AfPIC method with Forward-Backward Lagrangian reconstructions Martin Campos Pinto1 Merlin Pelz2 and Pierre-Henri Tournier3

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Aδf PIC method with Forward-Backward Lagrangian

reconstructions

Martin Campos Pinto1, Merlin Pelz2, and Pierre-Henri Tournier3

1Max-Planck-Institut f¨ur Plasmaphysik, Boltzmannstraße 2, D-85748 Garching b.

M¨unchen, Germany

2Department of Mathematics, University of British Columbia, Vancouver, British

Columbia, V6T 1Z2, Canada

3Sorbonne Universit´e, CNRS, Universit´e de Paris, Laboratoire Jacques-Louis Lions

(LJLL), F-75005 Paris, France

February 20, 2023

Abstract

In this work we describe a δf particle simulation method where the bulk density is

periodically remapped on a coarse spline grid using a Forward-Backward Lagrangian (FBL)

approach. This method is designed to handle plasma regimes where the densities strongly

deviate from their initial state and may evolve into general proﬁles. We describe the method

in the case of an electrostatic particle-in-cell scheme and validate its qualitative properties

using a classical two-stream instability subject to a uniform oscillating drive.

1 Introduction

In order to reduce the statistical noise in numerical simulations of kinetic plasma problems,

particle-in-cell (PIC) methods often follow a so-called δf approach [22, 14, 29] which consists

of decomposing the transported density in two parts, a bulk density f0given by an analytical

formula and a variation δf represented with numerical particles. In Ref. [2] this approach was

revisited as a variance reduction method in the scope of Monte Carlo algorithms, with f0playing

the role of a control variate, and since then several techniques have been devised to improve the

reduction of statistical error, in particular for gyrokinetic simulation models [20, 27, 26, 19] and

collisional models [8, 36, 34].

In many practical problems the plasma either remains close to an equilibrium state [20, 5],

or evolves as a small perturbation of some analytically known background [23, 3, 31] which can

be used as a bulk density. In some cases however the plasma evolves in an unpredictable way

and f0needs to be updated by a self-consistent algorithm to better follow the total density.

Typical examples are problems where full-fsimulations are needed. In the physics of tokamak

plasmas, one instance is the modeling of E×Bstaircases [13] which are long-lived patterns

of quasiregular step-like proﬁles that develop spontaneously in turbulent plasmas. As they

slowly move in the radial direction over large time scales, the separation assumption between an

arXiv:2210.03726v2 [physics.comp-ph] 17 Feb 2023

analytical background and ﬂuctuations is no longer valid after some time and simulations need

to involve full-fmethods such as the semi-Lagrangian scheme used in the GYSELA code [18].

Another example is the modelling of the tokamak edge region where the plasma density may

strongly deviate from local Maxwellian distributions, with large and intermittent ﬂuctuations.

This has motivated the development of various Eulerian full-fschemes such as those presented

in Ref. [15, 25], where large plasma blob structures can be seen propagating from the core region

towards the tokamak edge.

If one desires to model such problems with a δf PIC method, it is thus necessary to allow

for general updates of the bulk density over time. An interesting approach in this direction was

proposed in Ref. [1]: it consisted of projecting the particle density δf on a coarse spline basis

and add the resulting smoothed distribution to the bulk density.

In this article we consider a variant of this approach where the bulk density is updated using

a semi-Lagrangian approach based on the Forward-Backward Lagrangian (FBL) method [7].

Inspired by Ref. [12], the core of the FBL method is to compute backward trajectories on

arbitrary nodes by local inversions of the particle trajectories: as these describe the forward

transport ﬂow in phase space and are naturally provided by the PIC code, their local inversion

allows to perform semi-Lagrangian updates of a smooth density represented on a coarse grid.

In particular the novelty of our approach is that it does not primarily rely on an accurate

particle approximation of the density itself, but rather on an accurate description of the particle

trajectories. As these are in general much less noisy than the phase space density, we believe

that this new paradigm can lead to eﬃcient low-noise particle methods.

