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Numerical methods for the Vlasov equation
Eric Sonnendrucker
IRMAUniversite de Strasbourg
project-team CALVIINRIA Nancy Grand Est
September 13, 2011
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 1 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 2 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 3 / 107
Plasmas
When a gas is heated to a temperature of more than 104 K electronsleave their orbit around the nucleus to which they belong.
Gas becomes a globally neutral mixture of ions, electrons and neutralswhich is called plasma.
Plasmas are considered in addition to solid, liquid and gas to be thefourth state of matter.
99% of the mass of the universe consists of plasmas.
Many applications: satellite amplifiers, plasma etching inmicro-electronics, X-ray production, TVs, ...., thermonuclear fusion.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 4 / 107
Controlled thermonuclear fusion
Magnetic confinement (ITER)
Inertial confinement
laser (LMJ)heavy ions
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 5 / 107
The ITER project
International project involving European Union, China, India, Japan, SouthKorea, Russia and United States aiming to prove that magnetic fusion isviable source for energy.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 6 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 7 / 107
The Vlasov-Maxwell system
∂fs∂t
+ v · ∂fs∂x
+qs
ms(E + v × B) · ∂fs
∂v= 0, (1)
− 1
c2
∂E
∂t+∇× B = µ0 J,
∂B
∂t+∇× E = 0,
∇ · E =ρ
ε0,
∇ · B = 0.
with
ρ(x, t) =∑s
qs
∫fs(x, v, t) dv, J(x, t) =
∑s
qs
∫fs(x, v, t)v dv.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 8 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 9 / 107
One species Vlasov-Maxwell system
We consider only one species of particles and set all physical constants to1 for simpler computations. Domain periodic in x and whole space in vResults carry over to general model.
∂f
∂t+ v · ∇x f + (E + v × B) · ∇v f = 0, (2)
−∂E∂t
+∇× B = J, (3)
∂B
∂t+∇× E = 0, (4)
∇ · E = ρ, (5)
∇ · B = 0, (6)
with
ρ(x, t) =
∫f (x, v, t) dv, J(x, t) =
∫f (x, v, t)v dv.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 10 / 107
Characteristics of Vlasov equation
The solution of the Vlasov equation can be expressed with help ofcharacteristics which are the solutions of the differential system
dX
dt= V(t), (7)
dV
dt=
q
m(E(X(t), t) + V(t)× B(X(t), t)). (8)
Characteristics are denoted by
(X(t; x, v, s),V(t; x, v, s))
Solution of Vlasov equation then writes
f (x, v, t) = f0(X(0; x, v, t),V(0; x, v)).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 11 / 107
Conservative form of Vlasov equation
Vlasov equation can be written in conservative form
∂f
∂t+∇x,v · (Ff ) = 0, (9)
with F = (v,E + v × B)T such that ∇x,v · F = 0.
Conservation of total number of particles∫f (x, v, t) dx dv
follows by integrating in phase space.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 12 / 107
First conservation properties
Maximum principle
0 ≤ f (x, v, t) ≤ max(x,v)
(f0(x, v)). (10)
Follows fromf (x, v, t) = f0(X(0; x, v, t),V(0; x, v)).
Conservation of Lp norms
d
dt
(∫(f (x, v, t))p dx dv
)= 0 (11)
Multiply Vlasov eq. by f p−1 and integrate over phase space.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 13 / 107
Conservation of phase space volume
For any volume V of phase space∫V
f (x, v, t) dx dv =
∫F−1(V )
f0(y,u) dy du. (12)
Jacobian of flow is equal to one because phase space divergence ofadvection field
(v,E + v × B)T
vanishes.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 14 / 107
Conservation of momentum and energy
Momentum conservation
d
dt
[∫vf dxdv +
∫E× B dx
]=
d
dt
[∫J dx +
∫E× B dx
]= 0. (13)
Energy conservation
d
dt
[1
2
∫|v|2f dxdv +
1
2
∫(E 2 + B2) dx
]= 0. (14)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 15 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 16 / 107
Vlasov-Poisson model
The one species Vlasov equation reads
∂f
∂t+ v · ∇x f + E · ∇v f = 0. (15)
It is coupled to the Poisson equation
−∆φ = 1− ρ = 1−∫
f (t, x , v) dv , E = −∇φ.
1D Vlasov-Poisson equation first standard model.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 17 / 107
Specificities of Tokamaks
Large magnetic field.
Particle trajectories mostly confined along magnetic field lines.
Many instabilities of magnetized plasmas.
Toroidal geometry
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 18 / 107
The drift-kinetic model (in slab geometry)
Taking the limit of a large magnetic field we get a 4D model(r , θ, φ, v‖)
∂f
∂t+ vD · ∇x f + v‖ · ∇‖f +
q
mE‖ · ∇v f = 0,
with vD = E×BB2 . Poisson’s equation might be replaced by the
quasi-neutrality relation
−∇⊥ · (n0(r)
Bωc∇⊥φ) +
e n0(r)
Te(r)(φ− λ〈φ〉) = n − n0,
〈φ〉 representing the average of the potential over the magnetic fluxlines.
Guiding-center model as a starting point.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 19 / 107
The gyrokinetic model
It is a 5D model describing the evolution ofthe guiding center distributionf (r , θ, φ, v‖, µ)
∂f
∂t+ vD · ∇x f + v‖ · ∇‖f +
q
mE‖ · ∇v f = 0,
with vD = −∇J(φ)×BB2 , coupled with the quasi-neutrality equation
−∇⊥ · (n0(r)
Bωc∇⊥φ) +
e n0(r)
Te(r)(φ− λ〈φ〉) =
∫J(f ) dv‖dµ− n0.
