VLASOV1:One-dimensional (1x1v) Electrostatic Vlasov Code

ISSS-13 Hands-on Session for Vlasov code

Takayuki UmedaInstitute for Space-Earth Environmental Research

Nagoya Universityumeda@isee.nagoya-u.ac.jp

Computer Simulations

• Programming in Fortran/C• Compiling (with gfortran/gcc,

Intel ifort/icc, PGI pgf90/pgcc)• Running programs• Data output• Data import into

Visualization tool• Visualization

• For research

• Coding in script

• Running scripts

• Visualization

• For education

GeneralMATLAB/Octave/ScilabIDL/FL, python, ...

• Easy for debugging• Useful to beginners

KEMPO1 vs. VLASOV1

• Particle-In-Cell– Lagrangian + Eulerian

• Electromagnetic /Electrostatic

• 1x3v

• Self-consistent (enhanced) thermal fluctuations

• Eulerian

• Electrostatic only– Limited applications

• 1x1v– 1x3v too heavy...

• Thermal fluctuations imposed as initial noise.

• Various schemes

• Similar graphical user interface of MATLAB

Computational Resources: How heavy??

• 1x1v: Nx × Nv = 1000×1000 ~ 10MBSuitable for laptop computing

• 2x2v: Nx×Ny×Nvx×Nvy = 1004 ~ 4GB• 2x3v: Nx×Ny×Nvx×Nvy×Nvz = 405 ~ 4GBHigh-performance and parallel computing is essential for hyper-dimensional simulations.

Large gaps in both computational time and memory

Push “Preview” to check parameters.

Exercises

• Electrostatic Linear Dispersion Relation: omega-k_diagram.dat

• Propagation of Langmuir Wavesand Landau Damping: wave_propagation.dat

• Langmuir Decay Instability: Lamgmuir_decay.dat• Two-stream Instability: two-stream_instability.dat• Buneman Instability: Buneman_instability.dat• Harmonic Langmuir waves: harmonic_Langmuir.dat

ES dispersion relation

• Load “w-k_diagram.dat” and Preview parameters. • START the simulation run. • Change any parameter as you like…(^^)

• Change Interpolation scheme and check how the ω-k diagram changes.

4th Lagrange FFT

CIPSpline

Numerical dispersion

Numerical dissipation

Basic Equations (1D)• Vlasov equation

and• Poisson equation

or• Maxwell equation

0=∂∂

+∂∂

+∂∂

xxx v

fEmq

xfv

tf

0=∂∂

+∂∂

xJ

txρ

∫= xvfq dρ

00

=+∂∂

εxx J

tE

00

=−∂∂

ερ

xEx

Overview of Numerical Techniques for 1D ES code

0=∂∂

+∂∂

+∂∂

xxx v

fEmq

xfv

tf

0

0

=∂∂

+∂∂

=∂∂

+∂∂

xx

x

vfE

mq

tf

xfv

tf

“Splitting” Scheme [Cheng & Knorr 1976]Vlasov equation

Valid when vx and Ex are constant in time.

Use this as an approximation for Δt << 1.

Splitting Scheme[Cheng & Knorr, 1976]

i i+1

j

j+1vx

Ex

0

0

=∂∂

+∂∂

=∂∂

+∂∂

xfE

mq

tf

xfv

tf

x

x

Get ○

Splitting Scheme[Cheng & Knorr, 1976]

i i+1

j

j+1vx

Ex

0

0

=∂∂

+∂∂

=∂∂

+∂∂

xfE

mq

tf

xfv

tf

x

x

Get △

Splitting Scheme[Cheng & Knorr, 1976]

i i+1

j

j+1vx

Ex

0

0

=∂∂

+∂∂

=∂∂

+∂∂

xfE

mq

tf

xfv

tf

x

x

Get ☆

Time Stepping

t – ∆t/2

t

t +∆t/2

t + ∆t

KEMPO1(EM)v

x

J

B E1

2

3

4

6

5

Splitting Schemev

x

J

E3

1

5

12

4

Time Stepping

t – ∆t/2

t

t +∆t/2

t + ∆t

KEMPO1(EM)v

x

J

B E1

2

3

4

6

5

VLASOV1(ES)v

x

J

E2

3

4

1

v: velocity***.m x: position***.m

Numerical Integration in Time

• Euler : Low accuracy• Runge-Kutta (4th-order) : Higher accuracy

Needs more memory to keep intermediate data

• Semi-Lagrangian

xff

tt

∂∂~δ

𝑓𝑓𝑡𝑡+∆𝑡𝑡 = 𝑓𝑓𝑡𝑡 + ∆𝑡𝑡𝛿𝛿𝑓𝑓𝑡𝑡

𝑓𝑓𝑡𝑡+∆𝑡𝑡(𝑥𝑥) = 𝑓𝑓𝑡𝑡(𝑥𝑥 − 𝑣𝑣∆𝑡𝑡)

