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arXiv:1609.03053v1 [math.NA] 10 Sep 2016
GEMPIC: Geometric ElectroMagnetic
Particle-In-Cell Methods
Michael Kraus1,2, Katharina Kormann1,2, Philip J. Morrison3,
Eric Sonnendrücker1,2
1
Max-Planck-Institut für Plasmaphysik
Boltzmannstraße 2, 85748 Garching, Deutschland
2
Technische Universität München, Zentrum Mathematik
Boltzmannstraße 3, 85748 Garching, Deutschland
3
Department of Physics and Institute for Fusion Studies
The University of Texas at Austin, Austin, TX, 78712, USA
September 10, 2016
Abstract
We present a novel framework for Finite Element Particle-in-Cell methods based
on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell
system. We derive a semi-discrete Poisson bracket, which satisfies the Jacobi identity, and apply Hamiltonian splitting schemes for time integration. Techniques from
Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss’ law as well as stability of the field solver. The resulting
methods are gauge-invariant, feature exact charge conservation and show excellent
long-time energy and momentum behavior.
1
Contents
1 Introduction
3
2 The Vlasov-Maxwell System
2.1 Noncanonical Hamiltonian Structure . . . . . . . . . . . . . . . . . . . . .
2.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5
7
3 Finite Element Exterior Calculus
8
3.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Finite Element Spaces of Differential Forms . . . . . . . . . . . . . . . . . 9
3.3 Finite Element Discretization of Maxwell’s Equations . . . . . . . . . . . . 10
4 Discretization of the Hamiltonian Structure
4.1 Discretization of the Functional Field Derivatives .
4.2 Discretization of the Functional Particle Derivatives
4.3 Discrete Poisson Bracket . . . . . . . . . . . . . . .
4.4 Jacobi Identity . . . . . . . . . . . . . . . . . . . .
4.5 Discrete Hamiltonian and Equations of Motion . . .
4.6 Discrete Gauss’ Law . . . . . . . . . . . . . . . . .
4.7 Discrete Casimir Invariants . . . . . . . . . . . . . .
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13
14
15
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18
20
21
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5 Hamiltonian Splitting
22
5.1 Solution of the Sub-Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Splitting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Backward Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Example: Vlasov-Maxwell in 1D2V
27
6.1 Hamiltonian Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.2 Boris-Yee scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 Numerical Experiments
7.1 Weibel instability . . . . . .
7.2 Streaming Weibel instability
7.3 Strong Landau damping . .
7.4 Backward Error Analysis . .
7.5 Momentum conservation . .
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8 Summary
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31
31
33
33
35
36
38
2
1
Introduction
We consider a structure preserving numerical implementation of the Vlasov-Maxwell system, the system of kinetic equations, describing the dynamics of charged particles in a
plasma, coupled to Maxwell’s equations, describing electrodynamic phenomena arising
from the motion of the particles as well as from externally applied fields. While the
design of numerical methods for the Vlasov-Maxwell (and Vlasov-Poisson) system has attracted considerable attention since the early 1960s (see [70] and references therein), the
systematic development of structure-preserving or geometric numerical methods started
only recently.
The Vlasov-Maxwell system exhibits a rather large set of structural properties, which
should be considered in the discretization. Most prominently, the Vlasov-Maxwell system
features a variational [47, 83, 19] as well as a Hamiltonian [56, 81, 57, 50] structure. This
implies a range of conserved quantities, which by Noether’s theorem are related to symmetries of the Lagrangian and the Hamiltonian, respectively. In addition, the degeneracy
of the Poisson brackets in the Hamiltonian formulation implies the conservation of several
families of so-called Casimir functionals (see e.g. [59] for review).
Maxwell’s equations have a rich structure themselves. The various fields and potentials
appearing in these equations are most naturally described as differential forms [10, 79,
78, 5] (see also [33, 55, 29]). The spaces of these differential forms build what is called
a deRham complex. This implies certain compatibility conditions between the spaces,
essentially boiling down to the identities from vector calculus, curl grad = 0 and div curl =
0. It has been realised that it is of utmost importance to preserve this complex structure
in the discretization in order to obtain stable numerical methods. This goes hand-in-hand
with preserving the constraints on the electromagnetic fields, div B = 0 and Gauss’ law
div E = ρ, which constitute two more structural properties.
The compatibility problems of discrete Vlasov-Maxwell solvers has been widely discussed in the Particle-In-Cell (PIC) literature [77, 76, 6, 35, 34, 86] for exact charge
conservation. An alternative is to modify Maxwell’s equations by adding Lagrange multipliers to relax the constraint [62, 9, 49, 46, 63]. For a more geometric perspective
on charge conservation based on Whitney forms one can refer to [54]. Even though it
has attracted less interest the problem also exists for grid based discretizations of the
Vlasov equations and the same recipes apply there as discussed in [27, 69]. Note also
that the infinite-dimensional kernel of the curl operator has made it particularly hard
to find good discretization for Maxwell’s equations, especially for the eigenvalue problem [7, 8, 12, 18, 42].
Geometric Eulerian (grid-based) discretizations have been proposed based on spline
differential forms [4] as well as variational integrators [44]. Recently, also various discretizations based on discontinuous Galerkin methods have been proposed [68, 30, 31,
41, 22, 23, 24, 25, 48]. Even though these are usually not based on geometric principles,
they tend to show good long-time conservation properties with respect to e.g. momentum
and/or energy.
First attempts to obtain geometric integrators for Particle-in-Cell methods have been
made by Squire et al. [71], applying a discrete action principle to Low’s Lagrangian for
the Vlasov-Maxwell system [47] and discretising the electromagnetic fields via discrete
exterior calculus (DEC) [43, 32, 72]. This leads to gauge-invariant variational integrators
that satisfy exact charge conservation in addition to approximate energy conservation.
Evstatiev and Shadwick [36] performed a variational semi-discretization in space, but
3
used more standard finite difference and Finite Element discretizations of the fields and
an explicit symplectic integrator in time. As a result, their integrators preserve momentum
and energy but not charge. In this work there also appears the first integrator based on a
semi-discretization of the noncanonical Poisson bracket formulation of the Vlasov-Maxwell
system [56, 81, 57, 50], which automatically leads to gauge-invariant integrators solving
for the electromagnetic fields instead of the potentials. In a similar fashion, Xiao et al.
[82] suggest a Hamiltonian discretization using Whitney form interpolants for the fields.
As these preserve the deRham sequence structure of the involved spaces, their algorithm
is also charge conserving. However, both of these works lack a proof of the Jacobi identity
for the semi-discrete bracket, which is crucial for a Hamiltonian integrator. He et al. [40]
introduce a Hamiltonian discretization based on first order Finite Elements, which is a
special case of our framework.
In this work, we unify these ideas in a general, flexible and rigorous framework based on
Finite Element Exterior Calculus (FEEC) [2, 3, 26, 53]. We provide a semi-discretization
of the Vlasov-Maxwell Poisson structure, which preserves the defining properties of the
bracket, anti-symmetry and the Jacobi-identity. This semi-discrete bracket is used in
conjunction with splitting methods in order to obtain a fully discrete numerical scheme.
We proceed as follows. In Section 2, we provide a short review of the Vlasov-Maxwell
system and its Poisson bracket formulation, including a discussion of the Jacobi identity,
Casimir invariants and invariants commuting with the specific Vlasov-Maxwell Hamiltonian. In Section 3, we introduce the Finite Element Exterior Calculus framework using
the example of Maxwell’s equation, we introduce the deRham complex and Finite Element spaces of differential forms. The actual discretization of the Poisson bracket is
performed in Section 4. We prove the discrete Jacobi identity and the conservation of discrete Casimirs, including the discrete Gauss’ law. In Section 5, we introduce a splitting
for the Vlasov-Maxwell Hamiltonian, which leads to an explicit time-stepping scheme.
Various compositions are used in order to obtain higher order methods. Backward error
analysis is used in order to study the long-time energy behavior. In Section 6, we apply the
method to the Vlasov-Maxwell system in 1d2v using splines for the discretization of the
fields. Section 7 concludes the paper with numerical experiments, using nonlinear Landau
damping and the Weibel instability to verify the favorable properties of our scheme.
2
The Vlasov-Maxwell System
The non-relativistic Vlasov equation for a particle species s of charge qs and mass ms
reads
∂fs
qs
+ v · ∇x f s +
(E + v × B) · ∇v f = 0,
(1)
∂t
ms
and couples nonlinearly to the Maxwell equations,
∂E
− curl B = −J,
∂t
(2)
∂B
+ curl E = 0,
∂t
(3)
div E = ρ,
(4)
div B = 0.
(5)
4
These equations are to be solved with suitable initial and boundary conditions. Here, fs
is the phase space distribution function of particle species s, E and B are the electric
and magnetic fields, respectively, and we have scaled the variables, but retained the mass
ms and the signed charge qs to distinguish species. Observe that we use grad, curl, div
to denote ∇x , ∇x ×, ∇x ·, respectively, when they act on variables depending only on x.
The source for the Maxwell equations, the charge density ρ and the current density J, are
obtained from the Vlasov equations by
X Z
X Z
ρ=
qs fs dv, J =
qs fs vdv.
(6)
s
s
Taking the divergence of Ampère’s equation (2) and using Gauss’ law (4) gives the continuity equation for charge conservation
∂ρ
+ div J = 0.
∂t
(7)
Equation (7) serves as a compatibility condition for Maxwell’s equations, which are illposed when (7) is not satisfied. Moreover it can be shown that if the divergence constraints
(4)-(5) are satisfied at the initial time, they remain satisfied for all times by the solution of
Ampère’s equation (2) and Faraday’s law (3), which have a unique solution by themselves
provided adequate initial and boundary conditions are imposed. This follows directly
from the fact that the divergence of the curl vanishes and Eq. (7). The continuity equation follows from the Vlasov equation by integration over velocity space and using the
definitions of charge and current densities. However this does not necessarily remain true
when the charge and current densities are approximated numerically. The problem for
numerical methods is then to find a way to have discrete sources, which satisfy a discrete
continuity equation compatible with the discrete divergence and curl operators. Another
option is to modify the Maxwell equations, so that they are well posed independently of
the sources, by introducing two additional scalar unknowns that can be seen as Lagrange
multipliers for the divergence constraints. These should become arbitrarily small when
the continuity equation is close to being satisfied.
2.1
Noncanonical Hamiltonian Structure
The Vlasov-Maxwell system possesses a noncanonical Hamiltonian structure. The system
of equations (1)-(3) can be obtained from the following Poisson bracket, a biliniear, antisymmetric bracket that satisfies Leibniz’ rule and the Jacobi identity:
X Z fs δF δG {F, G} [fs , E, B] =
,
dx dv
ms δfs δfs
s
X qs Z δF δG
δG δF
f s ∇v
·
·
dx dv
− ∇v
+
ms
δfs δE
δfs δE
s
X qs Z
δF
δG
+
f s B · ∇v
dx dv
× ∇v
2
m
δf
δf
s
s
s
s
Z δF δG
δG δF
+
curl
dx,
(8)
·
− curl
·
δE δB
δE δB
where [f, g] = ∇x f · ∇v g − ∇x g · ∇v f . This bracket was introduced in [56], with a term
corrected in [50] (see also [81, 57]), and its limitation to divergence free magnetic fields
5
first pointed out in [57]. See also [20] and [60], where the latter contains the details of the
direct proof of the Jacobi identity
{F, {G, H}} + {G, {H, F }} + {H, {F, G}} = 0.