The outline is as follows. In Section 2 we present our general ansatz for the discrete density,

which may be seen as a hybrid discretization between particle and semi-Lagrangian density rep-

resentations. In Section 3 we recall the key steps of electrostatic particle-in-cell approximations,

and in Section 4 we describe the δf PIC method with FBL remappings of the bulk density. The

proposed method is summarized in Section 5, and in Section 6 we present a series of numerical

results involving two-stream instability test cases to illustrate the enhanced denoising properties

of our approach. We conclude in Section 7 with a summary of the proposed method, a discussion

on its novelty and two perspectives for future research.

2 Approximation ansatz

Our ansatz for the general density at a discrete time tnis a sum of two terms,

fn:= fn

∗+δfn(1)

(we shall often use := to highlight a deﬁnition) where the ﬁrst one will be seen as the bulk density,

a smooth approximation to the full solution, and the second one as the ﬁne scale variations.

Following the general principle of δf methods we require fn

∗to have a simple expression that is

easy to evaluate at arbitrary positions in a general d-dimensional phase space, and we represent

the variation δfnas an unstructured collection of numerical particles with coordinates zn

kand

weights δwn

δfn(z) :=

k=1

δwn

kϕε(z−zn

k).(2)

Here ϕεis a smooth shape function of scale εand z= (z1, . . . , zd) is a phase-space coordinate.

In typical problems where the transported density slightly deviates from an initial proﬁle, the

bulk density is often set to this initial value, fn

∗=f0or to some analytical equilibrium [20, 5].

In this note we investigate an alternate approach where the bulk density is represented as an

arbitrary collection of B-splines on a coarse grid with mesh-size h∗> ε, in the spirit of Ref. [1].

This yields an expression that is formally similar to that of δfn,

∗(z) = X

j∈d

∗,jϕh∗(z−jh∗) (3)

where ϕ∗(·−jh∗) is now the coarse B-spline shape centered on a general d-dimensional grid node

jh∗( is the set of integers) and wn

∗,jis its weight. In practice, only a ﬁnite number N∗of such

nodes is used, and as outlined in the introduction we will periodically update the coeﬃcients

of this spline bulk density using a Forward-Backward Lagrangian (FBL) reconstructions, which

involves a relatively small set of passive markers designed to track the characteristic ﬂow in

phase space.

2.1 Main numerical parameters and limit regimes

The main numerical parameters are as follows.

•Nris the remapping period, i.e., the number of time steps between two updates of the bulk

density. The limit value Nr=∞corresponds to a frozen bulk density, namely fn

∗=f0

∗

for every time step n.

•N∗is the number of coarse splines used to represent the bulk density in the computational

domain Ω ⊂d, it is on the order of Vol(Ω)(h∗)−d. The limit value of N∗= 0 (an empty

grid) corresponds to a “full-f” particle approximation.

•Npis the number of numerical particles describing the ﬁne scale structures. The limit

value of Np= 0 corresponds to a semi-Lagrangian ansatz [33] where the full density is

represented on a structured grid and can be updated in time with an ad-hoc scheme such

as the FBL method described in Ref. [7].

2.2 Weighted collections of spline shape functions

For simplicity, we consider spline shape functions for both the bulk density and the ﬁne scale

variations. Speciﬁcally, we set

ϕε(z) := 1

εdϕz

ε, z ∈d

with a reference shape function ϕdeﬁned as a centered cardinal B-spline of degree p,

ϕ(z) :=

i=1

Bp(zi) with support h−p+ 1

2,p+ 1

2id

involving standard univariate B-splines deﬁned recursively by

B0(x) := h−1

2,1

2i(x) and Bp(x) := ˆx+1

x−1

Bp−1for p≥1.

In the sequel, it will be convenient to denote an arbitrary collection of weighted splines as

Φε[W,Z](z) :=

k=1

wkϕε(z−zk)

where W= (wk)k=1...N ,Z= (zk)k=1...N . With this convention, the two components of our

general ansatz (1) read

∗= Φh∗[Wn

∗,Z∗] and δfn= Φε[δWn,Zn] (4)

where Z∗:= (jh∗)j∈dare the nodes of the spline grid.