𝑓𝑓𝑡𝑡+∆𝑡𝑡 = 𝑓𝑓𝑡𝑡 + ∆𝑡𝑡6 𝛿𝛿𝑓𝑓𝑡𝑡+2𝛿𝛿𝑓𝑓′+2𝛿𝛿𝑓𝑓′′+𝛿𝛿𝑓𝑓′′′

𝑓𝑓′ = 𝑓𝑓𝑡𝑡 + ∆𝑡𝑡2 𝛿𝛿𝑓𝑓

𝑡𝑡 , 𝑓𝑓′′ = 𝑓𝑓𝑡𝑡 + ∆𝑡𝑡2 𝛿𝛿𝑓𝑓

′

𝑓𝑓′′′ = 𝑓𝑓𝑡𝑡 + ∆𝑡𝑡𝛿𝛿𝑓𝑓′′

Semi-Lagrangian

0=∂∂

+∂∂

xfV

tf

),(),( ttVxfttxf ii ∆−=∆+

x

f(x,t)V∆t

1D advection equation

Shift of profiles with a numerical interpolation.

General solution

f(x,t+∆t)

)( Vtxff −≡

Interpolation Schemes

SchemeLinear2nd-order4th-orderCubic SplineFFTCIP3PIC4

AccuracyVery lowMiddleHighHighHighestVery highHigh

Conservation/Oscillation

Y/NY/YY/YN/YN/YN/YY/N

CPU loadLowLowMiddleHigherHighHighHigher

Conservative Form

• Non-conservative form

• Conservative form

For the periodic system (U1/2=UNx+1/2)

it

itt

i fff δ+=∆+

21

21 +−

∆+ −+= iit

itt

i UUff

∑∑ =∆+

i

ti

i

tti ff

∆t is included in δf and U.

Lagrange Interpolation• 1st-order (Linear)

• 2nd-order

• 4th-order

abxxg +=)(

abxcxxg ++= 2)(

Spuriousundershoot/overshoot

(extremum)

abxcxdxexxg ++++= 234)(

Lagrange Interpolation• 1st-order upwind

Conservative• 2nd-order central

Conservative

( )ti

ti

xti

tti ff

xtvff −

∆∆

+= −∆+

1

Ui+1/2Ui-1/2

+−

∆∆

+

−∆∆

+= +−+−∆+

22

211

211

ti

ti

tix

ti

tixt

itt

ifff

xtvff

xtvff

−

∆∆

−

+∆∆

−

−

∆∆

+

+∆∆

+=

++

−−∆+

22

22

12

1

12

1

ti

tix

ti

tix

ti

tix

ti

tixt

itt

i

ffxtvff

xtv

ffxtvff

xtvff Ui-1/2

Ui+1/2

Lagrange Interpolation• 4h-order centralConservative

+−+−

∆∆

+

+−−

∆∆

+

−+−

∆∆

+

−++−∆∆

=

−++

−++

−++

−+++

2433

12

241515

1277

1124

1123

1122

11221

ti

ti

ti

tix

ti

ti

ti

tix

ti

ti

ti

tix

ti

ti

ti

tix

i

ffffxtv

ffffxtv

ffffxtv

ffffxtvU

Cubic Spline32 xdxcxbff iii

ti

tti −+−=∆+

Non-conservativeδf

iiit

it

i

iiit

it

i

dcbffdcbff

+++=

−+−=

+

−

1

1

iiii

ti

iiii

ti

dcbbx

f

dcbbx

f

32

32

11

11

++==∂∂

−+−=−=∂∂

++

−−

xxtvx ≡

∆∆

x=-1 for “i+1”x=0 for “i”x=1 for “i-1”

iii

ti

iii

ti

dccxf

dccxf

622

622

121

2

121

2

+==∂∂

−==∂∂

++

−−

MATLAB has “spline” function.

Matrix operation necessary“Global” interpolation

FFT (Spectral Scheme)

tkx

tk

ttk FtjkvFF ∆+=∆+

]exp[ tjkvFF xt

ktt

k ∆−=∆+

[ ]∑ −=k

ikt

i tkxjFf )(exp ω

0=∂∂

+∂∂

xfv

tf

x

),(),( ttvxfttxf xii ∆−=∆+Non-conservativeGlobal interpolation

MATLAB has “fft” function.