The gyroaverage operator J transforms the guiding-center distributiononto the actual particle distribution enabling to take into account thefinite Larmor radius.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 20 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 21 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 22 / 107
Kelvin-Helmoltz instability in magnetized plasma (1/3)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 23 / 107
Kelvin-Helmoltz instability in magnetized plasma (2/3)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 24 / 107
Kelvin-Helmoltz instability in magnetized plasma (3/3)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 25 / 107
Numerical simulation of kinetic equations
Difficulties:
defined in phase space (4D, 5D or 6D).Appearance of small scales. Filamentation.
Particle methods: more efficient in high dimensions.
Good qualitative results at relatively low costNumerical noise and slow convergence → difficult to have accurateresults in some situations.
Methods using a grid of phase space:
Huge amount of computations in more than 3 dimensions.No numerical noise, but diffusion.Small scales at some point will not be resolved by the grid.Need to dissipate them for stability. How? Subgrid model?Entropy cannot be conserved for long time simulations
Fourier Transform in velocity space (Eliasson). Replace subgridresolution problem by boundary conditions to evacuate high modes.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 26 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 27 / 107
Particle Methods
Particle-In-Cell (PIC) method :
Idea: Follow particle trajectories, use grid field solve.Literature:
Physics (Birdsall-Langdon, Hockney-Eastwood)Mathematical analysis (Neunzert-Wick, Cottet-Raviart, Victory-Allen)
SPH type methods:
Idea: Compute interaction between finite sized macro-particles.Literature: (Bateson-Hewett)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 28 / 107
Discretization of the Vlasov equation by a particle method
Particle approximation of the Vlasov equation. Distribution functionis approximated by
fh(x , v , t) =∑k
wkδ(x − xk(t))δ(v − vk(t)).
Deterministic, pseudo-random or Monte-Carlo approximation of f0.
Once particles have been initialized, they are advanced usingdeterministic equations of motion
dxkdt
= vk ,dpk
dt= q(E (xk , t) + vk ∧ B(xk , t)).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 29 / 107
Discrete particle motion
Independent of mesh.
Once fields at particle positions are known, equations of motion aregenerally discretized using a Leap-Frog scheme.
xn+1k − xnk
∆t= v
n+ 12
k
vn+ 1
2k − v
n− 12
k
∆t=
q
m(En
k + vnk × Bnk)
Use Boris trick to avoid acceleration by magnetic field
v− = vn− 1
2k +
q
2mEn
v+ − v−
∆t=
q
2m(v+ + v−)× Bn
vn+ 12 = v+
k +q
2mEn
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 30 / 107
Coupling particles with fields through shape functions
Particle method defines point particles (Dirac masses).
Regularization by convolution with a finite width smoothing kernelgenerally called weighting function.
Splines of different orders on are generally used on structured grids
NGP CIC order 2
P1 Finite Element shape functions on unstructured grids.Generally fields obtained by Poisson or Maxwell field solver firstinterpolated at vertices of meshTruncated Gaussians have also been used by some authors(Jacobs-Hesthaven).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 31 / 107
Coupling with the Field solver: The Particle-In-Cell Method
Particle data scattered to surrounding grid pointsto compute charge and current densities usingweight function:
ρi = q∑k
S(xk − xi ), Ji = q∑k
S(xk − xi )vk .
Field solve performed on grid: E(xi ), B(xi ) arecomputed using some grid based field solver.
Fields are computed on particles usinginterpolation e.g. E(xk) =
∑S(xk − xi )E(xi ).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 32 / 107
Noise reduction techniques
Statistical noise reduction techniques can be applied:
δf methods. Decompose f = f 0 + δf and use particle approximationonly for perturbation. Decreases noise where δf << f 0.
Ef (Si )) = Ef 0(Si )) + Ef−f 0(Si ))
=
∫Si (x)f 0(x , v) dx dv +
∫Si (x)(f (x , v)− f 0(x , v)) dx dv
First part is known exactly, only second part is approximated bysample and its variance is a lot smaller if δf << f 0.
Very efficient when f stays close to known equilibrium as in sometokamak plasma simulations.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 33 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 34 / 107
The model
An abstract form of a kinetic equation reads
∂f
∂t+ a(z, t) · ∇f = 0,
z are the phase space coordinates, a depends non linearly and nonlocally on f , through e.g. the electromagnetic field.
The velocity field a is divergence free so that the equation can also bewritten in conservative form
∂f
∂t+∇ · (f a) = 0.
Mathematically for a given a the non conservative form is a transportequation which can be solved using the characteristics, which are thesolution of the ODE
dZ
dt= a(Z, t).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 35 / 107
Operator splitting
Consider e.g. the non relativistic Vlasov-Poisson equation
∂f
∂t+ v · ∇x f +
q
mE · ∇v f = 0.
We decompose the equation into the two following steps.
∂f
∂t+ v · ∇x f = 0, (16)
with v fixed and∂f
∂t+
q
mE(x, t) · ∇v f = 0, (17)
with x fixed.
We solve the two equations successively on one time step. At leastdimension reduction and in our example constant coefficientadvections for reduced equations.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 36 / 107
Analysis of splitting error
For constant coefficient differential operators A and B we consider
du
dt= (A + B)u
Formal solution on one time step reads
u(t + ∆t) = e∆t(A+B)u(t).
Split equations read
du
dt= Au,
du
dt= Bu.
with formal solutions on one time step
u(t + ∆t) = e∆tAu(t) et u(t + ∆t) = e∆tBu(t).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 37 / 107
Analysis of splitting error
If operators A and B commute split solution
u(t + ∆t) = e∆tBe∆tAu(t)
exact.
This is the case for constant coefficient transport not for generalVlasov.
Alternate solution of A and of B on substeps to get any given order.Negative time advance coefficients starting from third order splitting.
