Von-Neumann Stability Analysis of FD--TD methods in complex … · 2007-12-18 · Von-Neumann...

Von-Neumann Stability Analysis of FD–TD methodsin complex media

B. Bidégaray-Fesquet

Laboratoire de Modélisation et de CalculCNRS, Grenoble, France

Electric and Magnetic Fields 2006

B. Bidégaray-Fesquet (LMC) Von Neumann stability analysis EMF 2006 1 / 27

Outline

1 MotivationComplex optical materialsFinite difference schemes and stability

2 Towards an automation of the stability analysisMathematical toolsComputations by handAutomation

Outline

Maxwell Equations

Light propagation is described by Maxwell equations

curl E = −∂B∂t

curl H =∂D∂t

div D = 0,

div B = 0.Material properties are described by the constitution laws of thematerial

D = D(E, H),

B = B(E, H),

J = J(E, H),

M = M(E, H).

Material description

Simple materials: D = εE, B = µH, J = σE et M = 0.Examples of complex (linear) materials:

anisotropic materials: ε, µ and σ are tensors.cold plasmas, collision-less warm plasmas, magneto-ionic media,magnetic ferrites.

Debye and Lorentz dielectrics

B = µ0H, M = 0, J = ∂P/∂t et D = ε0ε∞E + P.

Debye: τ∂P∂t

+ P = ε0(εs − ε∞)E.

Lorentz:∂2P∂t2 + ν

∂P∂t

+ ω21P = ε0(εs − ε∞)ω2

Outline

Finite difference schemes

To have explicit and 2-order schemes, models are based on Yeescheme

Bn+1/2j+1/2 − Bn−1/2

δt= −

Enj+1 − En

ε0ε∞En+1

j − Enj

δt= −

Bn+1/2j+1/2 − Bn+1/2

j−1/2

µ0δx− Jn+1/2

together with a scheme for matter equations (here Debye–Young)

τPn+1/2

j − Pn−1/2j

δt= −

Pn+1/2j + Pn−1/2

2+ ε0(εs − ε∞)En

τJn+1/2j = −Pn+1/2

j + ε0(εs − ε∞)En+1

j + Enj

Former results

Definition of FD–TD schemesYee (1966);Luebbers, Hunsberger, Kunz, Standler, Schneider (1990);Kashiwa, Yoshida, Fukai (1990);Joseph, Hagness, Taflove (1991);Luebbers, Steich, Kunz (1993);Young (1995);Young, Kittichartphayak, Kwok, Sullivan (1995)

Other types of schemesFeise, Schneider, Bevelaqua (2004, pseudo-spectral);Stoykov, Kuiken, Lowery, Tavlove (2003, FE);

Scheme analysisPetropoulos (1994);Young (2000)

Aim of the study

Motivations Petropoulos performs a stability analysis:1 Explicit form, amplification matrix computation,2 Characteristic polynomial computation,3 Choice of physical (ε∞, ...) and numerical (δt , δx) parameters,4 Numerical computation of roots for the characteristic

polynomial (localization in the unit circle).

A closed subject? Young “analyzes” the stability of all theabove-mentioned schemes using two types of “methods”:

“the entries for [...] have not been rigorously established”“the entry for [...] follows directly from the semi-implicit natureof the scheme”

false,not really implicit, but written with symmetries.

Aim of the study

Outline

Schur and von Neumann polynomials

DefinitionA polynomial is a Schur polynomial if all its roots r satisfy |r | < 1.

DefinitionA polynomial is a von Neumann polynomial if all its roots r satisfy|r | ≤ 1.

DefinitionA polynomial is a simple von Neumann polynomial if it is a vonNeumann polynomial and its roots of modulus 1 are simple.

Von Neumann analysis

TheoremA necessary stability condition is that the characteristic polynomial is avon Neumann polynomial.

TheoremA sufficient stability condition is that the characteristic polynomial is asimple von Neumann polynomial.

Intermediate cases have to be dealt specifically coming back to thestructure of the amplification matrix.(

1 00 1

(1 00 1

) (1 10 1

(1 n0 1

)stable unstable

Sequences of polynomials, I

DefinitionLet φ(z) = c0 + c1z + · · ·+ cpzp be a polynomial of degree p withcomplex coefficients. Its conjugate polynomial isφ∗(z) = c∗p + c∗p−1z + · · ·+ c∗0zp.

From a polynomial φ0, a (finite) sequence of decreasing degree isconstructed via the relation

φm+1(z) =φ∗m(0)φm(z)− φm(0)φ∗m(z)

TheoremA polynomial φm is a Schur polynomial of exact degree d iff φm+1 is aSchur polynomial of exact degree d − 1 and |φm(0)| < |φ∗m(0)|.

Sequences of polynomials, II

TheoremA polynomial φm is a simple von Neumann polynomial iff

either φm+1 is a simple von Neumann polynomial and|φm(0)| < |φ∗m(0)|,or φm+1 is identically zero and φ′m is a Schur polynomial.

Difficult problem localize roots of polynomial of degree 3 to 20(according to applications and space dimensions) with coefficientsdepending on 5 to 10 parameters.

Many “simpler” problemsfind vanishing conditions for the leading coefficient,verify |φm(0)| < |φ∗m(0)|.

Outline

Computation step: Maxwell–Debye–Joseph 1D

1 Find an explicit and dimensionless formulation

Unj = (c∞Bn−1/2

j+1/2 , Enj , Pn

j /ε0ε∞) = exp(iξj)Un.

