Asymptotic preserving schemes (AP)for magnetically conﬁned...

C. Negulescu, June 2012

1

Asymptotic preserving schemes (AP) formagnetically confined plasmas

Lecture I

Claudia Negulescu

Institut de Mathématiques de Toulouse

Université Paul Sabatier


2Introduction/Motivation

Plasma

➠ gas of charged particles: electrons, ions, neutral atoms

➠ π λα σ µα → modulable substance

➠ 99% of the universe: stars, intergalactic medium,magnetosphere, ionosphere, aurora borealis, lightenings

➠ collective effects play an important role, interaction viaelectromagnetic forces

➠ diff. behaviour as neutral gases: short range interactions

➠ far from equilibrium, highly anisotropic, turbulent environ.

Earth plasmas/Applications

➠ fusion reactors (energy production)

➠ fluorescent lamps, plasma displays (light production)


3Introduction/Motivation

Earth plasmas/Applications

➠ satellites/rockets propulsion

➠ air flow control (aviation)

➠ sterilization, water cleaning (biology, medicine)

➠ semiconductor technology (switches, transistors)

➠ plasma lasers, weapons, gas discharges, ...

Mathematics

➠ modeling(link between mathematicians and physicists)

➠ analysis(ex/uniqueness/long time behaviour/properties)

➠ numerical analysis(error estimates/design of newschemes)

➠ scientific computing(implementation, phys. validation)


4Nuclear power

➠ nuclear force : binds the nucleons

➠ atomic nuclei mass < sum of the masses of the constitutingnucleons

➠ “lost” mass→ binding energy (E = mc2)

➠ binding energy/nucleon :→ low for light and heavy atomic nuclei→ high for average atomic nuclei (max. forFe)

➠ nuclear reactions to create average atomic nuclei→ production of nuclear energy

➠ two types of nuclear reactions :fusionandfission


5Thermonuclear fusion

→ Lawson’s criterion for successful fusion:

nτET ∼ 3 ∗ 1021m−3s keV

→ Under these conditions:Plasma tends to disperse and cool down

Requirements for fusion:

• suff. high temperaturesT (10− 15keV )

→ to overcome the repulsive Coulomb barrier

• suff. high densitiesn

→ to ensure that collisions take place

• suff. confinement timesτEτE := W

Ploss= plasma energy content

lossed power

→ to ensure the plasma heating by fusion

products


6The Sun

Three confinement principles:

➠ gravitational:stars of suff. high mass (Sun) are contractedso by gravitational forces, that the cond. for fusion occur

➠ magnetic:particles are trapped in a strong magn. field

➠ inertial: plasma/fuel capsule is compressed by laser beams

SUN:• highly complexe dynamical system (dynamoeffect/turbulences)

• solar activity has direct impact on Earth’smagnetosphere and ionosphere (solar wind)


7Magn. conf. fusion reactors

• helicoidal magnetic field linestwisted around toric mag. surfaces

• particle trajectories:gyration around the mag. field lineswith gyration radius :ρ = mv⊥

eB

• macroscopic quantitiesn, T , pare homogenous on a mag. surface

• confinement: equilibrium ofplasma pressure and mag. pressure

β = nTB2/2µ0

<< 1

→ superior bound on the density


8Instabilities / turbulence

• Plasma dynamics is governed by turbulent transport• Understanding plasma turbulences is crucial for the successof nuclear fusion→ more complexe than in neutral fluids: electromagn. fields

• Degradation of the confinementdue to the high density,temp., pressure gradients

➠ transverse transp. due to collisions between particles (1%)

➠ transverse transp. due to various micro-instabilities (99%)Rayleigh-Benard, Kevin-Helmholtz, ITG/ETG(ion/electron temperature gradient driven turbulence)

• Stabilizing effects:

➠ velocity shear, magnetic shear

➠ sheath conductivity


9Kinetic description of plasmas

• Vlasov eq.governs the dynamics of the particle distribution fct.

f(t, x, v) (6D phase-space)

∂tfα + v · ∇xfα +eα

m(E + v ×B) · ∇vfα = 0 , α : ions/electr.