Coefficients are found by equating first terms of Taylor expansion ofexponential.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 38 / 107
First order splitting
Successive solution on one time step is first order in time.
e∆t(A+B) = I + ∆t(A + B) +∆t2
2(A + B)2 + O(∆t3),
e∆tBe∆tA = (I + ∆tB +∆t2
2B2 + O(∆t3))(I + ∆tA +
∆t2
2A2 + O(∆t3))
= I + ∆t(A + B)∆t2
2(A2 + B2 + 2BA) + O(∆t3)).
If A and B do not commute we have(A + B)2 = A2 + AB + BA + B2 6= A2 + B2 + 2BA.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 39 / 107
Second order splitting
Strang splitting is second order: Solve A on half time step, B on full timestep, again A on half time step.
e∆t(A+B) = I + ∆t(A + B) +∆t2
2(A + B)2 + O(∆t3),
e∆t2Ae∆tBe
∆t2A =(I +
∆t
2A +
∆t2
4A2 + O(∆t3)))(I + ∆tB +
∆t2
2B2
+ O(∆t3))(I +∆t
2A +
∆t2
4A2 + O(∆t3)))
=I + ∆t(A + B) +∆t2
2(A2 + B2 + BA + AB) + O(∆t3).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 40 / 107
Development of methods based on phase space grid
The semi-Lagrangian method (generalization of Cheng and Knorrsplitting method).
Vlasov solver based on WENO method (Carrillo-Vecil, Qiu-Chrislieb,Qiu-Shu, Banks-Hittinger).
Discontinuous Galerkin (Qiu-Shu, Gamba-Morrison, Rossmanith-Seal,Crouseilles-Mehrenberger-Vecil)
Conservative flux based methods : (Boris-Book, Fijalkow,Filbet-ES-Bertrand, Crouseilles-Mehrenberger-ES).
Energy conserving Finite Difference (Arakawa).
Fourier in velocity space (Eliasson).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 41 / 107
History of semi-Lagrangian schemes for Vlasov
Cheng-Knorr (JCP 1976): splitting method for 1D Vlasov-Poisson.Cubic spline interpolation.
ES-Roche-Bertrand-Ghizzo (JCP 1998): general semi-Lagrangianframework for Vlasov-type equations.
Nakamura-Yabe (CPC 1999): semi-Lagrangian CIP method withHermite interpolation
Filbet-ES-Bertrand (JCP 2001) : semi-Lagrangian PFC method:positive and conservative
N. Besse - ES (JCP 2003) : semi-Lagrangian solver on unstructuredgrids.
V. Grandgirard et al. (JCP 2006): drift-kinetic SL scheme.
Crouseilles-Respaud-ES (CPC 2009): Forward semi-Lagrangian
Crouseilles-Mehrenberger-ES (JCP 2010): Equivalence of point basedand conservative methods for Vlasov-Poisson + new class of filters.
Qiu-Christlieb (JCP 2010): conservative SL WENO schemes.
Qiu-Shu (JCP 2011): DG SL schemes.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 42 / 107
Convergence of semi-Lagrangian schemes for Vlasov
Filbet (SINUM 2001): PFC method for Vlasov-Poisson
N. Besse (SINUM 2003): semi-Lagrangian method with linearinterpolation for Vlasov-Poisson.
Campos Pinto - Mehrenberger (Numer. Math. 2008): adaptative SLmethod for Vlasov-Poisson
N. Besse - Mehrenberger (Math of Comp 2008): split SL method forVlasov-Poisson for symmetric high order interpolation Lagrange andspline interpolation.
N. Besse (SINUM 2008): SL method with Hermite interpolation andpropagation of gradients.
Respaud - ES (Numer math, 2011): Forward SL for Vlasov-Poissonwith linear interpolation.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 43 / 107
The backward semi-Lagrangian Method
f conserved along characteristics
Find the origin of the characteristics ending atthe grid points
Interpolate old value at origin of characteristicsfrom known grid values → High orderinterpolation needed
Typical interpolation schemes.
Cubic spline (Cheng-Knorr)Cubic Hermite with derivative transport (Nakamura-Yabe)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 44 / 107
Comparison of interpolation operators
Amplification factor (left) and phase error (right)
Quadratic Lagrange (yellow), cubic Lagrange (green), cubic Hermite(blue), cubic spline (red).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 45 / 107
Cubic spline interpolation
Let f ∈ C k([a, b]), k ≥ 0 its cubic spline interpolant fh on the knots(xi )i∈[0,N] is defined by
fh(xi ) = f (xi ) for i = 0, . . . ,N, fh ∈ P3([xi , xi+1]), fh ∈ C 2([a, b]).
fh uniquely defined provided boundary conditions are given : Hermite(f ′ given), natural (f ′′ = 0) or periodic.
fh computed via its decomposition on B−splines. On uniform mesh
S3(x) =1
6h
(2− |x |h )3 if h ≤ |x | < 2h,
4− 6(xh
)2+ 3
(|x |h
)3if 0 ≤ |x | < h,
0 else.
Solution of triadiagonal matrix with additional terms on boundary.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 46 / 107
Interpolation
Cubic spline interpolation originally proposed by Cheng and Knorr isstill our method of choice.
Other methods have been tried: different variants of Lagrange,Hermite, higher order splines. None has proved superior to cubicsplines for our applications.
Features needed by interpolation: accuracy and robustness. Needs todegrade well when distribution is not resolved on the mesh.
New implementations. Local splines
Series approximation of derivative on boundary: Crouseilles, Latu, ES:JCP 2007.Fast algorithm by Unser IEEE Trans on Pattern Analysis and MachineIntelligence, vol 13 (3), 1991 for signal processing. Choleskydecomposition with constant coefficients on diagonal and off-diagonal.Iterations started with series using directly f .