(9)
The time evolution of any functional F [fs , E, B] is given by
d
F [fs , E, B] = {F, H} ,
dt
(10)
with the Hamiltonian H given as the sum of the kinetic energy of the particles and the
electric and magnetic field energies,
Z
X ms Z
1 2
H=
|v| fs (x, v) dx dv +
|E(x)|2 + |B(x)|2 dx.
(11)
2
2
s
In order to obtain the Vlasov equations, we consider the functional
Z
δxv [fs ] = fs (x′ , v′) δ(x − x′ ) δ(v − v′ ) dx′ dv′ = fs (x, v),
for which the equations of motion (10) are computed as
Z
h
i
∂fs
2
(x, v) = δ(x − x′ ) δ(v − v′ ) 12 |v′ | , fs (x′ , v′) dx′ dv′
∂t
Z
qs
δ(x − x′ ) δ(v − v′ ) ∇v fs (x′ , v′) · E(x′ ) dx′ dv′
−
ms
Z
qs
−
δ(x − x′ ) δ(v − v′ ) ∇v fs (x′ , v′) · (B(x′ ) × v′ ) dx′ dv′
ms
qs
E(x) + v × B(x) · ∇v fs (x, v).
= −v · ∇x fs (x, v) −
ms
(12)
(13)
For the electric field, we consider
δx [E] =
Z
E(x′ ) δ(x − x′ ) dx′ = E(x),
(14)
so that from (10) we obtain Ampère’s equation,
Z X
∂E
′
′
′
′
=
curl B(x ) −
qs fs (x , v ) v δ(x − x′ ) dx′ dv′ = curl B(x) − J(x),
∂t
s
(15)
where the current density J is given by
X Z
J(x) =
qs fs (x, v) v dv.
(16)
s
And for the magnetic field, we consider
Z
δx [B] = B(x′ ) δ(x − x′ ) dx′ = B(x),
and obtain the Faraday equation,
Z
∂B
= − (curl E(x′ )) δ(x − x′ ) dx = − curl E(x).
∂t
6
(17)
(18)
Our aim is to preserve this noncanonical Hamiltonian structure and its features at the discrete level. This can be done by taking only a finite number of initial positions for the particles instead of a continuum and by taking the electromagnetic fields in finite-dimensional
subspaces of the original function spaces. A good candidate for such a discretization is
the Finite Element Particle-In-Cell framework. In order to satisfy the continuity equation
as well as the identities from vector calculus and thereby preserve Gauss’ law and the divergence of the magnetic field, the Finite Element spaces for the different fields cannot be
chosen independently. The right framework is given by Finite Element Exterior Calculus
(FEEC).
Before describing this framework in more detail, we shortly want to discuss some
conservation laws of the Vlasov-Maxwell system. In Hamiltonian systems, there are two
kinds of conserved quantities, Casimir invariants and momentum maps.
2.2
Invariants
A family of conserved quantities are Casimir invariants (Casimirs), which originate from
the degeneracy of the Poisson bracket. Casimirs are functionals C(f , E, B) which Poisson
commute with every other functional G(f , E, B), i.e., {C, G} = 0. For the Vlasov-Maxwell
bracket, there are several such Casimirs [58, 61, 20]. First, the integral of any real function
hs of each distribution function fs is preserved, i.e.,
Z
Cs = hs (fs ) dx dv.
(19)
This family of Casimirs is a manifestation of Liouville’s theorem and corresponds to
conservation of phase space volume. Further we have two Casimirs related to Gauss’
law (4) and the divergence-free property of the magnetic field (5),
Z
CE = hE (x) div E − ρ dx,
(20)
Z
CB = hB (x) div B dx,
(21)
where hE and hB are arbitrary real functions of x. The latter functional, CB , is not a true
Casimir but should rather be referred to as pseudo-Casimir. It acts like a Casimir in that
it Poisson commutes with any other functional, but the Jacobi identity is only satisfied
when div B = 0 (see [57, 60]).
A second family of conserved quantities are momentum maps Φ, which arise from
symmetries that preserve the particular Hamiltonian H, and therefore also the equations
of motion. This means that the Hamiltonian is constant along the flow of Φ, i.e.,
{H, Φ} = 0.
(22)
From Noether’s theorem it follows that the generators Φ of the symmetry are preserved
by the time evolution, i.e.,
dΦ
= 0.
(23)
dt
If the symmetry condition (22) holds, this is obvious by the antisymmetry of the Poisson
bracket as
dΦ
= {Φ, H} = −{H, Φ} = 0.
(24)
dt
7
Therefore Φ is a constant of motion if and only if {Φ, H} = 0.
The complete set of constants of motion, the algebra of invariants, will be discussed
elsewhere. However, as an example of a momentum map we shall consider here the total
momentum
Z
XZ
P=
ms vfs dx dv + E × B dx.
(25)
s
By direct computations, assuming periodic boundary conditions, it can be shown that
Z
Z
dP
= {P, H} = E(ρ − div E) dx = E Q(x) dx.
(26)
dt
defining Q(x) := ρ − div E, which is a local version of the Casimir CE . Therefore, if
at t = 0 the Casimir Q ≡ 0, then momentum is conserved. If at t = 0 the Casimir
Q 6≡ 0, then momentum is not conserved and it changes in accordance with (26). For
a multi-species plasma Q ≡ 0 is equivalent to the physical requirement that Poisson’s
equation be satisfied. If for some reason it is not exactly satisfied, then we have violation
of momentum conservation.
For a single species plasma, say electrons, with a neutralizing positive background
charge ρB (x), say ions, Poisson’s equation is
div E = ρB − ρe .
(27)
The Poisson bracket for this case has the local Casimir
Qe = div E + ρe
(28)
and it does not recognize the background charge. Because the background is stationary,
the total momentum is
Z
Z
P = me v fe dxdv + E × B dx,
(29)
and it satisfies
dPe
= {Pe , H} = −
dt
Z
E ρB (x) dx .
(30)
We will verify this relation in the numerical experiments of Sec. 7.5.
3
Finite Element Exterior Calculus
Finite Element Exterior Calculus (FEEC) is a mathematical framework for mixed Finite
Element methods, which uses geometrical and topological ideas for systematically analyzing the stability and convergence of Finite Element discretizations of partial differential
equations. This proved to be a particularly difficult problem for Maxwell’s equation,
which we will use in the following as an example to review this framework.
8
3.1
Maxwell’s Equations
When Maxwell’s equations are used in some material medium, they are best understood
by introducing two additional fields. The electromagnetic properties are then defined by
the electric and magnetic fields, usually denoted by E and H, the displacement field D
and the magnetic induction B. For simple materials, the electric field is related to the
displacement field and the magnetic field to the magnetic induction by
D = εE,
B = µH,
where ε and µ are the permittivity and permeability tensors reflecting the material properties. In vacuum they become the scalars ε0 and µ0 , which are unity in our scaled variables,
while for more complicated media such as plasmas they can be nonlinear operators [60].
The Maxwell equations with the four fields read
∂D
− curl H = −J,
∂t
∂B
+ curl E = 0,
∂t
(31)
(32)
div D = ρ,
(33)
div B = 0.
(34)
The mathematical interpretation of these fields become clearer when interpreting them as
differential forms: E and H are 1-forms, D and B are 2-forms. The charge density ρ is a 3form and the current density J a 2-form. Moreover, the electrostatic potential φ is a 0-form
and the vector potential A a 1-form. The grad, curl, div operators represent the exterior
derivative applied respectively to 0-forms, 1-forms and 2-forms. To be more precise, there
are two kinds of differential forms, depending on the orientation. Straight differential
forms have an intrinsic (or inner) orientation, whereas twisted differential forms have an
outer orientation, defined by the ambient space. Faraday’s equation and div B = 0 are
naturally inner oriented, whereas Ampère’s equation and Gauss’ law are outer oriented.
This knowledge can be used to define a natural discretization for Maxwell’s equations.
For finite difference approximations a dual mesh is needed for the discretization of twisted
forms. This can already be found in Yee’s scheme [84]. In the Finite Element context,
only one mesh is used, but dual operators are used for the twisted forms. A detailed
description of this formalism can be found, e.g., in Bossavit’s lecture notes [11].
3.2
Finite Element Spaces of Differential Forms
The full mathematical theory for the Finite Element discretization of differential forms
is due to Arnold, Falk and Winther [2, 3] and is called Finite Element Exterior Calculus
(FEEC) (see also [53, 26]). Most Finite Element spaces appearing in this theory were
known before, but their connection in the context of differential forms was not made
clear. The first building block of FEEC is the following commuting diagram:
HΛ0(Ω)
d
Π0
V0
HΛ1 (Ω)
d
Π1
d
V1
HΛ2 (Ω)
d
Π2
d
9
V2
L2 (Ω)3
Π3
d
V3
(35)
where Ω R⊂ R3 , Λk (Ω) is the space of k-forms on Ω that we endow with the inner product
hα, βi = α ∧ ⋆β, ⋆ is the Hodge operator and d is the exterior derivative. Then we define
L2 Λk (Ω) = {ω ∈ Λk (Ω) | hω, ωi < +∞}
and the Sobolev spaces of differential forms
HΛk (Ω) = {ω ∈ L2 Λk (Ω) | dω ∈ L2 Λk+1 (Ω)}.
Obviously in a three-dimensional manifold the exterior derivative of a 3-form vanishes so
that HΛ3 (Ω) = L2 (Ω). This diagram can also be expressed using the standard vector
calculus formalism
H 1 (Ω)
grad
Π0
V0
H(curl, Ω)
curl
Π1
grad
H(div, Ω)
div
Π2
V1
curl
V2
L2 (Ω)3
Π3
div
(36)
V3
The first row of (36) represents the exact sequence of function spaces involved in Maxwell’s
equations in the sense that at each node, the kernel of the next operator is the image of the
previous: Im(grad) = Ker(curl), Im(curl) = Ker(div). And the power of the conforming
Finite Element framework is that this exact sequence can be reproduced at the discrete
level by choosing the appropriate finite-dimensional subspaces V0 , V1 , V2 , V3 . The order of
the approximation is dictated by the choice made for V0 and the requirement of having an
exact sequence at the discrete level. The projection operators Πi are the Finite Element
interpolants, which have the property that the diagram is commuting. This means for
example, that the grad of the projection on V0 is identical to the projection of the grad
on V1 . As proven by Arnold, Falk and Winther, their choice of Finite Elements naturally
leads to stable discretizations.
There are many known sequences of Finite Element spaces that fit this diagram. The
sequences proposed by Arnold, Falk and Winther are based on well-known Finite Element
spaces: on tetrahedra these are H 1 conforming Pk Lagrange Finite Elements for V0 , the
H(curl) conforming Nédélec elements for V1 , the H(div) conforming Raviart-Thomas
elements for V2 and Discontinuous Galerkin elements for V3 . A similar sequence can be
defined on hexahedra based on the H 1 conforming Q k Lagrange Finite Elements for V0 .
Other such exact sequences are available. Let us in particular cite the mimetic spectral
elements [45, 37, 65] and the spline Finite Elements [13, 14, 67] that we shall use in this
work, as splines are generally favored in PIC codes due to their smoothness properties
that enable noise reduction.