• Electromagn. fields are calculated selfconsist. fromMaxwell’s eq.

∇·E =1

ε0ρ , −

1

c2∂tE+∇×B = µ0j , ∇·B = 0 , ∂tB+∇×E = 0

Convenience:

➠ Plasmas are only slightly collisional⇒ far from equilib.

➠ Necessary to treat Landau resonances, trapped particles,

turbulence, plasma oscillations

Difficulties: High dimensionality→ 6D phase-space


10Fluid description of plasmas

• Plasma dynamics described by means of fluid variablesn(t, x), u(t, x),

T (t, x), satisfying theconservation lawscoupled toMaxwell’s eq.

∂tnα +∇ · (nαuα) = Snα , α : ions/electr.

mαnα [∂tuα + (uα · ∇)uα] = nαeα(E + uα ×B)−∇ · Pα +Rα ,

3

2nαkB [∂tTα + (uα · ∇)Tα] = −∇ · qα − Pα : ∇uα +Qα ,

Advantages:

➠ Reduce the dimensionality to3D space

Drawback:

➠ Fluid closures are not always adequate

➠ Fluid models over-estimate the transport level


11Hybrid Kinetic/Fluid models

Plasma dynamics is characterized by multi-scale phenomena

Kinetic modelsFluid models

Hybrid models

τpe τce τpi τci τa τcs τei

Lρe ρi

τa: Alfen wave period

τcs: Ion sound period

τei: Electr-ion collision time

τpe,pi : Inv. electr./ion plasma freq.

τce,ci: Electr./ion cyclotron period

λD : Debye length

ρe,i: Electr./ion Larmor radius

c: sound speed

δe δi

δe,i = c/ωpe,pi : Electr./ion skin depth

ωpe,pi: Electr./ion plasma frequency

λD


12Multi-scale problems

Exhibit a large varity of space and times scales

Cosmic

Hemisphere

wind

Time

Wind

Ray

15km

Ionosphere

Troposphere

100km

SouthernHemisphere

NorthernS

ea level

Ev. of natural plasma bubble

at altitude∼ 700km


13Multi-scale problems

A small-scale numerical simulation is out of reach

➠ requires mesh-sizes dependent on small scale param.ε ≪ 1

➠ excessive computational time and memory space are needed tocapture small scales

It is not always of interest to resolve the details at the smallscale. Multi-scale strategies are much more adequate!

➠ homogeneisation, domain decomposition, multi-grids, multi-scalemethods based on wavelets or finite elements, multi-scalevariational methods

Essential feature of these methods

➠ capture efficiently the large scale behavior of the solution, withoutresolving the small scale features


14Asymptotic Preserving schemes

Difficulty: Resolution of multiscale pb. can be very difficult, ifthe pb. becomes singular, as one of the parametersε → 0

➠ (P ε) sing. perturbed pb. of sol.fε

➠ the seq.fε converges towardsf0, sol. of a limit pb.(P 0)

➠ the limit pb. (P 0) is different in type from the initial(P ε)

➠ standard schemes would require∆t,∆x ∼ ε for stability

Definition: A schemeP ε,h is AP iff it is convergent forh → 0uniformely inε, i.e.

P ε,h P ε

P 0,h P 0

ε→

0

ε→

0

h → 0

h → 0


15Asymptotic Preserving schemes

AP-procedure:

➠ requires that the limit problem(P 0) is identified andwell-posed

➠ requires a sufficient degree of implicitness (not obvious)

➠ consists in trying to mimic at discrete level the asymptoticbehaviour of the sing. perturbed pb. sol.fε

Advantages:

➠ gives accurate and stable results, with no restrictions onthe computational mesh

➠ enables to capture automatically the Limit modelP 0, ifε → 0 (micro-macro transition)

➠ no more coupling needed, ifε(x) is variable


16Kinetic models and specific limit regimes

• Fundamental kinetic model: Vlasov/Boltzmann equation

∂tf + v · ∇xf +q

m(E + v ×B) · ∇vf = Q(f)

Several small scales/parameters occur, leading to diff. regimes:• Hydrodynamic scaling[Filbet/Jin; Dimarco/Pareschi]

∂tf + v · ∇xf =1

εQ(f)

➠ 0 < ε ≪ 1: mean free path (Knudsen nbr.)