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 47 / 107
Computation of the origin of the characteristics
Transport equation∂f
∂t+ a · ∇f = 0,
CharacteristicsdX
dt= a
Computation of the origin of the characteristics :
Explicit solution if a does not depend on x and tElse, numerical algorithm needed.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 48 / 107
Splitting for exact computation of characteristics
In many cases splitting can enable to solve a constant coefficientadvection at each split step. Ideal case.
E.g. separable Hamiltonian H(q,p) = U(q) + V (p).
Vlasov equation in canonical coordinates reads
∂f
∂t+∇pH · ∇qf −∇qH · ∇pf = 0.
Split equations then become
∂f
∂t+∇pV · ∇qf = 0,
∂f
∂t−∇qU · ∇pf = 0,
where U does not depend on p and V does not depend on q⇒ characteristics can be solved explicitly.Vlasov-Poisson falls into this category with q = x, p = v,H(x, v) = 1
2 mv2 + qφ(x, t).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 49 / 107
Backward computation of characteristics in general case.
Consider general case. Characteristics defined by
dX
dt= a(X, t).
Backward solution: Xn+1 is known and an known on the grid.Does not allow standard ODE integrator.
Numerical method for EDO can be derived using a quadratureformula on RHS of system by integrating on one time step, e.g. leftor right rectangle rule for 1st order:
Xn+1 − Xn = ∆t an(Xn) or Xn+1 − Xn = ∆t an+1(Xn+1).
No explicit solution:
Fixed point procedure needed in first case (e.g. Newton).Predictor-corrector method on a needed in second case.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 50 / 107
A two step second order method
Solve characteristics defined by dXdt = a(X, t).
Centered quadrature on two time steps:
Xn+1 − Xn−1 = 2∆t an(Xn), Xn+1 + Xn−1 = 2Xn + O(∆t2).
Use fixed point procedure to compute Xn−1 such that
Xn+1 − Xn−1 = ∆t an(Xn+1 + Xn−1
2).
Problem: compute f n+1 from f n−1. Even and odd order timeapproximations become decoupled after some time. Artificial couplingneeds to be introduced.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 51 / 107
A one step predictor-corrector second order method
Solve characteristics defined by dXdt = a(X, t).
Centered quadrature on one time step:
Xn+1 − Xn = ∆t an+ 12 (Xn+ 1
2 ), Xn+1 + Xn = 2Xn+ 12 + O(∆t2).
Now an+ 12 is unknown. But can be computed with first order scheme
(like in Runge-Kutta methods) for global second order accuracy.Requires two updates of distribution function per time step.
Use fixed point procedure to compute Xn such that
Xn+1 − Xn = ∆t an+ 12 (Xn+1 + Xn
2).
In practice use linear interpolation for evaluation of an+ 12 (X ) to get
explicit solution for Xn.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 52 / 107
Conservativity
Consider abstract Vlasov equation where z are all the phase spacevariables
∂f
∂t+ a(z, t) · ∇z f = 0 with ∇ · a = 0.
The equation is conservative: ddt
∫f dz = 0.
Consider splitting the equations by decomposing the variables into z1
and z2. Then the split equations read
∂f
∂t+ a1(z, t) · ∇z1f = 0, and
∂f
∂t+ a2(z, t) · ∇z2f = 0.
We have ∇ · a = ∇z1 · a1 +∇z2 · a2 = 0, but in general ∇z1 · a1 and∇z2 · a2 do not vanish separately.
One or more of the split equations may not be conservative.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 53 / 107
Example 1: Vlasov-Poisson
In this case the Vlasov equation reads
∂f
∂t+ v · ∇x f + E · ∇v f = 0.
So a = (v,E(x, t))
Standard splitting yields:
∂f
∂t+ v · ∇x f = 0 and
∂f
∂t+ E · ∇v f = 0.
So that a1 = v and a1 = E(x, t).
In this case ∇x · a1 = 0 and ∇v · a2 = 0.
Splitting yields two conservative equations.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 54 / 107
Example 2: guiding-center model
Classical model for magnetized plasmas. Describes motion in planeperpendicular to magnetic field.
∂ρ
∂t+ vD · ∇ρ = 0, −∆φ = ρ,
vD =−∇φ× B
B2=
(−∂φ∂y∂φ∂x
)if B = ez unit vector in direction z .
The model is conservative: ∇ · vD = − ∂2φ∂x∂y + ∂2φ
∂y∂x = 0.
Split equations become
∂ρ
∂t− ∂φ
∂y
∂ρ
∂x= 0,
∂ρ
∂t+∂φ
∂x
∂ρ
∂x= 0.
In general ∂2φ∂x∂y 6= 0.
The split equations are not conservative.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 55 / 107
Problem with non conservative Vlasov solver
When non conservative splitting is used for the numerical solver, thesolver is not exactly conservative.
Does generally not matter when solution is smooth and well resolvedby the grid. The solver is still second order and yields good results.
However: Fine structures develop in non linear simulations and are atsome point locally not well resolved by the phase space grid.
In this case a non conservative solvers can exhibit a large numericalgain or loss of particles which is totally unphysical.
Lack of robustness.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 56 / 107
Vortex in Kelvin-Helmholtz instability
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 57 / 107
Conservative semi-Lagrangian method
Start from conservative form of Vlasov equation
∂f
∂t+∇ · (f a) = 0.∫
V f dx dv conserved along characteristics
Three steps:
High order polynomial reconstruction.Compute origin of cellsProject (integrate) on transported cell.
Efficient with splitting in 1D conservative equations as cells are thendefined by their 2 endpoints. A lot more complex for 2D (or more)transport.
Splitting on conservative form: always conservative.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 58 / 107
High order polynomial reconstruction
We only use the method with 1D splitting with equations inconservative form.