3.3
Finite Element Discretization of Maxwell’s Equations
This framework is enough to express discrete relations between all the straight (or primal
forms), i.e., E, B, A and φ. The commuting diagram yields a direct expression of the
discrete Faraday equation. Indeed projecting all the components of the equation onto V2
yields
∂Π2 B
+ Π2 curl E = 0,
∂t
10
which is equivalent, due to the commuting diagram property, to
∂Π2 B
+ curl Π1 E = 0.
∂t
Denoting with an h index the discrete fields, Bh = Π2 B, Eh = Π1 E, this yields the
discrete Faraday equation,
∂Bh
+ curl Eh = 0.
(37)
∂t
In the same way, the discrete electric and magnetic fields are defined exactly as in the
continuous case from the discrete potentials, thanks to the compatible Finite Element
spaces,
∂A
∂Π1 A
∂Ah
= − grad Π0 φ −
= − grad φh −
,
∂t
∂t
∂t
Bh = Π2 B = Π2 curl A = curl Π1 A = curl Ah ,
Eh = Π1 E = −Π1 grad φ − Π1
(38)
(39)
so that automatically we get
div Bh = 0.
(40)
On the other hand, Ampère’s equation and Gauss’ law relate expressions involving twisted
differential forms. In the Finite Element framework, these should be expressed on the dual
complex, which inherits the commuting diagram properties from the primal complex:
2
L (Ω)
3
div∗
(H(div, Ω))
Π0
V0∗
∗
curl∗
(H(curl, Ω))
Π1
div∗
V1∗
∗
grad∗
(H 1(Ω))∗
Π2
curl∗
Π3
grad∗
V2∗
(41)
V3∗
Recalling that the dual operator denoted by T ∗ of an operator T in L2 (Ω) is defined, using
a test function v, by hT ∗ u, vi = hu, T vi, we find
Z
Z
Z
∗
grad u v dx =
u · grad v dx = − div u v dx,
(42)
Ω
Ω
Ω
using Green’s formula and assuming periodic boundary conditions, so that grad∗ = − div.
Similarly,
Z
Z
Z
Ω
curl∗ u · v dx =
Ω
u · curl v dx =
Ω
curl u · v dx,
(43)
so that curl∗ = curl. Due to these properties the discrete dual spaces are such that
V0∗ = V3 , V1∗ = V2 , V2∗ = V1 and V3∗ = V0 .
In the Finite Element framework the dual operators and spaces are not explicitly
needed. They are most naturally used seamlessly by keeping the weak formulation of the
corresponding equations. The weak form of Ampère’s equation is found by taking the dot
product of (2) with a test function F and applying a Green identity. Assuming periodic
boundary conditions, the weak solution of Ampère’s equation (E, B) ∈ H(curl, Ω) ×
H(div, Ω) is characterized by
Z
Z
Z
d
E · v dx − B · curl v dx = − J · v dx ∀v ∈ H(curl, Ω).
(44)
dt Ω
Ω
Ω
11
The discrete version is obtained by replacing the continuous spaces by their finite-dimensional
subspaces. The approximate solution (Eh , Bh ) ∈ V1 × V2 is characterized by
Z
Z
Z
d
Eh · vh dx −
Bh · curl vh dx = − Jh · vh dx ∀vh ∈ V1 .
(45)
dt Ω
Ω
Ω
In the same way the weak solution of Gauss’ law E ∈ H(curl, Ω) is characterized by
Z
Z
E · ∇v dx = − ρv dx ∀v ∈ H 1 (Ω),
(46)
Ω
Ω
its discrete version being Eh ∈ V1 characterized by
Z
Z
Eh · ∇vh dx = − ρh vh dx ∀vh ∈ V0 .
Ω
(47)
Ω
The last step for the Finite Element discretization is to define a basis for each of the
finite-dimensional spaces V0 , V1 , V2 , V3 , with dim Vk = Nk and to find equations relating
the coefficients on these bases. Let us denote by {Λ0i }i=1...N0 and {Λ3i }i=1...N3 a basis
of V0 and V3 , respectively, which are spaces of scalar functions, and {Λ1i }i=1...N1 a basis
of V1 ⊂ H(curl, Ω) and {Λ2i }i=1...N2 a basis of V2 ⊂ H(div, Ω), which are vector valued
functions. We shall also need for each basis Λki the dual basis denoted by σik defined by
σik (Λkj ) = δij , where δij is the Kronecker symbol, whose value is one for i = j and zero
else. The dual basis corresponds to the degrees of freedom of Finite Element terminology.
Elements of the finite-dimensional spaces can be expanded on their respective bases, e.g,
elements of V1 and V2 , respectively, as
Eh =
N1
X
ei Λ1i ,
Bh =
i=1
N2
X
bi Λ2i ,
(48)
i=1
denoting by e = (e1 , . . . , eN1 )⊤ and b = (b1 , . . . , bN2 )⊤ the corresponding degrees of freedom, with ei = σi1 (Eh ) and bi = σi2 (Bh ). Due to the exact sequence property we have
that curl Eh ∈ V2 for all Eh ∈ V1 , so that curl Eh can be expressed in the basis of V2 by
curl Eh =
N2
X
ci Λ2i .
i=1
Let us also denote by c = (c1 , . . . , cN2 )⊤ . On the other hand
!
N1
N1
N1
X
X
X
1
1
2
curl Eh = curl
ei Λj =
ej curl Λj ,
σi (curl Eh ) =
ej σi2 (curl Λ1j ).
j=1
j=1
j=1
Denoting by C the discrete curl matrix,
C = (σi2 (curl Λ1j ))1≤i≤N2 ,1≤j≤N1 ,
(49)
the degrees of freedom of curl Eh in V2 are related to the degrees of freedom of Eh in V1
by c = Ce. In the same way we can define the discrete gradient matrix G and the discrete
divergence matrix D, given by
G = (σi1 (grad Λ0j ))1≤i≤N1 ,1≤j≤N0
and
12
D = (σi3 (div Λ2j ))1≤i≤N3 ,1≤j≤N2 ,
(50)
respectively. Denoting by ϕ = (ϕ1 , . . . , ϕN0 )⊤ and a = (a1 , . . . , aN1 )⊤ the degrees of
freedom of the potentials φh and Ah , with ϕi = σi0 (φh ) and ai = σi1 (Ah ), the relation (38)
between the discrete fields (48) and the potentials can be written using only the degrees
of freedom as
e = −Gϕ −
da
,
dt
b = Ca.
(51)
Finally, we need the ”mass” matrices in each of the discrete spaces Vi , which define the
discrete Hodge operator linking
the primal exact sequence with the dual exact sequence.
R
1
We denote by (M1 )i,j = Ω Λi · Λ1j dx the mass matrix in V1 and similarly M0 , M2 ,
M3 the mass matrices in V0 , V2 and V3 , respectively. Using these definitions as well as
̺ = (̺1 , . . . , ̺N3 )⊤ and j = (j1 , . . . , jN2 )⊤ with ̺i = σi3 (ρh ) and ji = σi2 (Jh ), we obtain a
system of ordinary differential equations for each of the continuous equations, namely
M1
de
− C ⊤ M2 b = −j,
dt
db
+ Ce = 0,
dt
G ⊤ M1 e = ρ,
Db = 0.
(52)
(53)
(54)
(55)
Moreover the exact sequence properties can also be expressed at the matrix level. The
primal sequence being
R N0
G
R N1
C
R N2
D
R N3 ,
(56)
with Im G = Ker C, Im C = Ker D, and the dual sequence being
R N3
D⊤
R N2
C⊤
R N1
G⊤
R N0 ,
(57)
with Im D ⊤ = Ker C ⊤ , Im C ⊤ = Ker G ⊤ .
4
Discretization of the Hamiltonian Structure
The continuous bracket (8) is for the Eulerian (as opposed to Lagrangian) formulation
of the Vlasov equation, and operates on functionals of the distribution function and the
electric and magnetic fields. We incorporate a discretization that uses a finite number
of characteristics instead of the continuum particle distribution function. This is done
by localizing the distribution function on particles, which amounts to a Monte Carlo
discretization of the first three integrals in (8) if the initial phase space positions are
randomly drawn. Moreover instead of allowing the fields E and B to vary in H(curl, Ω) and
H(div, Ω), respectively, we keep them in the discrete subspaces V1 and V2 . This procedure
yields a discrete Poisson bracket, from which one obtains the dynamics of a finite (large)
number of degrees of freedom: the particle phase space positions za = (xa , va ), where
a = 1, ..., Np , with Np the number of particles, and the coefficients of the fields in the
Finite Element basis, where we denote by ei the degrees of freedom for Eh and by bi
the degrees of freedom for Bh , as introduced in Sec. 3. The FEEC framework of Sec. 3
13
automatically provides the following discretization spaces for the potentials, the fields and
the densities:
φ h ∈ V0 ,
A h , E h ∈ V1 ,
B h , Jh ∈ V 2 ,
ρh ∈ V3 .
Recall that the coefficient vectors of the fields are denoted e and b. In order to also get
a vector expression for the particle quantities, we denote by
V = (v1 , . . . , vNp )⊤ .
X = (x1 , . . . , xNp )⊤ ,
(58)
We use this setting to transform (8) into a discrete Poisson bracket for the dynamics of
the coefficients e, b, X and V.
4.1
Discretization of the Functional Field Derivatives
Upon inserting (48), any functional F [Eh ] can be considered as a function F̂ (e) of the
Finite Element coefficients,
F [Eh ] = F̂ (e).
(59)
Therefore, we can write the first variation of F [E],
δF [E]
δF [E] =
, δE ,
δE
(60)
as
δF [Eh ]
, Ēh
δE
=
L2
*
+
∂ F̂ (e)
, ē
∂e
,
(61)
R N1
with
Ēh (x) =
N1
X
ēi (t) Λ1i (x),
ē = (ē1 , . . . , ēN1 )⊤ .
(62)
i=1
Expressing the functional derivative on the dual basis σ 1i (x) of Λ1i (x), such that
Z
Λ1i (x) · σ 1j (x) dx = δij ,
(63)
we find using (61) for Ēh = Λ1i (x) that
N
1
δF [Eh ] X
∂ F̂ (e) 1
=
σ i (x).
δE
∂ei
i=1
(64)
On the other hand expanding the dual basis in the original basis,
σ 1i (x)
=
N1
X
j=1
14
aij Λ1i (x),
(65)
and taking the L2 inner product with Λ1i (x) we find that the matrix A = (aij ) verifies
AM1 = IN1 , where IN1 denotes the N1 × N1 identity matrix, so that A is the inverse of
the mass matrix M1 and
N1
X
δF [Eh ]
∂ F̂ (e)
=
(M1−1 )ij Λ1j (x).
δE
∂ei
i,j=1
(66)
In full analogy we find
N2
X
∂ F̂ (b)
δF [Bh ]
=
(M2−1 )ij Λ2j (x).
δB
∂b
i
i,j=1
(67)
Next, using (49), we find
curl
N1
X
∂ F̂ (e)
δF [Eh ]
=
(M1−1 )ij C jk Λ2k (x).