Semi-Lagrangian

CIP Scheme: Constrained Interpolation Profile

xfg

xgv

tg

xfv

tf

x

x

∂∂

≡=∂∂

+∂∂

=∂∂

+∂∂

,0

0

One of “multi-data (multi-moment)” schemes

2

32

32

∆∆

+∆∆

+=

∆∆

+

∆∆

+∆∆

+=

∆+

∆+

xtvd

xtvcgg

xtvd

xtvc

xtvgff

xi

xi

ti

tti

xi

xi

xi

ti

tti

CIP Scheme: Constrained Interpolation Profile

xfg

xgv

tg

xfv

tf

x

x

∂∂

≡=∂∂

+∂∂

=∂∂

+∂∂

,0

0

2

32

32

∆∆

+∆∆

+=

∆∆

+

∆∆

+∆∆

+=

∆+

∆+

xtvd

xtvcgg

xtvd

xtvc

xtvgff

xi

xi

ti

tti

xi

xi

xi

ti

tti

)(2)(32

11

11

−−

−−

−−+=−−+=

iiiii

iiiii

ffggdffggc

3data (𝑓𝑓, 𝜕𝜕𝑓𝑓/𝜕𝜕𝑥𝑥, 𝜕𝜕𝑓𝑓/𝜕𝜕𝑣𝑣) for 1x1v

One of “multi-data (multi-moment)” schemes

“PIC” Scheme: Conservative,Non-oscillatory and Positive

( )( )( ) ( )( )( )( ) ( )( )( )( ) ( ) cffxxx

bffxxxaffxxxxfU

ti

ti

ti

ti

ti

ti

tii

12

1

1

2112132112

21

++

+

−+

−−−+−−−−++

−−++−=

Another conservative form of the 4th-degree Lagrange interpolation.

Modification (correction) of the gradients.

Details are given in Umeda EPS 2008 & Umeda et al. CPC 2012

Gradients

0 ≤ 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 ≤ 1

“PIC” Scheme: Conservative,Non-oscillatory and Positive

Details are given in Umeda EPS 2008 & Umeda et al. CPC 2012

Detect local extrema (fmax and fmin) and limit the gradient to (fmax-fi) or (fmin-fi)

Langmuir wave propagation

• Load “wave_propagation.dat” and Previewparameters. Also, preview the initial noise.

• START the simulation run.

Electron fluid eqs.

γ = 3 for 1D velocity space.

Langmuir wave propagation

• Change initial noise and check the wave propagation.

• Change interpolation scheme and check how the result changes.

Linear interpolation

Numerical diffusion (dissipation)

Langmuir wave propagation

• Change initial noise and check the wave propagation.

• Change interpolation scheme and check how the result changes.

• Change E-field scheme and check how the result changes.– Charge+Poisson and Charge Conservation

should be same.

i-1 i i+1ρ

f(v)ρ

f(v)ρ

f(v)

ExJx

Staggered Grid SystemSame as KEMPO1Ex

Jx

Poisson Maxwell0=

∂∂

+∂∂

xJ

txρ

∫= xvfq dρ

Charge conservation

i-1 i i+1ρ

f(v)ρ

f(v)ρ

f(v)

ExJx

Collocated Grid SystemExJx

Maxwell

∫= xvfq dρ

ExJx

∫= xxx vfvqJ d

Can we use this “non-conservative” scheme??Check by Current+E-field

Landau damping

• Change Initial noise to a higher mode number and check the wave propagation.

Mode 7 + FFT

exp(-0.096*2 t)

Recurrence [Cheng & Knorr 1976]

(back to initial condition)

dv ~ 0.2kx = 2*pi*7/100tR = 2*pi/(kx*dv)

~ 71.4

Langmuir decay• Load “Langmuir_decay.dat” and Preview parameters. • START the simulation run. • Make the thermal velocity of ions (Species 2) smaller

and check how the wave propagate. VT(2) = 0.05VT(2) = 0.2

Langmuir decay• Make the amplitude of the initial noise smaller and

check how the wave mode propagate.• Check that large-amplitude Langmuir waves are

stable when ions are absent.

VT(2) = 0.05, PAMP(4) = 0.5VT(2) = 0.05, Pamp(4) = 0.05

Two-stream instability

• Load “two-stream_instability.dat” and Previewparameters.

• START the simulation run. • Change Interpolation scheme and check how

the result changes. – Compare the results between 4th Lagrange

and PIC4, and check how the limiter of PIC4 scheme affects.

CIP3

4thLagrange

PIC4

Buneman instability• Load “Buneman_instability.dat” and Preview

parameters. • START the simulation run. • Change the thermal velocity of ions and check

how the result changes. VT(2) = 0.2VT(2) = 0.1

Harmonic Langmuir waves• Load “harmonic_Langmuir.dat” and Preview

parameters. • START the simulation run.

Yoon et al. & Umeda et al. PoP 2003

For further studies…• If you want install your own scheme, edit

“velocity.m” and “position.m” files. – Numerical schemes for Vlasov simulations

are still developing. • Try to implement recent high-accuracy

schemes into VLASOV1.– MMA [Minoshima+ CPC 2015] is a high-accuracy

multi-data scheme. Please compare with CIP.– Monotonicity Preserving scheme [Tanaka+ ApJ

2017] is a higher-order one. Please compare with PIC4.

For further studies…• For scientific computing, the VLASOV1/MATLAB

code should be ported to Fortran/C. – Fortran/C codes run much faster than MATLAB

scripts.• Parallel computing is also necessary for solving

multiple velocity dimensions. • High-performance computing techniques are

required for large-scale simulations. • If you find any bug, please report to

umeda@isee.nagoya-u.ac.jp