Unknowns are cell averages: fj = 1∆x f (x) dx .
At time step tn let f nj known average value of f n on cell [xj− 1
2, xj+ 1
2]
of length hj = xj+ 12− xj− 1
2.
Construct polynomial pm(x) of degree m such that
1
hj
∫ xj+ 1
2
xj− 1
2
pm(x) dx = f nj .
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 59 / 107
Reconstruction by primitive
To this aim look for pm(x) such that ddx pm(x) = pm(x). Then
hj fnj =
∫ xj+ 1
2
xj− 1
2
pm(x) dx = pm(xj+ 12)− pm(xj− 1
2).
Let W (x) =∫ xx 1
2
f n(x) dx a primitive of function f n with value f nj on
[xj− 12, xj+ 1
2]. Then W (xj+ 1
2) =
∑jk=1 hk f n
k and
W (xj+ 12)−W (xj− 1
2) = hj f
nj = pm(xj+ 1
2)− pm(xj− 1
2).
Take for pm interpolating polynomial at points xj+ 12
of W
1
hj
∫ xj+ 1
2
x 12
pm(x) dx =1
hj(pm(xj+ 1
2)− pm(xj− 1
2))
=1
hj(W (xj+ 1
2)−W (xj− 1
2)) = f n
j ,
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 60 / 107
Choice of interpolation
What interpolation should be chosen for primitive?
Lagrange interpolation with centered stencil (used in PFC Filbet, ES,Bertrand JCP 2001).
ENO type interpolation. Lagrange with varying stencil. Not efficientfor Vlasov. WENO has proven a good choice.
Cubic spline interpolation: cubic polynomial on each cell, globally C 2
→ reconstructed function is then locally a quadratic polynomial andglobally C 1. Linked to cubic spline interpolation for classicalsemi-Lagrangian method.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 61 / 107
Origin of cells and projection
Compute cell origins:
In 1D cell and its origin determined by end points. Compute origin ofend points like in classical semi-Lagrangian method.Need to make sure end points do not cross → restriction on time step.
Compute average value of f n+1 on cells using∫ xi+ 1
2
xi− 1
2
f n+1(x) dx =
∫ X (tn;xi+ 1
2,tn+1)
X (tn;xi− 1
2,tn+1)
f n(x) dx ,
where f n(x) is the high order reconstruction.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 62 / 107
Link between classical and conservative semi-Lagrangianmethods
For constant coefficient advections it can be shown that
C-Lag(2d) ⇐⇒ SL-Lag(2d+1)PSM ⇐⇒ SPL
Consequences :1 Classical and conservative semi-Lagrangian methods equivalent for
constant coefficients split equations.2 The PFC method (Filbet-ES-Bertrand, JCP 2001) corresponds for the
Vlasov-Poisson (or Vlasov-Maxwell) systems to a classicalsemi-Lagrangian method with cubic Lagrange interpolation.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 63 / 107
Positive reconstruction
Physical distribution function is always positive.
High-order interpolation can lead to negative values in some zones.
Reconstructed polynomial can be locally modified to remain positive.
Performed for Lagrange reconstruction in PFC method.
Introduces a little more dissipativity, but far less as monotonicreconstructions performed in fluid dynamics.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 64 / 107
Simulations
Evolution of L2 norm for N = 128 for SLD
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 65 / 107
Simulations
Evolution of L1 norm for N = 128 for SLD
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 66 / 107
Simulations
Evolution of total energy for N = 128 for SLD
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 67 / 107
Simulations
Evolution of total energy for N = 512 for SLD
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 68 / 107
The forward semi-Lagrangian method
f conserved along characteristics
Characteristics advanced with same timeschemes as in PIC method.
Leap-Frog Vlasov-Poisson
Runge-Kutta for guiding-center or gyrokinetic
Values of f deposited on grid of phase space using convolution kernel.
Identical to PIC deposition scheme but in whole phase space insteadof configuration space only.
Similar to PIC method with reconstruction introduced by Denavit(JCP 1972).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 69 / 107
Discrete distribution function
Function projected on partition of unity basis for conservativity.Linear B-splines very diffusive. Not useful in practice.Good choice is cubic B-splines.f is reprojected on mesh at each time step.Between tn and tn+1
fh(x , v , t) =∑i ,j
wi ,jB(x − X (t; xi , vj , tn))B(v − X (t; xi , vj , tn)).
The weight wi ,j associated to the particle starting from grid point(xi , vj) at tn is coefficient coefficient of spline satisfying interpolationconditions
fh(xk , vl , tn) =∑i ,j
wi ,jB(xk − xi )B(vk − vj).
Projection on phase space mesh is obtained with formula
f n+1(xk , vl) =∑i ,j
wi ,jB(xk−X (tn+1, xi , vj , tn))B(vl−V (tn+1, xi , vj , tn)).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 70 / 107
Time advance for Vlasov-Poisson
As opposite to BSL, trajectories are advanced forward in time.
Advection field is known at initial time. Standard EDO algorithmscan be applied.
For Vlasov-Poisson, we have z(tn) = (xn, vn), andE (tn, zn) = E (tn, xn). Separable Hamiltonian.