δE
∂e
i
i,j=1
(68)
Finally, we can re-express the following term in the Poisson bracket
Z
δF [Eh ] δG[Bh ]
·
dx =
curl
δE
δB
Z
N1 X
N2 X
N2
X
∂ F̂ (e)
∂ Ĝ(b)
−1
−1
=
(M1 )ij C jk
(M2 )mn Λ2k (x) · Λ2n (x) dx
∂e
∂b
i
m
i,j=1 k=1 m,n=1
=
N1 X
N2
X
∂ F̂ (e)
i,j=1 k=1
∂ei
N
(M1−1 )ij C jk
N
1 X
2
∂ F̂ (e)
∂ Ĝ(b) X
∂ Ĝ(b)
=
(M1−1 C)ik
.
∂bk
∂e
∂b
i
k
i=1 k=1
(69)
The other terms in the bracket involving functional derivatives with respect to the fields
are handled similarly. In the next step we need to discretize the distribution function and
the corresponding functional derivatives.
4.2
Discretization of the Functional Particle Derivatives
We proceed by assuming a particle-like distribution function for Np particles labeled by a,
fh (x, v, t) =
Np
X
a=1
wa δ x − xa (t) δ v − va (t) ,
(70)
with mass ma , charge qa , weights wa , particle positions xa and particle velocities va . Here,
Np denotes the total number of particles of all species, with each particle carrying its own
mass and signed charge. Functionals of the distribution function, F [f ], can be considered
as functions of the particle phase space trajectories (X, V) upon inserting (70),
F [fh ] = F̂ (X, V).
15
(71)
Variation of the left-hand side gives,
Z
δF
δF [fh ] =
δf dx dv
δf
Z
Np
X
δF
wa
=−
δ v − va (t) ∇x δ x − xa (t) · δxa
δf
a=1
!
+ δ x − xa (t) ∇v δ v − va (t) · δva dx dv
=
Np
X
a=1
wa
!
δF δF · δxa + ∇v · δva .
∇x δf (xa ,va )
δf (xa ,va )
Upon equating this expression with the variation of the right-hand side of (71),
!
Np
X
∂ F̂
∂ F̂
δxa +
δva ,
δ F̂ (X, V) =
∂xa
∂va
a=1
we obtain
δF ∂ F̂
= w a ∇x ∂xa
δf (xa ,va )
and
∂ F̂
δF .
= w a ∇v ∂va
δf (xa ,va )
(72)
(73)
(74)
Now we have all the ingredients necessary to perform the discretization of the Poisson
bracket.
4.3
Discrete Poisson Bracket
Replacing all functional derivatives in (8) as outlined in the previous two sections, we
obtain the semi-discrete Poisson bracket
X 1 ∂ F̂ ∂ Ĝ
∂ Ĝ ∂ F̂
{F̂ , Ĝ}[X, V, e, b] =
·
−
·
ma wa ∂xa ∂va ∂xa ∂va
a
!
Np N1
X
X qa ∂ Ĝ
∂
F̂
∂
F̂
∂
Ĝ
+
(M1−1 )ij Λ1j (xa ) ·
−
(M1−1 )ij Λ1j (xa ) ·
m
∂e
∂v
∂e
∂va
a
i
a
i
a=1 i,j=1
!
Np N2
X
X qa
∂
Ĝ
∂
F̂
+
bi (t) Λ2i (xa ) ·
×
2w
m
∂va ∂va
a a
a=1 i=1
!
N1 X
N2
X
∂
Ĝ
∂
Ĝ
∂
F̂
∂ F̂
−1
−1
, (75)
(M1−1 )ij C ⊤
−
(M1−1 )ij C ⊤
+
jk (M2 )kl
jk (M2 )kl
∂e
∂b
∂e
∂b
i
l
i
l
i,j=1
k,l=1
with the curl matrix C as given in (49). In order to express the semi-discrete Poisson
bracket (75) in matrix form, we denote by 1 (X) the 3Np × N1 matrix with generic term
Λ1i (xa ), where 1 ≤ a ≤ Np and 1 ≤ i ≤ N1 , and by B(X, b) the 3Np × 3Np block diagonal
matrix with generic block


2,3
2,2
N
0
Λ
(x
)
−Λ
(x
)
2
a
a
i
i
X
b h (xa , t) =
(76)
B
bi (t) −Λ2,3
0
Λi2,1 (xa ) .
i (xa )
2,2
2,1
i=1
Λi (xa ) −Λi (xa )
0
16
Further, let us introduce a mass matrix Mp and a charge matrix Mq for the particles.
Both are diagonal Np × Np matrices with elements (Mp )aa = ma wa and (Mq )aa = qa wa ,
respectively. Additionally, we will need the 3Np × 3Np matrices
M p = Mp ⊗ I3 ,
M q = Mq ⊗ I3 ,
(77)
where I3 denotes the 3 × 3 identity matrix. This allows us to rewrite
!
N p N2
X
X qa
∂
Ĝ
∂
F̂
=
bi (t) Λ2i (xa ) ·
×
2w
m
∂va ∂va
a
a
a=1 i=1
Np
N2
X
1 ∂ Ĝ
∂ F̂ qa X
·
bi (t) Λ2i (xa ) ×
=−
∂va ma i=1
ma wa ∂va
a=1
⊤
∂ F̂
−1
−1 ∂ Ĝ
=−
.
M p M q B(X, b) M p
∂V
∂V
Here, the derivatives are represented by the 3Np vector
!⊤
!⊤
∂ F̂
∂ F̂
∂ F̂
∂ F̂
∂ F̂
∂ F̂
∂ F̂ ∂ F̂ ∂ F̂
,
,
=
, ...,
,
,
,
, ...,
=
1
2
3
∂V
∂v1
∂vNp
∂v11 ∂v12 ∂v13
∂vN
∂vN
∂vN
p
p
p
(78)
(79)
and correspondingly for ∂ Ĝ/∂V, ∂ F̂ /∂e, ∂ F̂ /∂b, etc. Thus, the discrete Poisson bracket
(75) becomes
∂ Ĝ −1 ∂ F̂
∂ F̂ −1 ∂ Ĝ
Mp
−
M
∂X
∂V ∂X p ∂V
⊤
∂ F̂
−1
1
⊤
−1 ∂ Ĝ
+
M p M q (X) M1
∂V
∂e
⊤
∂ F̂
−1 ∂ Ĝ
−1 1
(X) M q M p
−
M1
∂e
∂V
⊤
∂ F̂
−1
−1 ∂ Ĝ
M p M q B(X, b) M p
+
∂V
∂V
⊤
⊤
∂ F̂
∂ F̂
−1 ⊤ ∂ Ĝ
−1 ∂ Ĝ
+
−
.
M1 C
CM1
∂e
∂b
∂b
∂e
{F̂ , Ĝ}[X, V, e, b] =
(80)
The action of this bracket on two functionals F̂ and Ĝ can also be expressed as
{F̂ , Ĝ} = DF̂ ⊤ J (u) DĜ,
denoting by D the derivative with respect to the dynamical variables
u = (X, V, e, b)⊤ ,
and by J the Poisson matrix, given by


0
M −1
0
0
p
−1
1
−M −1
M −1
M −1
(X)M1−1
0 
p
p M q B(X, b) M p
p Mq
.
J (u) = 
−1
−1
1
⊤
−1
 0
−M1
(X) M q M p
0
M1 C ⊤ 
0
0
−CM1−1
0
(81)
(82)
We immediately see that J (u) is anti-symmetric, but it remains to be shown that it
satisfies the Jacobi identity.
17
4.4
Jacobi Identity
The discrete Poisson bracket (80) satisfies the Jacobi identity if and only if the following
condition holds (see e.g. [59, Section IV] or [38, Section VII.2, Lemma 2.3]):
X ∂Jij (u)
∂Jjk (u)
∂Jki (u)
Jlk (u) +
Jli (u) +
Jlj (u) = 0 for all i, j, k.
(83)
∂ul
∂ul
∂ul
l
To simplify the verification of (83), we start by identifying blocks of the Poisson matrix
J whose elements contribute to the above condition. Therefore, we write


J11 (u) J12 (u) J13 (u) J14 (u)
J21 (u) J22 (u) J23 (u) J24 (u)

(84)
J (u) = 
J31 (u) J32 (u) J33 (u) J34 (u) .
J41 (u) J42 (u) J43 (u) J44 (u)
The Poisson matrix J only depends on X and b, so in (83) we only need to sum l over
the corresponding indices, 1 ≤ l ≤ 3Np and 6Np + N1 < l ≤ 6Np + N1 + N2 , respectively.
Considering the terms Jlk (u), Jli (u) and Jlj (u), we see that in these index ranges, only
−1
−1
J12 = M −1
are non-vanishing, so that we have to account only
p and J43 = −M2 CM1
for those two blocks. Note that J12 is a diagonal matrix, therefore (J12 )ij = (J12 )ii δij .
Further, only J22 , J23 and J32 depend on b and/or X, so only those blocks have to be
considered when computing derivatives with respect to u. In summary, we have


0
J12
0
0
J21 J22 (X, b) J23 (X) 0 
,
(85)
J (u) = 
0
J32 (X)
0
J34 
0
0
J43
0
and we obtain two conditions. The contributions involving J22 and J12 are
!
3Np
X
∂ J22 (X, b) ij
∂ J22 (X, b) jk
∂ J22 (X, b) ki
J12 lk +
J12 li +
J12 lj = 0,
∂Xl
∂Xl
∂Xl
l=1
(86)
for 1 ≤ i, j, k ≤ 3Np , which corresponds to (83) for 3Np < i, j, k ≤ 6Np . Inserting the
actual values for J12 and J22 and using that Mp is diagonal, (86) becomes
−1
∂ M −1
p M q B(X, b) M p
∂Xk
ij
−1
Mp
kk
+
−1
∂ M −1
p M q B(X, b) M p
jk
M −1
p
ii
∂Xi
−1
∂ M −1
p M q B(X, b) M p
ki
M −1
= 0. (87)
+
p
jj
∂Xj
All outer indices of this expression belong to the inverse matrix Mp−1 . As this matrix is
constant, symmetric and positive definite, we can contract the above expression with Mp
on all indices, to obtain
∂ M q B(X, b) ij ∂ M q B(X, b) jk ∂ M q B(X, b)
ki
+
+
= 0,
1 ≤ i, j, k ≤ 3Np . (88)
∂Xk
∂Xi
∂Xj
18
If in the first term of (88) one picks a particular index k, then this selects the σ component
of the position xa of some particle. At the same time, in the second and third term, this
selects a block of (76), which is evaluated at the same particle position xa . This means
that the only non-vanishing contributions in (88) will be those indexed by i and j with Xi
and Xj corresponding to components µ and ν of the same particle position xa . Therefore,
the condition (76) reduces further to
!
bνσ (xa ) ∂ B
bσµ (xa )
bµν (xa ) ∂ B
∂B
+
= 0,
1 ≤ a ≤ Np , 1 ≤ µ, ν, σ ≤ 3, (89)
+
qa
∂xσa
∂xµa
∂xνa
bµν denotes the components of the matrix in (76). When all three indices are equal,
where B
b h (xa , t) which vanish. When two of
this corresponds to diagonal terms of the matrix B
the three are equal, it cancels because of the skew-symmetry of the same matrix and
for all three indices distinct, this condition corresponds to div Bh = 0. Choosing initial
conditions such that div Bh (x, 0) = 0 and using a discrete deRham complex guarantees
div Bh (x, t) = 0 for all times t. Note that this was to be expected, because it is the
discrete version of the div B = 0 condition for the continuous Poisson bracket (8) [57].
