SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Semi-Lagrangian Formulations for LinearAdvection Equations and Applications to
Kinetic Equations
Jingmei Qiu
Department of Mathematical and Computer ScienceColorado School of Mines
joint work w/ Chi-Wang ShuSupported by NSF and AFOSR.
CSCAMM, University of Maryland College Park
1 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Outline
• Background: numerical methods for kinetic equations
• Semi-Lagragian finite difference methods for linearadvection equations
• Simulation results
• Ongoing and future work
2 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
VP system
The Vlasov-Poisson (VP) system,
∂f
∂t+ p · ∇xf + E(t, x) · ∇pf = 0, (1)
E(t, x) = −∇xφ(t, x), −4xφ(t, x) = ρ(t, x). (2)
f (t, x,p): the probability of finding a particle with velocity p atposition x at time t.ρ(t, x) =
∫f (t, x,p)dp - 1: charge density
3 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Numerical approach:Lagrangian vs. Eulerian vs.
semi-Lagrangian
• Lagrangian: tracking a finite number of macro-particles.– e.g., PIC (Particle In Cell)
dx
dt= v,
dv
dt= E (3)
• Eulerian: fixed numerical mesh– e.g., finite difference WENO, finite volume, finiteelement, spectral method.
• Semi-Lagrangian:–e.g., finite difference, finite volume, finite element,spectral method.
4 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Strang splitting for solving theVlasov equation
∂f
∂t+ v · ∇xf + E(t, x) · ∇vf = 0,
Nonlinear Vlasov eqation⇒ a sequence of linear equations.
• 1-D in x and 1-D in v :
∂f
∂t+ v
∂f
∂x= 0 , (4)
∂f
∂t+ E (t, x)
∂f
∂v= 0 . (5)
5 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
SL schemes
1 Solution space: point values, or cell averages, or piecewisepolynomials living on fixed Eulerian grid.
2 Evolution: tracking characteristics backward in time.
3 Project the evolved solution back onto the solution space.
Remark: The only error in time comes from trackingcharacteristics backward in time. The scheme is not subject toCFL time step restriction.
6 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Various formulationsof SL finite difference schemes
SL finite difference (point values)
• Scheme I: advective schemethe solution is evolved along characteristics (in Lagrangianspirit) approximating advective form of equation
ft + afx = 0.
• Scheme II: convective schemethe solution is evolved over fixed point (in Eulerian spirit)approximating conservative form of equation
ft + (af )x = 0.
? a being a constant, with possible extension to a = a(x , t)(relativistic Vlasov equation).
7 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Scheme I: advective scheme
f n+1i = f (xi , t
n+1) = f (x?i , tn) = f (xi − ξ0∆x , tn), ξ0 =
a∆t
∆x
When ξ0 ∈ [0, 12 ]
u r r r u u u u u r r r u tn+1
u r r r u u
u u u r r r u tn
x0 xi−2 xi−1 xi xi+1 xi+2 xNξ0 ∈ [0, 1
2]
8 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Third order example
f n+1i = f n
i + (−1
6f ni−2 + f n
i−1 −1
2f ni −
1
3f ni+1)ξ0
+ (1
2f ni−1 − f n
i +1
2f ni+1)ξ2
0
+ (1
6f ni−2 −
1
2f ni−1 +
1
2f ni −
1
6f ni+1)ξ3
0 , (6)
? Method 1: apply WENO interpolation on (6).? No mass conservation.
9 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Matrix vector form
f n+1i = f n
i + ξ0(f ni−2, f
ni−1, f
ni , f
ni+1) · AL
3 · (1, ξ0, ξ20)′, (7)
with matrix
AL3 =
−1/6 0 1/6
1 1/2 −1/2−1/2 −1 1/2−1/3 1/2 −1/6
=
−1/6 0 1/65/6 1/2 −1/31/3 −1/2 1/6
0 0 0
−
0 0 0−1/6 0 1/65/6 1/2 −1/31/3 −1/2 1/6
.