➠ in the limit ε → 0, one gets the compressible Euler eq.

➠ AP-scheme: Decomposition of the source term in stiff-and non-stiff part

Q(f)

ε=

Q(f)− P (f)

ε+

P (f)

ε



• Drift-Diffusion scaling[Klar; Lemou/Mieussens]

∂tf +1

ε(v · ∇xf +∇xΦ · ∇vf) +

1

ε2Q(f) = G

➠ 0 < ε ≪ 1: mean free path

➠ in the limit ε → 0, one gets the Drift-Diffusion model

➠ AP-scheme: Micro-Macro decomp.f = ρM + εg

• Vlasov-Poisson quasi-neutral limit[Belaouar;Crouseilles;Degond;Sonnendrucker;Navoret;Vignal]

∂tf + v∂xf + ∂xΦ∂vf = 0

−λ2∂xxΦ = 1− ρ

➠ 0 < λ ≪ 1: rescaled Debye length

➠ AP-scheme: Reformulation of the Poisson equation



• Vlasov-Maxwell quasi-neutral limit[Degond/Deluzet/Doyen]

• High-field limit, strong magn. fields[Bostan, Golse, Saint-Raymond]

∂tf +1

εv(p) · ∇xf −

1

ε(E + v(p)×

B

ε) · ∇vf = 0

➠ 0 < ε ≪ 1: rescaled cyclotronic period

➠ in the limit ε → 0, one gets the guiding-center approx.

➠ asymptotical analysis:• Study of the dominant operatorT := (v(p)× B) · ∇v

• Projection of the eq. onker T = averaging along thecharact. flow associated toT

➠ construction of AP-scheme mimics this asymp. analysis


19Fluid models and specific limit regimes

• Euler-Poisson quasi-neutral limit[Crispel/Degond/Vignal]

∂tn+∇ · (nu) = 0

∂t(nu) +∇ · (nu⊗ u) +∇p(n) = n∇Φ

−λ2∆Φ = 1− n

➠ 0 < λ ≪ 1: rescaled Debye length

• High-field limit, Euler-Lorentz[Brull;Degond;Deluzet;Mouton;Sangam;Vignal]

∂tn+∇ · (nu) = 0

∂t(nu) +∇ · (nu⊗ u) +1

τ∇p(n) =

1

τn (E + u× B)

➠ 0 < τ ≪ 1: rescaled gyro-period


20Fluid models and specific limit regimes

• Low Mach-nbr. limit [Degond/Tang; Cordier/Degond/Kumbaw]

∂tn+∇ · (nu) = 0

∂t(nu) +∇ · (nu⊗ u) +1

ε2∇p(n) = 0

➠ 0 < ε ≪ 1: rescaled Mach-nbr.

➠ in the limit ε → 0, one gets the incompressible Euler eq.

➠ AP-scheme: Stiff term is decomposed as

1

ε2∇p(n) = α∇p(n) +

1− αε2

ε2∇p(n)

• Highly anisotropic temperature eq.[Lozinski/Mentrelli/Narski/Negulescu]

∂tT −1

ε∇|| · (K||∇||T )−∇⊥ · (K⊥∇⊥T ) = 0


21Outline of the talk

• Kinetic models

➠ Boltzmann eq. in drift-diffusion limit

➠ Boltzmann eq. in hydrodynamic limit

➠ Vlasov-Poisson eq. in quasi-neutral limit

➠ Vlasov eq. in high field limit, variable Larmor radii

• Fluid models

➠ anisotropic elliptic eq. (electric potential)

➠ anisotropic, nonlinear, degenerate parabolic eq.(temperature)

Asymptotic preserving schemes (AP)for magnetically conﬁned...

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