From the contributions involving J22 , J23 , J32 and J43 we have
3Np
X
∂ J23 (X, b)
∂Xl
l=1
jk
J12
∂ J32 (X, b)
+
li
∂Xl
ki
J12
+
lj
!
N2
X
∂ J22 (X, b)
∂bl
l=1
ij
J43
lk
= 0, (90)
for 1 ≤ i, j ≤ 3Np and 1 ≤ k ≤ N1 , which corresponds to (83) for 3Np < i, j ≤ 6Np <
k ≤ 6Np + N1 . Writing out (90) and using that Mp is diagonal gives
∂ M1−1
1
(X)⊤ M q M −1
p
∂Xj
ki
−1
Mp
−
jj
∂ M −1
p Mq
=
1
(X)M1−1
jk
M −1
p
=
ii
∂Xi
−1
N2
X
∂ M −1
p M q B(X, b) M p
ij
l=1
∂bl
CM1−1
lk
. (91)
Again, we can contract this with the matrix Mp on the indices i and j, in order to remove
Mp−1 , and with M1 on the index k, in order to remove M1−1 . This results in the simplified
condition
N2
∂ M q B(X, b) ij ∂ 1 (X)⊤ M q ki ∂ M q 1 (X) jk X
C lk .
(92)
−
=
∂Xj
∂Xi
∂bl
l=1
The bl derivative of B results in the 3Np × 3Np block diagonal matrix
block


2,2
0
Λ2,3
l (xa ) −Λl (xa )
2
b (xa ) = −Λ2,3 (xa )
,
Λ
0
Λ2,1
l
l
l (xa )
2,2
2,1
Λl (xa ) −Λl (xa )
0
19
2
l (X)
with generic
(93)
so that (92) becomes
∂
1
(X)⊤ M q
∂Xj
ki
−
1
∂ Mq
(X)
∂Xi
jk
N2
X
=
Mq
2
l (X) ij
l=1
C
lk
.
(94)
Similarly as before, picking a particular index j in the first term, selects the ν component
of the position xa of some particle. At the same time, in the second term, this selects the
ν component of Λ1 , evaluated at the same particle position xa . The only non-vanishing
derivative of this term is therefore with respect to components of the same particle position, so that Xi denotes the µ component of xa . Hence, condition (94) simplifies to
qa
∂Λ1,µ
∂Λ1,ν
k (xa )
k (xa )
−
ν
∂xa
∂xµa
= qa
N2
X
l=1
b 2 (xa )
Λ
l
µν
C
lk
,
(95)
for 1 ≤ a ≤ Np , 1 ≤ µ, ν ≤ 3 and 1 ≤ k ≤ N1 . This conditions states that the curl of the
one-form basis, evaluated at some particle’s position, equals the two-form basis, evaluated
at the same particle’s position.
This concludes the verification of the Jacobi identity (83) for the discrete bracket (80).
4.5
Discrete Hamiltonian and Equations of Motion
The Hamiltonian is discretized by inserting (70) and (48) into (11),
1
Hh =
2
Z
Np
X
2
|v|
a=1
ma wa δ x − xa (t) δ v − va (t) dx dv
Z X
N1
N2
2 X
2 1
1
2
bj (t) Λj (x) dx
ei (t) Λi (x) + +
2
i=1
j=1
(96)
which in matrix notation becomes
Ĥ =
1
2
V⊤ M p V + 21 e⊤ M1 e + 12 b⊤ M2 b.
(97)
The semi-discrete equations of motion are given by
Ẋ = {X, Ĥ},
V̇ = {V, Ĥ},
ė = {e, Ĥ},
ḃ = {b, Ĥ},
(98)
which are equivalent to
u̇ = J (u) DH(u).
(99)
With DH(u) = (0, M p V, M1 e, M2 b)⊤ , we obtain
Ẋ = V,
V̇ = M −1
p Mq
ė =
M1−1
⊤
1
(X)e + B(X, b)V ,
C M2 b(t) −
ḃ = −Ce(t),
1
(X) M q V ,
⊤
(100a)
(100b)
(100c)
(100d)
where the first two equations describe the particle dynamics and the last two equations
describe the evolution of the electromagnetic fields.
20
4.6
Discrete Gauss’ Law
Multiplying (100c) by G ⊤ M1 on the left, we get
G ⊤ M1 ė = G ⊤ C ⊤ M2 b(t) − G ⊤
1
(X)⊤ M q V.
(101)
As CG = 0 from (56), the first term on the right-hand side vanishes. Observe that
1
0
(X)Gψ = grad
(X)ψ
∀ψ ∈ RN0 ,
(102)
and using dxa /dt = va we find that
dΨh (xa (t))
dxa (t)
=
· grad Ψh (xa (t)) = va · grad Ψh (xa (t)),
dt
dt
for any Ψh ∈ V0 with grad Ψh ∈ V1 , so that we get
(103)
d 0 (X)⊤
(104)
M q 1Np ,
dt
where 1Np denotes the column vector with Np terms all being unity, needed for the sum
over the particles when there is no velocity vector. This shows that the discrete Gauss’
law is conserved,
(105)
G ⊤ M1 e = − 0 (X)⊤ M q 1Np .
G ⊤ M1 ė = −G ⊤
4.7
1
0
(X)⊤ M q V = − grad
(X)⊤ M q V = −
Discrete Casimir Invariants
Let us now find the Casimirs of the semi-discrete Poisson structure. These are functionals
C(X, V, e, b) such that {C, F } = 0 for any function F . In terms of the Poisson matrix
J , this can be expressed as J (u) DC(u) = 0. Upon writing this for each of the lines of
J of (82), we find for the first line
Mp−1 DV C = 0.
(106)
This implies that C does not depend on V, which we shall use in the sequel. Then the
third line simply becomes
M1−1 C ⊤ Db C = 0 ⇒ Db C ∈ Ker(C ⊤ ).
(107)
Then, because of the exact sequence property, there exists b̃ ∈ RN3 such that DbC = D ⊤ b̃.
Hence all functions of the form
C(b) = b⊤ D ⊤ b̃ = b̃⊤ Db,
b̃ ∈ RN3
(108)
are Casimirs, which means that Db, the matrix form of div Bh , is conserved.
The fourth line, using that C does not depend on V, becomes
CM1−1 De C = 0 ⇒ M1−1 De C ∈ Ker(C),
(109)
so because of the exact sequence property there exists ẽ ∈ RN1 such that De C = M1 Gẽ.
Finally the second line couples e and X and reads, upon multiplying by Mp
DX C = Mq
1
(X)M1−1 De C = Mq
1
(X)Gẽ = Mq grad
0 (X)ẽ,
(110)
using the expression for De C derived previously and (102). So it follows that all functions
of the form
C(X, e) = 1⊤
N Mq
0
(X)ẽ + e⊤ M1 Gẽ = ẽ⊤
are Casimirs, which means that G ⊤ M1 e +
form of Gauss’ law (105).
0
0
(X)⊤ Mq 1N + ẽ⊤ G ⊤ M1 e,
ẽ ∈ RN0 , (111)
(X)⊤ Mq 1N is conserved. This is the matrix
21
5
Hamiltonian Splitting
Following [28, 66, 39], we split the discrete Hamiltonian (97) into three parts,
Ĥ = Ĥp + ĤE + ĤB ,
(112)
with
Ĥp =
1
2
V⊤ M p V,
ĤE = 12 e⊤ M1 e,
ĤB = 21 b⊤ M2 b.
(113)
Writing u = (X, V, e, b)⊤ , we split the discrete Vlasov-Maxwell equations (100a)-(100d)
into three subsystems,
u̇ = {u, Ĥp },
u̇ = {u, ĤE },
u̇ = {u, ĤB }.
(114)
The exact solution to each of these subsystems will constitute a Poisson map. Because a
composition of Poisson maps is itself a Poisson map, we can construct Poisson structure
preserving integration methods for the Vlasov-Maxwell system by composition of the
exact solutions.
5.1
Solution of the Sub-Systems
The discrete equations of motion for ĤE are
Ẋ = 0,
M p V̇ = M q
ė = 0,
(115a)
1
(X)e,
(115b)
(115c)
ḃ = −Ce(t).
(115d)
For initial conditions X(0), V(0), e(0), b(0) the exact solutions at time ∆t are given by
X(∆t) = X(0),
M p V(∆t) = M p V(0) + ∆t M q 1 (X(0))e(0),
e(∆t) = e(0),
b(∆t) = b(0) − ∆t Ce(0).
(116a)
(116b)
(116c)
(116d)
The discrete equations of motion for ĤB are
Ẋ = 0,
(117a)
V̇ = 0,
(117b)
⊤
M1 ė = C M2 b(t),
(117c)
ḃ = 0.
(117d)
For initial conditions X(0), V(0), e(0), b(0) the exact solutions at time ∆t are given by
X(∆t) = X(0),
V(∆t) = V(0),
M1 e(∆t) = M1 e(0) + ∆t C ⊤ M2 b(0),
b(∆t) = b(0).
22
(118a)
(118b)
(118c)
(118d)
The discrete equations of motion for Ĥp are
Ẋ = V,
(119a)
M p V̇ = M q B(X, b)V,
1
(119b)
⊤
M1 ė = − (X) M q V,
ḃ = 0,
(119c)
(119d)
For general magnetic field coefficients b, this system cannot be exactly integrated [39].
Note that each component V̇µ of the equation for V̇ does not depend on Vµ , where
Vµ = (v1,µ , v2,µ , . . . , vNp ,µ )⊤ , etc., with 1 ≤ µ ≤ 3. Therefore we can split this system
once more into
Ĥp = Ĥp1 + Ĥp2 + Ĥp3 ,
(120)
with
Ĥpµ = 21 Vµ⊤ Mp Vµ
for
1 ≤ µ ≤ 3.