3 Particle approximations to transport equations

Our method may be described for general non-linear transport problems of the form

∂tf(t, z) + U[f]· ∇zf(t, z) = 0 (5)

where z∈dis the phase-space variable and U[f] the generalized velocity ﬁeld associated to

the solution f.

3.1 Characteristic ﬂows

The characteristic trajectories associated to Eq. (5) are the curves Z(t) = Z(t;s, z)∈d,

solution to the ODEs d

dtZ(t) = U[f](t, Z(t)), Z(s) = z

for arbitrary s, t ∈[0, T ] and z∈d, see e.g. Ref. [30]. The (forward) characteristic ﬂow

between two times tn=n∆tand tn+1 = (n+ 1)∆tis then deﬁned as

Fn,n+1

ex (z) := Z(tn+1;tn, z) (6)

and the inverse mapping Bn,n+1

ex := Fn,n+1

ex −1is the backward ﬂow

Bn,n+1

ex (z) = Z(tn;tn+1, z).(7)

Using the backward ﬂow we can write the solution to Eq. (5) over the time interval [tn, tn+1] as

f(tn+1, z) = f(tn,Bn,n+1

ex (z)).(8)

Here we may restrict ourselves to divergence-free velocity ﬁelds: divzU[f] = 0. The character-

istic ﬂows are then measure-preserving and the transport is conservative.

3.2 1D1V Vlasov-Poisson equation

A simple example is the periodic Vlasov-Poisson equation in a two dimensional phase-space, i.e.,

z= (x, v) with a periodic space coordinate x∈[0, L] and velocity v∈

∂tf(t, x, v) + v∂xf(t, x, v)−E(t, x)∂vf(t, x, v) = 0 (9)

for t≥0, (x, v)∈[0, L]×, with a normalized initial density ´L

0´f0(x, v) dvdx= 1 and a

periodic electric ﬁeld E=E[f] deﬁned by







E(t, x) = −∂xφ(t, x)

−∆φ(t, x) = ρ(t, x) = 1

L−ˆf(t, x, v) dv. (10)

This corresponds to an electrostatic, normalized (ε0=qe=m= 1) periodic electron plasma in

1D, with constant neutralizing background ion density, so that ´L

0ρdx= 0. Here the generalized

velocity ﬁeld is

U[f](t, z) = v, −E(t, x)with z= (x, v).

Due to the non-linear nature of this transport equation, the characteristic ﬂow has no explicit

expression.

3.3 Full-fparticle approximation

Particle approximations represent the transported density fn(z)≈f(tn, z) as a sum of numerical

particles of the form

fn(z) = Φε[Wn,Zn](z) (11)

with weights initially set to

k:= f0(z0

Npg0(z0

k), k = 1, . . . , Np,(12)

where g0is the sampling distribution of the initial markers Z0= (z0

k)k=1,...,Np, see e.g. Ref. [34].

As the problem is conservative the weights are kept constant in time, Wn+1 =Wn, and the

markers are pushed forward

Zn+1 =Fn(Zn) (13)

using some approximation to the forward ﬂow in Eq. (6) which takes as parameter the numerical

solution at time tn,

Fn(Z) = F∆t[Wn,Zn](Z).

3.4 Electrostatic full-fleap-frog ﬂow

For the Vlasov-Poisson equation (9), a standard numerical ﬂow is given by a leap-frog (Strang

splitting) scheme,

Flf,pic

∆t,∆x[Wn,Zn](Z) = Fx

∆t

2◦ Fv,pic

∆t,∆x[Wn+1

2,Zn+1

2]◦ Fx

∆t

(Z) (14)

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AfPICmethodwithForward-BackwardLagrangianreconstructionsMartinCamposPinto1,MerlinPelz2,andPierre-HenriTournier31Max-Planck-InstitutfurPlasmaphysik,Boltzmannstrae2,D-85748Garchingb.Munchen,Germany2DepartmentofMathematics,UniversityofBritishColumbia,Vancouver,BritishColumbia,V6T1Z2,Canada3Sorbonne...

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AfPIC method with Forward-Backward Lagrangian reconstructions Martin Campos Pinto1 Merlin Pelz2 and Pierre-Henri Tournier3

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