Natural scheme is Verlet algorithm, which is second order accurate intime
Step1 : vn+ 12 − vn =
∆t
2E (tn, xn),
Step2 : xn+1 − xn = ∆t vn+1/2,
Step3 : vn+1 − vn+ 12 =
∆t
2E (tn+1, xn+1).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 71 / 107
Time advance for guiding-center
Explicit Euler and Runge Kutta have been implemented.Second-order Runge Kutta method
Step1 : X n+1 − X n = ∆tE⊥(tn,X n)
Step2 : Compute E⊥(tn+1,X n+1)
Step3 : X n+1 − X n =∆t
2
[E⊥(tn,X n) + E⊥(tn+1,X n+1)
]Fourth order Runge-Kutta
Step1 : k1 = E⊥(tn,X n)
Step2 : Compute k2 = E⊥(tn+ 12 ,X n +
∆t
2k1)
Step3 : Compute k3 = E⊥(tn+ 12 ,X n +
∆t
2k2)
Step4 : Compute k4 = E⊥(tn+1,X n + ∆tk3)
Step5 : X n+1 − X n =∆t
6[k1 + 2k2 + 2k3 + k4]
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 72 / 107
Landau damping
f0(x , v) = (1 + 0.001 cos(kx))1√2π
e−v2
2 , L = 4π.
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 73 / 107
Bump on tail (BSL (top) vs. FSL (bottom))
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 74 / 107
Bump on tail: potential energy (BSL (top) vs. FSL(bottom))
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 75 / 107
Energy conservation for Kelvin-Helmoltz instability
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Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 76 / 107
FSL vs. BSL
Forward semi-Lagrangian method very promising.
Accurate description of whole phase-space in particular tail ofdistribution function and small perturbations.
A little more diffusive than BSL. Better with smaller time steps.
Some advantages: classical explicit EDO solver can be used, inparticular high order if needed. No need for predictor-corrector orfixed point algorithm.
Can benefit from charge conserving PIC algorithms(Villasenor-Buneman).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 77 / 107
A L2-norm conserving Finite Difference scheme
Introduced by Arakawa (1966) for equations of the form
∂f
∂t+ J(ψ, f ) = 0,
for 1D Vlasov ψ = ϕ− v2
2 and J(ψ, f ) = ∂ψ∂x
∂f∂v −
∂ψ∂v
∂f∂x ,
Second and fourth order implemented:
Particle conservation: ddt
∫R2 f (t) dx dv = 0.
Energy conservation: ddt
∫R2 f (t)ψ(t) dx dv = 0.
Conservation of ‖f ‖L2 .
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 78 / 107
Stabilization of the method
Strong oscillations in presence of filamentation.
Collision model of the form ∂f∂t = ∂J
∂v , where J is chosen so as tomaximize entropy and conserve as many moments
∫fvk dv as desired
(Robert-Sommeria).
A Lagrange multiplier technique yields (for k = 2):
∂f
∂t+ J(ψ, f ) = α
∂
∂v
(∂f
∂v+ A1f − A2fv
),
with A1 = u0
ε−u20/n
, and A2 = nε−u2
0/n.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 79 / 107
Evolution of L1 and L2 norms
Figure: Time development of numerical L1 and L2 norms for the non linearLandau damping test.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 80 / 107
Evolution of L2 norm and kinetic entropy
Figure: Time development of numerical L2 norm and entropy of f (t) for the twostream instability test.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 81 / 107
Computation time
Num. method 32 × 32 pts 32 × 64 pts 32 × 128 pts
FBM 03.33 sec. 05.39 sec. 10.80 sec.
PFC 03.56 sec. 06.28 sec. 11.20 sec.
FDM 17.22 sec. 35.27 sec. 71.20 sec.
SPECTRAL 04.10 sec. 08.25 sec. 16.90 sec.
CIP 13.83 sec. 21.40 sec. 43.24 sec.
SL SPLINE 06.12 sec. 10.55 sec. 20.90 sec.
SL HERMITE 03.60 sec. 06.90 sec. 11.00 sec.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 82 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 83 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 84 / 107
General ideas
Monitor conservation of invariants:
Number of particles:∫
f (x , v , t) dxdv should be conserved exactlyTotal energyLp norms:
∫|f (x , v , t)|p dxdv in particular L1, L2, L∞.
Analytical solution of equations linearized around an equilibrium.
Analytical solution of linear Vlasov equation: only external field, noself-consistent field.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 85 / 107
Outline
1 Plasma physics
2 ModelsConservation properties of Vlasov-Maxwell systemReduced models
3 Numerical solution of the Vlasov equationIntroductionParticle MethodsPhase space grid-based methods
4 Code validationGeneral considerationsSolution of linearized Vlasov-Poisson
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 86 / 107
Maxwellian equilibrium
1D Vlasov-Poisson. Periodic domain of length L = 2π/k0 in x . WholeR in v . −e charge of electron and m its mass.
Equilibrium distribution solution of Vlasov-Poisson
∂f 0
∂t+ v
∂f 0
∂x− e
mE 0(x)
df 0
dv= 0 (18)
dE 0
dx=
e
ε0(n0 −
∫ +∞
−∞f 0(x , v) dv) (19)
with intial condition f 0(x , v , t) = f0(x , v) and where
n0 = 1L
∫ L0
∫ +∞−∞ f 0(x , v) dx dv is the uniform background neutralizing
density.
Equilibrium f 0(x , v , t) = f0(x , v) = f 0(v). Solution
f 0(v) =n0
2πvthe− v2
2v2th .
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 87 / 107
Linearization of Vlasov-Poisson around equilibrium
Hilbert expansion around equilibriumf (x , v , t) = f 0(x , v) + εf 1(x , v , t), E (x , t) = E 0(x) + εE 1(x , t) (withE 0(x) = 0).
Plug into Vlasov-Poisson, use fact that f 0 is solution and neglectO(ε2) terms.
Linearized Vlasov-Poisson equation around Maxwellian equilibriumreads
∂f 1
∂t+ v
∂f 1
∂x− e
mE 1(x)
df 0
dv= 0, (20)
dE 1
dx= − e
ε0
∫ +∞
−∞f 1(x , v , t) dv , (21)
with initial condition f 1(x , v , 0) = f 10 (x , v).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 88 / 107
Fourier series in x
Consider x 7→ g(x) continuous and L−periodic. It can be expandedinto a Fourier series defined by
g(x) =+∞∑
k ′=−∞g(k)e ikx , g(k) =
1
L
∫ L
0g(x)e−ikx dx , k =
2π
Lk ′.