(121)
For concise notation we introduce the Np × N1 matrix 1µ (X) with generic term Λ1i,µ (xa ),
P 2
2
and the Np × Np diagonal matrix 2µ (b, X) with entries N
i=1 bi (t)Λi,µ (xa ), where 1 ≤ µ ≤
3, 1 ≤ a ≤ Np , 1 ≤ i ≤ N1 , 1 ≤ j ≤ N2 . Then, for Ĥp1 we have
Ẋ1 =
V1 (t),
2
⊤
3 (b(t), X(t)) V1 (t),
Mq 22 (b(t), X(t))⊤ V1 (t),
− 11 (X(t))⊤ Mq V1 (t),
Mp V̇2 = −Mq
Mp V̇3 =
M1 ė =
(122a)
(122b)
(122c)
(122d)
for Ĥp2 we have
Ẋ2 =
Mp V̇1 =
Mp V̇3 =
M1 ė =
V2 (t),
2
⊤
3 (b(t), X(t)) V2 (t),
−Mq 21 (b(t), X(t))⊤ V2 (t),
− 12 (X(t))⊤ Mq V2 (t),
Mq
(123a)
(123b)
(123c)
(123d)
and for Ĥp3 we have
Ẋ3 =
V3 (t),
2
⊤
2 (b(t), X(t)) V3 (t),
Mq 21 (b(t), X(t))⊤ V3 (t),
− 13 (X(t))⊤ Mq V3 (t),
Mp V̇1 = −Mq
Mp V̇2 =
M1 ė =
23
(124a)
(124b)
(124c)
(124d)
For Ĥp1 and initial conditions X(0), V(0), e(0), b(0) the exact solutions at time ∆t are
given by
X1 (∆t) = X1 (0) + ∆t V1 (0),
Z∆t
Mp V2 (∆t) = Mp V2 (0) − Mq 23 (b(0), X(t)) V1(0) dt,
Mp V3 (∆t) = Mp V3 (0) +
M1 e(∆t) = M1 e(0) −
0
Z∆t
0
∆t
Z
Mq
2
2 (b(0), X(t)) V1 (0) dt,
1
⊤
1 (X(t)) Mq V1 (0) dt,
(125a)
(125b)
(125c)
(125d)
0
for Ĥp2 by
X2 (∆t) = X2 (0) + ∆t V2 (0),
Z∆t
1
Mp V (∆t) = Mp V1 (0) + Mq 23 (b(0), X(t)) V2(0) dt,
(126a)
(126b)
0
Mp V3 (∆t) = Mp V3 (0) −
M1 e(∆t) = M1 e(0) −
Z∆t
Mq
2
1 (b(0), X(t)) V2 (0) dt,
(126c)
0
Z∆t
1
⊤
2 (X(t)) Mq V2 (0) dt,
(126d)
0
and for Ĥp3 by
X3 (∆t) = X3 (0) + ∆t V3 (0),
Z∆t
Mp V1 (∆t) = Mp V1 (0) − Mq 22 (b(0), X(t)) V3(0) dt,
(127a)
(127b)
0
Mp V2 (∆t) = Mp V2 (0) +
Z∆t
Mq
2
1 (b(0), X(t)) V3 (0) dt,
(127c)
0
M1 e(∆t) = M1 e(0) −
Z∆t
⊤
1
3 (X(t)) Mq V3 (0) dt,
(127d)
0
respectively, where all components not specified are constant. The only challenge in
solving these equations is the exact computation of line integrals along the trajectories [17,
71, 54]. However, as only one component of the particle positions xp is changing in each
step of the splitting, and moreover the trajectory is given by a straight line, this is not
very complicated.
24
5.2
Splitting Methods
⊤
Given initial conditions u(0) = X(0), V(0), e(0), b(0) , a numerical solution of the discrete Vlasov-Maxwell equations (100a)-(100d) at time ∆t can be obtained by composition
of the exact solutions of all the subsystems. A first order integrator can be obtained by
the Lie-Trotter composition [75],
u(∆t) = exp(∆t XE ) exp(∆t XB ) exp(∆t Xp1 ) exp(∆t Xp2 ) exp(∆t Xp3 ) u(0),
(128)
where XE , etc., denote the Hamiltonian vector fields corresponding to ĤE , etc., that is
e.g. XE = J (·, DĤE ). A second order integrator can be obtain by the symmetric Strang
composition [73],
u(∆t) = exp(∆t/2 XE ) exp(∆t/2 XB ) exp(∆t/2 Xp1 ) exp(∆t/2 Xp2 ) exp(∆t/2 Xp3 )
exp(∆t/2 Xp3 ) exp(∆t/2 Xp2 ) exp(∆t/2 Xp1 ) exp(∆t/2 XB ) exp(∆t/2 XE ) u(0).
(129)
The corresponding maps can be written as
ϕ∆t,L = ϕ∆t,p3 ◦ ϕ∆t,p2 ◦ ϕ∆t,p1 ◦ ϕ∆t,B ◦ ϕ∆t,E ,
(130)
ϕ∆t,S2 = ϕ∆t/2,L ◦ ϕ∗∆t/2,L ,
(131)
and
respectively, where ϕ∗∆t,L denotes the adjoint of ϕ∆t,L , defined as
ϕ∗∆t,L = ϕ−1
−∆t,L .
(132)
Let us note that the Lie splitting ϕ∆t,L and the Strang splitting ϕ∆t,S2 are conjugate
methods by the adjoint of ϕ∆t,L [52], i.e.,
ϕ∆t,S2 = (ϕ∗∆t/2,L )−1 ◦ ϕ∆t,L ◦ ϕ∗∆t/2,L = ϕ−∆t/2,L ◦ ϕ∆t,L ◦ ϕ∗∆t/2,L = ϕ∆t/2,L ◦ ϕ∗∆t/2,L .
(133)
The last equality holds by the group property of the flow, but is only valid when the exact
solution of each subsystem is used in the composition (and not some general symplectic
integrator). This implies that the Lie splitting shares many properties with the Strang
splitting which are not found in general first order methods.
Another second order integrator with smaller error constant than ϕ∆t,S2 can be obtained by the following composition,
ϕ∆t,L2 = ϕα∆t,L ◦ ϕ∗(1/2−α)∆t,L ◦ ϕ(1/2−α)∆t,L ◦ ϕ∗α∆t,L .
(134)
Here, α is a free parameter which can be used to reduce the error constant. A particular
small error is obtained for α = 0.1932 [51]. Fourth order time integrators can easily be
obtained from a second order integrator like ϕ∆t,S by the following composition[74, 85],
ϕ∆t,S4 = ϕγ1 ∆t,S2 ◦ ϕγ2 ∆t,S2 ◦ ϕγ1 ∆t,S2 ,
with
γ1 =
1
,
2 − 21/3
γ2 = −
25
21/3
.
2 − 21/3
(135)
Alternatively, we can compose the first order integrator ϕ∆t,L together with its adjoint
ϕ∗∆t,L as follows [51],
with
ϕ∆t,L4 = ϕa5 ∆t,L ◦ ϕ∗b5 ∆t,L ◦ . . . ◦ ϕa2 ∆t,L ◦ ϕ∗b2 ∆t,L ◦ ϕa1 ∆t,L ◦ ϕ∗b1 ∆t,L ,
(136)
√
√
146 + 5 19
−2 + 10 19
1
a1 = b5 =
,
a2 = b4 =
,
a3 = b3 = ,
540 √
135
5
√
14 − 19
−23 − 20 19
,
a5 = b1 =
.
a4 = b2 =
270
108
For higher order composition methods see e.g. [38] and [52] and references therein.
5.3
Backward Error Analysis
In the following, we want to compute the modified Hamiltonian for the Lie-Trotter composition (130). For a splitting in two terms, H = HA + HB , the Lie-Trotter composition
can be written as
ϕ∆t = ϕ∆t,B ◦ ϕ∆t,A = exp(∆t XA ) exp(∆t XB ) = exp(∆t X̃),
(137)
H̃ = H + ∆tH̃1 + ∆t2 H̃2 + O(∆t3 ),
(138)
where XA and XB are the Hamiltonian vector fields corresponding to HA and HB , respectively, and X̃ is the modified vector field corresponding to the modified Hamiltonian
H̃. Following Hairer et al. [38, Chapter IX], the modified Hamiltonian H̃ is given by
with
1
(139)
H̃1 = {HA , HB },
2
1
H̃2 =
{{HA , HB }, HB } + {{HB , HA }, HA } .
(140)
12
In order to compute the modified Hamiltonian for splittings with more than two terms,
we have to apply the Baker–Campbell–Hausdorff formula recursively. For the Lie-Trotter
splitting (130), expressed in terms of the corresponding Hamiltonian vector fields as
ϕ∆t = exp(∆t XE ) exp(∆t XB ) exp(∆t Xp1 ) exp(∆t Xp2 ) exp(∆t Xp3 ),
we split ϕ∆t = exp(∆t X̃) into
exp(∆t XE ) exp(∆t XB ) = exp(∆t X̃EB ),
exp(∆t Xp1 ) exp(∆t Xp2 ) = exp(∆t X̃p1,2 ),
exp(∆t X̃p1,2 ) exp(hXp3 ) = exp(∆t X̃p1,2,3 ),
exp(∆t X̃EB ) exp(∆t X̃p1,2,3 ) = exp(∆t X̃),
where the corresponding Hamiltonians are given by
h
H̃EB = ĤE + ĤB + {ĤE , ĤB } + O(∆t2 ),
2
h
H̃p1,2 = Ĥp1 + Ĥp2 + {Ĥp1 , Ĥp2 } + O(∆t2 ),
2
h
H̃p1,2,3 = H̃p1,2 + Ĥp3 + {H̃p1,2 , Ĥp3 } + O(∆t2 )
2
h
h
= Ĥp1 + Ĥp2 + Ĥp3 + {Ĥp1 , Ĥp2 } + {Ĥp1 + Ĥp2 , Ĥp3 } + O(∆t2 ).
2
2
26
(141)
The Hamiltonian H̃ corresponding to X̃ is given by
H̃ = ĤE + ĤB + Ĥp1 + Ĥp2 + Ĥp3 + hH̃1 + O(∆t2 ),
with the first order correction H̃1 obtained as
i
1h
{ĤE , ĤB } + {Ĥp1 , Ĥp2 } + {Ĥp1 + Ĥp2 , Ĥp3 } + {ĤE + ĤB , Ĥp } .
H̃1 =
2
The various Poisson brackets are computed as follows,
(142)
(143)
{ĤE , ĤB } = e⊤ C ⊤ b,
{ĤE , Ĥp } = −e⊤
1
(X)⊤ M q V,
{ĤB , Ĥp } = 0,
{Ĥp1 , Ĥp2 } = V1Mq B3 (b, X)⊤ V2,
{Ĥp2 , Ĥp3 } = V2Mq B1 (b, X)⊤ V3,
{Ĥp3 , Ĥp1 } = V3Mq B2 (b, X)⊤ V1,
where Bµ (X, b) denotes Np × Np diagonal matrix with elements Bhµ (xa ). The Lie-Trotter
integrator (130) preserves the modified energy H̃ = Ĥ+hH̃1 to O(∆t2 ), while the original
energy Ĥ is preserved only to O(∆t).
6
Example: Vlasov-Maxwell in 1D2V
In one spatial dimension (x = x1 ) and two velocity dimensions v = (v1 , v2 ), the Vlasov
equation takes the following form
∂f (x, v, t) qs
∂f (x, v, t)
qs
v2
· ∇v f (x, v, t) = 0,
+ v1
+
E(x, t) · ∇v f (x, v, t) +
B3 (x, t)
−v1
∂t
∂x
ms
ms
(144)
while Maxwell’s equations become
∂B3 (x, t)
∂E2 (x, t)
=−
,
(145)
∂t
∂x
∂E1 (x, t)
= −j1 (x),
(146)
∂t
∂B3 (x, t)
∂E2 (x, t)
=−
− j2 (x),
(147)
∂t
∂x
∂E1 (x, t)
= ρ.
(148)
∂x
Here, we consider the components of the electromagnetic fields separately and we have
that E1 is a one-form, E2 is a zero-form and B3 is again a one-form. We denote the
semi-discrete fields by Dh , Eh and Bh respectively, and write
Dh (x, t) =
Eh (x, t) =
Bh (x, t) =
N1
X
i=1
N0
X
i=1
N1
X
i=1
27
di (t) Λ1i (x),
ei (t) Λ0i (x),
bi (t) Λ1i (x).
(149)
Next we introduce an equidistant grid in x and denote the spline of degree p with support
starting at xi by Nip . We can express the derivative of Nip as follows
d p
1
p−1
Nip−1 (x) − Ni+1
(x) .