λ = c∞δt/δx , δ = δt/2τ, ηs = εs/ε∞, σ = λ(eiξ − 1), q = |σ|2.2 Compute the amplification matrix: Un+1 = GUn

1 −σ 0(1+δ)σ∗

1+δηs

(1−δηs)−(1+δ)q1+δηs

2δ1+δηs

σ∗ −q 1

3 Compute the characteristic polynomial

P(Z ) ∝ φ0(Z ) = [1 + δηs]Z 3 − [3 + δηs − (1 + δ)q]Z 2

+[3− δηs − (1− δ)q]Z − [1− δηs].

Analysis step: Maxwell–Debye–Joseph 1D

4 Sequence of polynomials

φ1(Z ) = 2δ{2ηsZ 2 − [4ηs − (ηs + 1)q]Z + [2ηs − (ηs − 1)q]},φ2(Z ) = 24δ2(ηs − 1)q{[4ηs − (ηs + 1)q]Z − [4ηs − (ηs + 1)q]}.

5 Check |φm(0)| < |φ∗m(0)| and the degreem = 0: OK,m = 1: q ∈ [0, 4] is needed (CFL condition λ ≤ 1 for Yee scheme in1D). Equality if q = 0 or ηs = 1.m = 2: Equality if q = 4.

6 Specific casesq = 0: 1 is a double eigenvalue of G in two stable vectorsubspaces.ηs = 1: φ1 has two distinct complex roots if q ∈]0, 4[.q = 4 and ηs 6= 1: φ2 has -1 as only root.q = 4 and ηs = 1: G has only one eigendirection associated to thedouble eigenvalue -1 ⇒ instability!

Conclusion on the computation by hand

ResultsResults in accordance with the partial results of Petropoulos,but not with all those intuitive of Young.1D is tractable by hand.

Model Scheme CFL (1D)B_ED δt ≤ δx/c∞

Debye B_EP δt ≤ min(δx/c∞, 2τ)BP_E δt ≤ min(δx/c∞, 2τ)

B_ED δt ≤ δx/√

2c∞Lorentz B_EPJ δt < δx/c∞

BJ_EP δt ≤ min(δx/√

2c∞, 2/ω1√

2ηs − 1)2D is tractable by hand, thanks to the experience of 1D: 1D isa factor of the 2D.3D is not reasonable by hand.

New goals

Automate computations to grapple the 3D.

Explain factorizations of polynomials.

Outline

What is automated?

A MAPLE toolbox (collection of routines) has been written. Itautomates:

Maxwell equations.

Matter equations, generalization to 1D, 2D and 3D.

Suppress redundant equations.

Dimensionless formulation.

Explicit formulation.

Amplification matrix.

Characteristic polynomial.

Sequence of polynomials.

Tests |φm(0)| < |φ∗m(0)|.

What is not (already) automated?

Coupling Maxwell–Debye and Maxwell–LorentzVanishing leading coefficients.and hence : Choice of specific treatments.

Other couplingsDimensionless formulation.

Imperfect automationTests |φm(0)| < |φ∗m(0)|.Find the sign of a “high” degree polynomial with a largenumber of variables, positive (ηs, δ) or lying in an interval (q).According to the run, the result is always true but notnecessarily optimal.

Example: Maxwell–Debye–Young 1D, I

# Load toolboxesrestart; with(linalg);read Maxwell; read ChVar1D; read vonNeumann;# Definition of the modelDim := 1; Polar := ""; Formula := "B_EP"; Model := "MD";Eq[1] := Faraday1D;Eq[2] := Ampere1D_EBP;Eq[3] := tau*(P[n+1]-P[n])/dt+1/2*P[n+1]

-1/2*eps0*(epss-epsinfini)*(E[n+1]+E[n]);nbeq := 3;# Dimensionless formulationVar := CalcVar(Dim, Polar, Formula, Model);# Characteristic polynomialG := Amplif(Eq, nbeq, Var);phi[0] := Ampli2Poly(G, Z);save phi, "MD_1D_BEP.m";

Example: Maxwell–Debye–Young 1D, II

# Von Neumann SequencevonNeumann(phi, Z, nbeq);fact := facteurs(phi, Z, nbeq);rest := restes(phi, facts, Z, nbeq);# Sign study in the general case# Vanishing if delta=0, q=0 et etas=1verif(phi, Z, nbeq, [0 < delta, 1 < etas, 0 < q, q < 4]);# Specific case: delta = 0phidelta0[0] := eval(phi[0], delta = 0);psidelta0[0] := diff(phidelta0[0], Z);vonNeumann(psidelta0, Z, nbeq-1);verif(psidelta0, Z, nbeq-1, [1 < etas, 0 < q, q < 4]);# Specific case: q=0phiq0[0] := eval(phi[0], q = 0);vonNeumann(phiq0, Z, nbeq);psiq0[0] := diff(phiq0[1], Z);verif(psiq0, Z, 1, [0 < delta, 1 < etas]);Gq0 := eval(G, q = 0);

Conclusion

Stability conditions for FD–TD schemes for Maxwell–Debye andMaxwell–Lorentz systems.

Definition of a general strategy to analyze the stability of FD–TDschemes.

FutureExplain factorizations.Application to other complex models and other fields ofapplications.Distribute the toolbox via a web page.

Thanks for your attention!

Conclusion

Stability conditions for FD–TD schemes for Maxwell–Debye andMaxwell–Lorentz systems.

Definition of a general strategy to analyze the stability of FD–TDschemes.

FutureExplain factorizations.Application to other complex models and other fields ofapplications.Distribute the toolbox via a web page.

Thanks for your attention!

Von-Neumann Stability Analysis of FD--TD methods in complex … · 2007-12-18 · Von-Neumann...

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