10 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Conservative formulation
Rewrite the advective scheme into a conservative form
f n+1i = f n
i − ξ0((f ni−1, f
ni , f
ni+1) · CL
3 · (1, ξ0, ξ20)′
−(f ni−2, f
ni−1, f
ni ) · CL
3 · (1, ξ0, ξ20)′)
= f ni − ξ0(f n
i+1/2 − f ni−1/2)
with
CL3 =
−16 0 1
656
12 −1
313 −1
216
.
? Method 2: apply WENO reconstructions on flux functionsf ni+1/2? The scheme is locally conservative. However, such splittingcan NOT be generalized to equations with variable coefficients.
11 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Scheme II: conservative scheme
Integrating the conservative form of PDE
ft + (af )x = 0,
over [tn, tn+1] at xi gives
f n+1i = f n
i − (
∫ tn+1
tn
af (x , τ)dτ)x |x=xi
.= f n
i −Fx |x=xi
?= f n
i −1
∆x(Fi+ 1
2− Fi− 1
2) +O(∆xk)
where F(x) =∫ tn+1
tn af (x , τ)dτ , and Fi± 12
are numerical fluxes.
12 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
F(x)
F(xi ) =
∫ tn+1
tn
af (xi , τ)dτ =
∫ xi
x?i
f (ξ, tn)dξ,
by applying the Divergence Theorem on∫
Ω(ft + (af )x)dx = 0.
u r r r u u u u u r r r u tn+1
u r r r u u u u u r r r u tn
x0 xi−2 xi−1 xi xi+1 xi+2 xN
13 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
F(x)
F(xi ) =
∫ tn+1
tn
af (xi , τ)dτ =
∫ xi
x?i
f (ξ, tn)dξ,
by applying the Divergence Theorem on∫
Ω(ft + (af )x)dx = 0.
u r r r u u u u u r r r u tn+1
u r r r u u u
Ω
x?i6
u u r r r u tn
x0 xi−2 xi−1 xi xi+1 xi+2 xN
13 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Fi± 12
Let H(x) to be a function s.t.
F(x) =1
∆x
∫ x+ ∆x2
x−∆x2
H(ξ)dξ
then
Fx |x=xi =1
∆x(H(xi+ 1
2)−H(xi− 1
2)).
Let numerical fluxes Fi± 12
= H(xi± 12).
14 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
A first order scheme
f n+1i = f n
i − (
∫ tn+1
tn
af (x , τ)dτ)x
.= f n
i −Fx
= f ni −
1
∆x(F(xi )−F(xi−1)), if a > 0
= f ni −
1
∆x(
∫ xi
x?i
f (ξ, tn)dξ −∫ xi−1
x?i−1
f (ξ, tn)dξ)
Remark. When a is a constant and cfl < 1, the schemereduces to
f n+1i = f n
i −a∆t
∆x(f n
i − f ni−1)
which is the first order upwind scheme.
15 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
High order WENO reconstructions
High order WENO reconstructions are applied in the followingreconstructions
f (xi , tn)Ni=1
WENO−→
∫ xi
x?i
f (ξ, tn)dξ
N
i=1
WENO−→Fi+ 1
2
N
i=1
• Computational expensive: two weno reconstructionprocedures
• The reconstruction stencil is widely spread (not compact),leading to instability of algorithm.
16 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
f (xi , tn)Ni=1
WENO−→
∫ xi
x?i
f (ξ, tn)dξ
N
i=1
WENO−→Fi+ 1
2
N
i=1
f (xi , tn)Ni=1
WENO/ENO
−−−−−−−−−−−−−−−−−−−−−−→Fi+ 1
2
N
i=1
17 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Third order WENO reconstructionwhen a = 1, dt < dx .
Consider two substencils
S1 = xi−1, xi, S2 = xi , xi+1.