(150)
Ni (x) =
dx
∆x
In the F inite Element field solver, we represent Eh by an expansion in splines of order p
and Dh , Bh by an expansion in splines of order p − 1, that is
Dh (x, t) =
Eh (x, t) =
Bh (x, t) =
N1
X
i=1
N0
X
i=1
N1
X
di (t) Nip−1 (x),
ei (t) Nip (x),
(151)
bi (t) Nip−1 (x).
i=1
The Hamiltonian is given in terms of the degrees of freedom u = (X, V1, V2 , d, e, b) by
Ĥ =
1
2
V1⊤ Mp V1 + 21 V2⊤ Mp V2 + 21 d⊤ M1 d + 21 e⊤ M0 e + 21 b⊤ M1 b.
(152)
The discrete Poisson bracket (80) for this system reads
∂ F̂
∂ Ĝ −1 ∂ F̂
∂ Ĝ
Mp−1
−
M
∂X1
∂V1 ∂X1 p ∂V1
⊤
∂ F̂
−1
1
⊤
−1 ∂ Ĝ
+
Mp Mq (X) M1
∂V1
∂d
⊤
∂ Ĝ
∂ F̂
−1 1
−1
M1
(X)Mq Mp
−
∂d
∂V1
⊤
∂ F̂
−1
0
⊤
−1 ∂ Ĝ
+
Mp Mq (X) M0
∂V2
∂e
⊤
∂ Ĝ
∂ F̂
−1 0
−1
M0
(X)Mq Mp
−
∂e
∂V2
⊤
∂ F̂
∂ Ĝ
−1
−1
+
Mp Mq B(X, b)Mp
∂V1
∂V2
⊤
∂ Ĝ
∂ F̂
−1
−1
Mp B(X, b)Mq Mp
−
∂V2
∂V1
⊤
∂ F̂
∂ Ĝ
+
M0−1 C ⊤ M1−1
∂e
∂b
⊤
∂ F̂
−1
−1 ∂ Ĝ
−
.
M1 CM0
∂b
∂e
{F̂ , Ĝ}[X, V1, V2 , d, e, b] =
(153)
Here, we denote by 0 (X) the Np ×N0 matrix with generic term Λ0i (xa ), where 1 ≤ a ≤ Np
and 1 ≤ i ≤ N0 , and by 1 (X) the Np × N1 matrix with generic term Λ1i (xa ), where
1 ≤ a ≤ Np and 1 ≤ i ≤ N1 . Further, B(X, b) denotes the Np × Np diagonal matrix with
entries
N1
X
Bh (xa , t) =
bi (t) Λ1i (xa ).
(154)
i=1
28
The equations of motion are obtained as follows,
Ẋ = V1 ,
V̇2 = Mp−1 Mq
ḋ = −M1−1
(X)d + B(X, b)V2 ,
0
(X)e − B(X, b)V1 ,
1
V̇1 = Mp−1 Mq
1
(X)⊤ Mq V1 ,
ė = M0−1 C ⊤ b(t) −
ḃ = −Ce(t),
0
(155)
(X)⊤ Mq V2 ,
which is seen to be in direct correspondence with (144)-(148).
6.1
Hamiltonian Splitting
The solution of the discrete equations of motion for ĤD + ĤE at time ∆t is
Mp V1 (∆t) = Mp V1 (0) + ∆t Mq
Mp V2 (∆t) = Mp V2 (0) + ∆t Mq
b(∆t) = b(0) − ∆t Ce(0).
1
0
(X(0)) d(0),
(X(0)) e(0),
(156)
The solution of the discrete equations of motion for ĤB is
M0 e(∆t) = M0 e(0) + ∆t C ⊤ b(0).
(157)
The solution of the discrete equations of motion for Ĥp1 is
X(∆t) = X1 (0) + ∆t V1 (0),
Z∆t
Mp V2 (∆t) = Mp V2 (0) − Mq B(X(t), b(0))V1(0) dt,
M1 d(∆t) = M1 d(0) −
0
Z∆t
1
(158)
(X(t))⊤ Mq V1 (0) dt,
0
and for Ĥp2 it is
Mp V1 (∆t) = Mp V1 (0) +
M0 e(∆t) = M0 e(0) −
Z∆t
0
∆t
Z
Mq B(X(0), b(0))V2(0) dt,
1
(159)
(X(0))⊤ Mq V2 (0) dt,
0
respectively.
6.2
Boris-Yee scheme
As an alternative discretization scheme, we consider a Boris-Yee scheme [9, 84] with our
conforming Finite Elements. The scheme uses a time staggering working with the variables
29
Xn+1/2 = X(tn + ∆t/2), dn+1/2 = d(tn + ∆t/2), en+1/2 = e(tn + ∆t/2), Vn = V(tn ), and
bn = b(tn ) in the nth time step tn = t0 + n∆t. The Hamiltonian at time tn is defined as
Ĥ =
1
2
(V1n )⊤ Mp V1n + 12 (V2n )⊤ Mp V2n + 21 (dn−1/2 )⊤ M1 dn+1/2
+ 12 (en−1/2 )⊤ M0 en+1/2 + 21 (bn )⊤ M1 bn . (160)
Given Xn−1/2 , dn−1/2 , en−1/2 , Vn−1, bn−1 the Vlasov-Maxwell system is propagated by
the following time step:
1. Compute bn according to
bni
=
bin−1
and bn−1/2 = (bn−1 + bn )/2.
∆t n−1/2
n−1/2
−
e
− ei−1
.
∆x i
(161)
2. Propagate vn−1 → vn by equation
∆t qs n−1/2 n−1/2
E
(xa
),
2 ms
1 − α2 −
2α −
+
va,1
=
v
+
v ,
a,1
1 + α2
1 + α2 a,2
−2α −
1 − α2 −
+
va,2
=
v
+
v ,
1 + α2 a,1 1 + α2 a,2
∆t qs n−1/2 n−1/2
van = va+ +
E
(xa
),
2 ms
va− = van−1 +
where α =
(162)
(163)
(164)
(165)
qa ∆t n−1/2 n−1/2
B
(x
).
ma 2
3. Propagate xn−1/2 → xn+1/2 by
n
xan+1/2 = xan−1/2 + ∆t va,1
.
(166)
and accumulate jn1 , jn2 by
jn1
=
Np
X
a=1
Np
jn2 =
X
a=1
n
wa va,1
Λ1 (xan−1/2 + xan+1/2 )/2 ,
n
wa va,2
Λ0 (xan−1/2 + xan+1/2 )/2 .
(167)
(168)
4. Compute dn+1/2 according to
and en+1/2 according to
M1 dn+1/2 = M1 dn−1/2 − ∆t jn1 ,
M0 en+1/2 = M0 en−1/2 +
∆t T n
C b − ∆t jn2 .
∆x
(169)
(170)
For the initialisation, we sample X0 and V0 from the initial sampling distribution, set e0 ,
b0 from the given initial fields, and solve Poisson’s equation for d0 . Then, we compute
X1/2 , d1/2 , and e1/2 from the corresponding equations of the Boris-Yee scheme for a half
time step, using b0 , V0 instead of the unknown values at time ∆t/4. Note that the error
in this step is of order (∆t)2 . But since we only introduce this error in the first time step,
the overall scheme is still of order two.
30
7
Numerical Experiments
We have implemented the Hamiltonian splitting scheme as well as the Boris-Yee scheme
as part of the SeLaLib library [1]. In this section, we present results for various test cases
in 1d2v, comparing the conservation properties of the total energy and of the Casimirs
for the two schemes. We simulate the electron distribution function in a neutralizing
ion background. In all experiments, we have used splines of order three for the 0-forms.
The particle loading was done using Sobol numbers and antithetic sampling (symmetric
around the middle of the domain in x and around the mean value of the Gaussian from
which we are sampling in each velocity dimension). We sample uniformly in x and from
the Gaussians of the initial distribution in each velocity dimension.
7.1
Weibel instability
We consider the Weibel instability studied in Weibel [80] in the form simulated in [28]. We
study a reduced 1d2v model with a perturbation along x1 , a magnetic field along x3 and
electric fields along the x1 and x2 directions. Moreover, we assume that the distribution
function is independent of v3 . The initial distribution and fields are of the form
v22
1 v12
1
+
(1 + α cos(kx)) ,
exp −
f (x, v, t = 0) =
πσ1 σ2
2 σ12 σ22
B3 (x, t = 0) = β cos(kx),
E2 (x, t = 0) = 0,
x ∈ [0, 2π/k),
(171)
(172)
(173)
and E1 (x, t = 0) is computed from Poisson’s equation.
√
choice of parameters, σ1 = 0.02/ 2, σ2 =
√ In our simulations, we use the following
12σ1 , k = 1.25, α = 0, and β = −10−4 . Note that these are the same parameters as in
[28] except for the fact that we sample from the Maxwellian without perturbation in x1 ,
relying on perturbations due to numerical noise.
The dispersion relation from [80] applied to our model reads
2 2
ω
σ2
ω
σ2
2
2
,
(174)
−1−
φ
D(ω, k) = ω − k +
σ1
σ1
σ1 k σ1 k
Rz
where φ(z) = exp − 21 z 2 −i∞ exp 21 ξ 2 dξ. For our parameter choice, this gives a growth
rate of 0.02784. In Fig. 1, we show the electric and magnetic energies together with the
analytic growth rate. We see that the growth rate is verified in the numerical solution.
This simulation was performed with 100,000 particles, 32 grid points, splines of degree 3
and 2 and ∆t = 0.05.
In Table 1, we show the conservation properties of our splitting with various order of
the splitting (cf. Sec. 5.2) and compare them also to the Boris–Yee scheme. The other
numerical parameters are kept as before.
We can see that the Poisson equation is satisfied in each time step for the Hamiltonian splitting. This is a Casimir (cf. Sec. 2.2) and therefore naturally conserved by the
Hamiltonian splitting. On the other hand, this is not the case for the Boris–Yee scheme.
We can also see that the energy error improves with the order of the splitting; however, the Hamiltonian splitting method as well as the Boris-Yee scheme are not energy
conserving. The time evolution of the total energy error is depicted in Fig. 2 for the
various methods.
31
Propagator
Lie
Strang
2nd, 4 Lie
4th, 3 Strang
4th, 10 Lie
Boris
total energy
4.9E-7
6.3E-7
9.8E-7
2.1E-9
2.1E-13
3.4E-10
Poisson
8.7E-15
1.5E-14
1.6E-14
2.2E-14
3.9E-14
1.0E-4
Table 1: Weibel instability: Maximum error in the total energy and Poisson’s equation
until time 500 for simulation with various integrators: Lie–Trotter splitting from (130)
(Lie), Strang splitting from (131) (Strang), second order splitting with 4 Lie parts defined
in (134) (2nd, 4 Lie), fourth order splitting with 3 Strang parts defined in (135) (4th, 3
Strang) and 10 Lie parts defined in (136) (4th, 10 Lie).
103
101
10-1
1
2
E1
2
1
2
E2
2
1
2
B3
2
growth rate
10-3
10-5
10-7
10-9
10-11
10-13
10-15
0
100
200
300
400
500
time
|H(time) − H(0)|/H(0)
Figure 1: Weibel instablity: The two electric and the magnetic energies together with the
analytic growth rate.