Apply to f 1. Multiply linearized Vlasov and Poisson by e−ikx andintegrate on one period:
∂ f
∂t(k , v , t) + ikv f (k , v , t)− e
mE (k, t)
df 0
dv= 0, (22)
ikE (k, t) = − e
ε0
∫ +∞
−∞f (k , v , t) dv . (23)
Initial condition satisfies f (k , v , 0) = f0(k, v).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 89 / 107
Laplace transform in time
Let ω ∈ C and define Laplace transform by
f (ω) =
∫ +∞
0f (t)e iωt dt for =(ω) > R, (24)
Inverse Laplace transform defined by
f (t) =
∫ +∞+iu
−∞+iuf (ω)e−iωt dω ∀u > R, (25)
such that u analytical in the half space =(ω) > R
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 90 / 107
Application to linearized Vlasov-Poisson
Multiplying ∂ f∂t (k , v , t) by e iωt and integrating with respect to t
between 0 and +∞, we have∫ +∞
0
∂ f
∂t(k , v , t)e iωt dt = [f (k , v , t)e iωt ]+∞0 − iω
∫ +∞
0f (k , v , t)e iωt dt
= −f (k, v , 0)− iωf (k, v , ω),
where we denote by f (k , v , ω) the Laplace transform of f (k , v , t).
Whence, Laplace transform of Vlasov and Poisson
(−iω + ikv)f (k, v , ω)− e
mE (k, ω)
df 0
dv= f0(k , v), (26)
E (k, ω) =ie
kε0
∫ +∞
−∞f (k, v , ω) dv . (27)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 91 / 107
Expression of E (k , ω)
Plug the value of f from Vlasov into Poisson
E =ie
kε0
∫ +∞
−∞
f0(k , v) + em E df 0
dv
−iω + ikvdv =
ie2
kε0mE
∫ +∞
−∞
df 0
dv
−iω + ikvdv +
ie
kε0
∫ +∞
−∞
f0(k , v)
−iω + ikvdv .
Let
D(k , ω) = 1− e2
k2ε0m
∫ +∞
−∞
df 0
dv
v − ωk
dv , N(k , ω) =e
k2ε0
∫ +∞
−∞
f0(k , v)
v − ωk
dv .
Then
E (k , ω) =N(k , ω)
D(k , ω), f (k , v , ω) = i
em E (k , ω)df
0
dv + f0(k , v)
(ω − kv).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 92 / 107
Computation of velocity integrals
In the velocity integrals in D and N with have terms of the form
G (ω) =
∫ +∞
−∞
g(v)
v − ωk
dv .
G (ω) is analytical (assuming g is) for =(ω) > 0.
Inverse Laplace transform well defined in this case.
Pole for =(ω) = 0. We need to extend the function to negative valuesof =(ω) by analytical continuation.
Integral in complex plane does not depend on contour as long as thereis no pole between chose contours → instead of choosing real line ascontour for =(ω) > 0 any line unerneath the real line can be chosen.
For analytical continuation for =(ω) ≤ 0 just take integration lineparallel to real line strictly below ω or any deformation of this contourstaying underneath pole.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 93 / 107
Examples of contours
Contour for pole on real line (left), underneath real line (right)
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 94 / 107
The plasma dispersion function
As plasma equilibria are strongly linked to Maxwellians, there is a functionthat naturally appears in calculation of dispersion equations for kineticequations.Fried and Conte called it the plasma dispersion function:
Z (ζ) =1√π
∫γ
e−z2
z − ζdz , (28)
where γ is any contour passing below the pole ζ.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 95 / 107
Properties of Z function
Z verifies the following properties
Z (ζ) =1√π
[Pr
∫ +∞
−∞
e−(u+ζ)2
udu + iπe−ζ
2
], (29)
=√πe−ζ
2[i − erfi(ζ)], (30)
where erfi(ζ) = 2√π
∫ ζ0 et
2dt is the complex error function. For b ∈ R
Pr
∫ +∞
−∞
g(u)
u − bdu = lim
δ→0
[∫ b−δ
−∞
g(u)
u − bdu +
∫ +∞
b+δ
g(u)
u − bdu
]is the Cauchy principal value.The derivatives of Z verify:
Z ′(ζ) = −2(1 + ζZ (ζ)),
Z ′′(ζ) = 4ζ − 2(1− 2ζ2)Z (ζ).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 96 / 107
Expression of E (x , t)
We got an exact expression for N and D and therefore of E (and f ).
The electric field can be computed by inverse Laplace and Fouriertransform of this expression.
Inverse Laplace transform
E (k , t) =1
2iπ
∫ +∞+iu
−∞+iuE (k , ω)e−iωt dω.
Can be computed using the residue theorem assuming that E (k , ω) isanalytical apart from a finite number of poles. Then
E (k , t) =∑j
Resω=ωj (E (k , ω))e−iωj t ,
the sum being made over the poles which can be computednumerically.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 97 / 107
Computation of the residue
E (k , t) =∑j
Resω=ωj E (k , ω)e−iωt , with E (k , ω) =N(k , ω)
D(k, ω)
and the ωj are the roots of D(k , ω) = 0 for fixed k. There are in generalseveral roots of D(k , ω) for a given k . Only the one with largest imaginarypart matters after some time.The residue can be computed thanks to the Taylor expansion of D(k , ω) ina neighborhood of ωj
D(k , ω) = D(k , ωj) + (ω − ωj)∂D
∂ω(k , ωj) + O((ω − ωj)
2),
and so, if ωj is a simple root, we have D(k, ωj) = 0 and ∂D∂ω (k , ωj) 6= 0. So
that
Resω=ωj
(N(k , ω)
D(k, ω)e−iωt
)= lim
ω→ωj
((ω−ωj)N(k , ω)
D(k , ω)e−iωt) =
N(k , ωj)∂D∂ω (k , ω)
e−iωj t .
