1
(∆x)2+
1
(∆y)2≤ 2
∆t⇒ ∆t ≤ 1
c√
1(∆x)2 + 1
(∆y)2
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 20
Enforcing stability of Yee algorithm
2c
√
1
(∆x)2+
1
(∆y)2≤ 2
∆t⇒ ∆t ≤ 1
c√
1(∆x)2 + 1
(∆y)2
When ∆x = ∆y = ∆:
∆t =∆
c√
2
● Maxwell’s Equations
● Properties of materials
● Integral form
● Equations in scalar form
● Transverse mode
● Maxwell’s equations in 2D
FDTD methods
Divergence-free
Numerical stability
● TM mode
● Time eigenvalue problem
● Condition of temporal stability
● Space eigenvalue problem
● Condition of spatial stability
● Enforcing stability of Yee
algorithm
18th and 25th February, 2014 UCD - p. 20
Enforcing stability of Yee algorithm
2c
√
1
(∆x)2+
1
(∆y)2≤ 2
∆t⇒ ∆t ≤ 1
c√
1(∆x)2 + 1
(∆y)2
When ∆x = ∆y = ∆:
∆t =∆
c√
2
In practical applications, we always intend to keep ∆t as large as possible tominimize the computational time, but its value should be always less (at leastslightly) than the upper bound to avoid instability caused by the “numericaljunk” accumulation.
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