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16
Lie
Strang
2nd, 4 Lie
4th, 3 Strang
4th, 10 Lie
Boris
0
100
200
300
400
500
time
Figure 2: Weibel instablity: Difference of total energy and its initial value as a function
of time for various integators.
32
Propagator
Lie
Strang
2nd, 4 Lie
4th, 3 Strang
4th, 10 Lie
Boris
total energy
6.4E-5
1.4E-6
1.5E-8
1.7E-10
5.7E-13
1.1E-7
Poisson
8.3E-15
1.4E-14
2.0E-14
9.4E-15
1.0E-14
5.8E-4
Table 2: Streaming Weibel instability: Maximum error in the total energy and Poisson’s
equation until time 500 for simulation with various integrators.
7.2
Streaming Weibel instability
As a second test case, we consider the streaming Weibel instability. We study the same
reduced model as in the previous section, but following [16, 25] the initial distribution
and fields are prescribed as
(v2 − v0,1 )2
v12
1
δ exp −
exp − 2
f (x, v, t = 0) =
πσ
2σ
2σ 2
(v2 − v0,2 )2
,
(175)
+ (1 − δ) exp −
2σ 2
B3 (x, t = 0) = β sin(kx),
(176)
E2 (x, t = 0) = 0,
(177)
and E1 (x, t = 0) is computed from Poisson’s equation.
√
We set the parameters to the following values σ = 0.1/ 2, k = 0.2, β = −10−3 ,
v0,1 = 0.5, v0,2 = −0.1 and δ = 1/6. The parameters are chosen as in the case 2 of [25].
The growth rate of energy of the second component of the electric field was determined
to be 0.03 in [16]. In Fig. 3, we show the electric and magnetic energies together with the
analytic growth rate. We see that the growth rate is verified in the numerical solution.
This simulation was performed on the domain x ∈ [0, 2π/k) with 2,000,000 particles, 128
grid points, splines of degree 3 and 2 and ∆t = 0.01. Observe the energy of E1 emerges
at times earlier than in [16], which is caused by particle noise.
As for the Weibel instability, we compare the conservation properties in Table 2 for
various integrators. Again we see that the Hamiltonian splitting conserves the Poisson
equation as opposed to the Boris–Yee scheme. The energy conservation properties of the
various scheme show about the same behavior as in the previous test case (see also Fig. 4
for the time evolution of the energy error).
7.3
Strong Landau damping
Finally, we also study the electrostatic example of strong Landau damping with initial
distribution
2 2
v +v
1 + α cos(kx) .
(178)
f (x, v) = exp − 12σ2 2
The physical parameters are chosen as σ = 1, k = 0.5, α = 0.5 and the numerical
parameters as x ∈ [0, 2π/k), v ∈ R2 , ∆t = 0.01, nx = 32. The fields B3 and E2 are
initialized to zero and remain zero over time. In this example, we essentially solve the
33
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
1
2
E1
2
1
2
E2
2
1
2
B3
2
growth rate
0
50
100
150
200
time
|H(time) − H(0)|/H(0)
Figure 3: Streaming Weibel instablity: The two electric and the magnetic energies together with the analytic growth rate.
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16
Lie
Strang
2nd, 4 Lie
4th, 3 Strang
4th, 10 Lie
Boris
0
50
100
150
200
time
Figure 4: Streaming Weibel instablity: Difference of total energy and its initial value as
a function of time for various integrators.
10 1
γ1 =
− 0. 285473
10 0
1
2
||E1 (time, x)||2
γ2 = 0. 086671
10 -1
10 -2
10 -3
10 -4
10 -5
0
10
20
30
40
50
time
Figure 5: Landau damping: Electric energy with fitted growth rates.
34
Integrator
γ1
GEMPIC
−0.286
viVlasov1D [44]
−0.286
Cheng & Knorr [21]
−0.281
Nakamura & Yabe [64]
−0.280
Ayuso & Hajian [30]
−0.292
Heath, Gamba, Morrison, Michler [41] −0.287
Cheng, Gamba, Morrison [22]
−0.291
γ2
+0.087
+0.085
+0.084
+0.085
+0.086
+0.075
+0.086
Table 3: Damping and growth rates for strong Landau damping.
10-2
10-3
|H(time) − H(0)|/H(0)
10-4
10-5
10-6
10-7
lie
strang
2nd, 4 lie
4th, 3 strang
4th, 10 lie
Boris
10-8
10-9
10-10
10-11
10-12
0
10
20
30
40
50
time
Figure 6: Landau damping: Electric energy with fitted growth rates.
Vlasov–Ampère equation with results equivalent to the Vlasov–Poisson equations. In
Fig. 5 we show the time evolution of the electric energy associated with E1 . We have also
fitted a damping and growth rate (using the marked local maxima in the plot). These are
in good agreement with other codes (see Table 3). Again the energy conservation for the
various method is visualized as a function of time in Fig. 6. And again we see that the
fourth order methods give excellent energy conservation.
7.4
Backward Error Analysis
For the Lie-Trotter splitting, the error in the Hamiltonian Ĥ is of order ∆t. However,
using backward error analysis (cf. Section 5.3), modified Hamiltonians can be computed,
which are preserved to higher order. Accounting for first order corrections H̃1 , the error
in the modified Hamiltonian,
H̃ = Ĥ + ∆tH̃1 + O(∆t2 ),
(179)
is of order (∆t)2 . For the 1d2v example, this correction is obtained as
H̃1 =
i
1h
{ĤD + ĤE , ĤB } + {Ĥp1 , Ĥp2 } + {ĤD + ĤE + ĤB , Ĥp1 + Ĥp2 } ,
2
35
(180)
10-5
10-6
error
10-7
energy max
energy corrected max
energy l2
energy corrected l2
10-8
10-9
∆t
∆t 2
10-10
10-2
10-1
∆t
Figure 7: Weibel instability: Error (maximum norm and l2 norm) in the total energy for
simulations with ∆t = 0.01, 0.02, 0.05.
with the various Poisson brackets computed as
{ĤD , ĤB } = 0,
{ĤE , ĤB } = e⊤ C ⊤ b,
{Ĥp1 , Ĥp2 } = V1 Mq B(b, X)⊤ V2,
{ĤD , Ĥp1 } = −d⊤
1
(X)⊤ Mq V1 ,
{ĤD , Ĥp2 } = 0,
{ĤE , Ĥp1 } = 0,
{ĤE , Ĥp2 } = −e⊤
0
(X)⊤ Mq V2,
{ĤB , Ĥp1 } = 0,
{ĤB , Ĥp2 } = 0.
In Fig. 7, we show the maximum and l2 error of the energy and the corrected energy
for the Weibel instability test case with the parameters in Sec. 7.1. The simulations
were performed with 100,000 particles, 32 grid points, splines of degree 3 and 2 and
∆t = 0.01, 0.02, 0.05. We can see that the theoretical convergence rates are verified in the
numerical experiments. Figure 8 shows the energy as well as the modified energy for the
Weibel instability test case simulated with a time step of 0.05.
7.5
Momentum conservation
Finally, let us discuss conservation of momentum for our one species 1d2v equations. In
this case, Eq. (30) becomes
Z
dPe,1,2
= − E1,2 (x, t) dx,
(181)
dt
Let us consider Pe,1 (t). From (146), we get
Z
Z
d
E1 (x, t) dx = − j1 (x, t) dx.
dt
36
(182)
10-4
|H(time) − H(0)|/H(0)
10-5
10-6
10-7
10-8
10-9
10-10
energy
energy corrected
10-11
10-12
0
100
200
300
400
500
time
Figure 8: Weibel instability: Energy and first-order corrected energy for simulation with
∆t = 0.05.
R
Since in general j1 (x, t) dx 6= 0, momentum is not conserved. As a means of testing
moment conservation, Crouseilles et al. [28] replaced (146) by
Z
∂E1 (x, t)
= −j1 (x, t) + j1 (x, t) dx.
(183)
∂t
R
Thus, for this artificial dynamics, momentum will be conserved if E1 (x, 0) dx = 0.
Instead, we do not modify the equations but check the validity of (181): We define a
discrete version of (181), integrated over time, in the following way
Z
n Z
∆t X
n
0
m−1
m
P̃e,1 = Pe,1 −
Dh (x) dx + Dh (x) dx ,
(184)
2 m=1
Z
n Z
∆t X
m−1
m
n
0
Eh (x) dx + Eh (x) dx .
(185)
P̃e,2 = Pe,2 −
2 m=1
R
Note that in all our examples, jR1,2 (x, t) dx = 0. For a Gaussian initial distribution
our antithetic sampling ensures that j1,2 (x, t) dx = 0 holds in the discrete sense. Our
numerical results verify that momentum is indeed conserved up to roundoff error for the
Weibel and Landau test cases.
For the streaming Weibel instability, on the other hand, we have a sum of two Gaussians in the second component of the velocity. Since in our sampling method we draw the
particles based on Sobol pseudo-random numbers, the fractions Rdrawn from each of the
Gaussians are not exactly given by δ and 1 − δ as in (175). Hence j2 (x, t) dx is small but
nonzero. Comparing the discrete momentum defined by Eq. (185) with the discretization
of its definition (25),
Z
X
n
n
Pe,2 =
va,2 ma wa − Dhn (x)Bhn (x) dx,
(186)
a
we see a maximum deviation over time of 1.6×10−16 for a simulation with Strang splitting,
2, 000, 000 particles, 128 grid points, and ∆t = 0.01. This shows that our discretization
with a Strang splitting conserves (181) in the sense of (185) up to roundoff error.
37
8
Summary
In this work, a general framework for geometric Finite Element Particle-in-Cell methods
for the Vlasov-Maxwell system was presented. The discretization proceeded in two steps.
First, a semi-discretization of the noncanonical Poisson bracket was obtained, which preserves the Jacobi identity and important Casimirs. Then, the system was discretized
in time by Hamiltonian splitting methods, still retaining exact conservation of Casimirs.
Energy was not preserved, but backward error analysis showed that the energy error does
not depend on the degrees of freedom, the number of particles or the number of time steps.
The favorable properties of the method were verified in various numerical experiments.
The generality of the framework opens up several new paths for subsequent research.
Instead of splines, other Finite Element spaces that form a deRham sequence could be
used, e.g., mimetic spectral elements or Nédélec elements for one-forms and RaviartThomas elements for two-forms. Further, it also should be possible to apply this approach
to other systems like the gyrokinetic Vlasov-Maxwell system [15], although in this case the
necessity for new splitting schemes might arise. Energy-preserving time stepping methods might provide an alternative to Hamiltonian splitting algorithms, where a suitable
splitting cannot be easily found. This is a topic currently under investigation.
Acknowledgments
Useful discussion with Omar Maj and Marco Restelli are gratefully appreciated. MK
has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie grant agreement No 708124. PJM received support from the U.S. Dept. of Energy Contract # DE-FG05-80ET-53088 and a
Forschungspreis from the Humboldt Foundation. This work has been carried out within
the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053.
The views and opinions expressed herein do not necessarily reflect those of the European
Commission. A part of this work was carried out using the HELIOS supercomputer system at Computational Simulation Centre of International Fusion Energy Research Centre
(IFERC-CSC), Aomori, Japan, under the Broader Approach collaboration between Euratom and Japan, implemented by Fusion for Energy and QST.
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