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 98 / 107
Landau damping
Initial condition
f0(x , v) = (1 + ε cos(kx))1√2π
e−v2
2 , L = 4π.
So that f 0(v) = 1√2π
e−v2
2 and initial perturbation
f 10 (x , v) = ε cos(kx)) 1√
2πe−
v2
2 Charge density is
ρ10(x) =
∫f 10 (x , v) dv = ε cos(kx) = ε
e ikx + e−ikx
2.
Only 1 and −1 modes are not 0. The electric field verifiesdEdx (x , 0) = ρ1
0(x), so that E (x , 0) = εk sin(kx). We have
D(k , ω) = 1− 1
2
ω2p
k2v 2th
Z ′(ω√
2vthk),
N(k , ω) = g(k)n0e
k2ε0
1√2vth
Z (1√
2πvth).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 99 / 107
So that
N(k , ω)∂D∂ω (k , ω)
= −2g(k)m
ekv 2
th
Z ( ω√2vthk
)
Z ′′( ω√2vthk
)(31)
= −g(k)m
ekv 2
th
Z ( ω√2vthk
)
2 ω√2vthk
− (1− ω2
v2thk
2 )Z ( ω√2vthk
). (32)
Dominant roots of dispersion relation
k ωjN(k,ωj )∂D∂ω
(k,ωj )
0.5 ±1.4156− 0.1533i 0.3677 e±i0.536245
0.4 ±1.2850− 0.0661i 0.424666 e±i0.3357725
0.3 ±1.1598− 0.0126i 0.63678 e±i0.114267
0.2 ±1.0640− 5.510× 10−5i 1.129664 e±i0.00127377
Denote by ωr = <(ωj), ωi = =(ωj), r the amplitude ofN(k,ωj )∂D∂ω
(k,ωj )and ϕ its
phase.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 100 / 107
Linear solution of Landau damping
Maple code for dispersion relation
k := 0.2;
dd := w− > 1 + (1/k ∗ ∗2) ∗ (1 + sqrt(Pi/2) ∗ w/k
∗ exp(−w ∗ ∗2/(2 ∗ k ∗ ∗2)) ∗ (−erfi(w/(sqrt(2) ∗ k)) + I ));
root2 := fsolve(dd(w),w = 1, complex);
Dominant solution for E
E (k , t) ≈ re iϕe−i(ωr+iωi )t + re−iϕe−i(−ωr+iωi )t ,
= reωi t(e−i(ωr t−ϕ) + e i(ωr t−ϕ)),
= 2reωi t cos(ωr t − ϕ).
Then as E (−k, t) = −E (k, t), we have
E (x , t) ≈ 4εreωi t sin(kx) cos(ωr t − ϕ).
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 101 / 107
Analytical solution and computed solution for k=0.4
f0(x , v) = (1 + 0.001 cos(kx))1√2π
e−v2
2 , L = 4π.
Dominant analytical solution
E (k , t) = 0.002× 0.424666e0.0661t cos(1.2850t − 0.3357725)
-20
-18
-16
-14
-12
-10
-8
-6
0 5 10 15 20 25 30 35 40 45 50
’Runs/thek.dat’log(0.002*0.42466*abs(cos(1.285*x-0.33577))*exp(-0.0661*x))
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 102 / 107
Two stream instability (1/2)
Dispersion relation
D(k , ω) = 1 +ω2p
2k2v 2th
[2+√π
2
(ω
vthk− v0
vth
)e−
( ωk−v0)2
2v2th (i − erfi(
ωk − v0√
2vth)
√π
2
(ω
vthk+
v0
vth
)e−
( ωk
+v0)2
2v2th (i − erfi(
ωk + v0√
2vth)
,
Some roots for k = 0.2
v0 ω ω
1.3 0.02115i 1.1648− 0.00104i2.4 0.2258i 1.3390− 0.00242i3.0 0.2845i 1.446− 0.00299i
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 103 / 107
Two stream instability (2/2)
Maple code
k := 0.2; v0 := 3;
dd := w− > 1 + (1/(2 ∗ k ∗ ∗2)) ∗ ((1 + sqrt(Pi/2) ∗ (w/k − v0)
∗ exp(−(w/k − v0) ∗ ∗2/2) ∗ (−erfi((w/k − v0)/sqrt(2)) + I ))
+ (1 + sqrt(Pi/2) ∗ (w/k + v0) ∗ exp(−(w/k + v0) ∗ ∗2/2)
∗ (−erfi((w/k + v0)/sqrt(2)) + I )));
root2 := fsolve(dd(w),w = 0, complex);
root3 := fsolve(dd(w),w = 1, complex);
Initial condition
f0(x , v) = (1 + 0.001 cos(kx))1
2√
2π(e−
(v−v0)2
2 + e−(v+v0)2
2 ), L =2π
k.
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 104 / 107
k=0.2; v0=1.3
-12
-11
-10
-9
-8
-7
-6
-5
0 10 20 30 40 50
’Runs/thek.dat’ u 1:2-0.001*x -6.2
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 105 / 107
k=0.2; v0=2.4
-12
-10
-8
-6
-4
-2
0
2
4
0 10 20 30 40 50
’Runs/thek.dat’ u 1:20.2258*x -8.4
Eric Sonnendrucker (Univ. Strasbourg) Numerical methods for the Vlasov equation September 13, 2011 106 / 107
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