Let ξ = dtdx , the third order reconstruction from the three point
stencil xi−1, xi , xi+1
Fi+ 12
= γ1F (1)
i+ 12
+ γ2F (2)
i+ 12
where the linear weights
γ1 =1− ξ
3, γ2 =
2 + ξ
3,
18 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
and
F (1)
i+ 12
= (−(3
2ξ +
1
2ξ2)f n
i + (1
2ξ +
1
2ξ2)f n
i−1)dx
F (2)
i+ 12
= ((−1
2ξ +
1
2ξ2)f n
i − (1
2ξ +
1
2ξ2)f n
i+1)dx
Idea of WENO: adjust the linear weighting γi to a nonlinearweighting wi , such that
• wi is very close to γi , in the region of smooth structures,
• wi weights little on a non-smooth sub-stencil.
19 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
• Smoothness indicator:
β1 = (f ni−1 − f n
i )2, β2 = (f ni − f n
i+1)2
• Nonlinear weights
w1 = γ1/(ε+ β1)2, w2 = γ2/(ε+ β2)2
• Normalized nonlinear weights wi :
w1 = w1/(w1 + w2), w2 = w2/(w1 + w2)
20 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Summary of scheme II
Method 3
1 Reconstruction numerical fluxes Fi± 12
from WENO/ENO
reconstruction of f ni Ni=1
2
f n+1i = f n
i −1
∆x(Fi+ 1
2− Fi− 1
2),
? Compare with the method of line approach (Runge Kuttatime discretization), the time integration of the PDE here isexact.? The scheme can be extended to accommodate extra largetime step evolution and to the case of variable coefficienta(x , t).
21 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Comparison
• Method 1 & 2:• Approximates the advective form of PDE• Frame of reference: Lagrangian• Method 2 is the conservative formulation of Method 1 for
equations with constant coefficients• Method 1 can be generalized to equations with variable
coefficients
• Method 3:• Approximates the conservative form of PDE• Frame of reference: Eulerian• conservative scheme• extends to equations with variable coefficients
22 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Numerical Simulations
Method 1, 2, 3 with fifth order WENO reconstruction
• Linear advection equation
• Rigid body rotation
• Vlasov Poisson system
23 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Linear transport: ut + ux + uy = 0
Table: Order of accuracy with u(x , y , t = 0) = sin(x + y) at T = 20.dt = 1.1dx = 1.1dy .
– method 1 method 2 method 3
mesh error order error order error order
20×20 6.03E-4 – 8.28E-4 – 7.94E-4 –
40×40 2.24E-5 4.75 2.62E-5 4.97 2.51E-5 4.98
60×60 3.10E-6 4.88 3.44E-6 5.00 3.29E-6 5.01
80×80 7.50E-7 4.93 8.16E-7 5.00 7.80E-7 5.00
24 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Rigid body rotation:ut − yux + xuy = 0
Figure: Initial profile and numerical solution of method 1 from thenumerical mesh 100× 100, dt = 1.1dx = 1.1dy at T = 12π
25 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Rigid body rotation:ut − yux + xuy = 0
Figure: Method 2 and 3 with the numerical mesh 100× 100,dt = 1.1dx = 1.1dy at T = 12π.
26 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Rigid body rotation:ut − yux + xuy = 0
Figure: Method 1 (left) and 3 (right).
27 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Rigid body rotation:ut − yux + xuy = 0
Figure: Method 1 (left) and 3 (right).
28 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Landau damping
Consider the VP system with initial condition,
f (x , v , t = 0) =1√2π
(1 + αcos(kx))exp(−v 2
2), (8)
29 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Weak Landau damping:α = 0.01, k = 2
Figure: Time evolution of L2 norm of electric field.
30 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Weak Landau damping (cont.)
Figure: Time evolution of L1 (upper left) and L2 (upper right) norm,discrete kinetic energy (lower left), entropy (lower right).
31 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Strong Landau damping:α = 0.5, k = 2
Figure: Time evolution of the L2 norm of the electric field.
32 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Strong Landau damping (cont.)
Figure: Time evolution of L1 (upper left) and L2 (upper right) norm,discrete kinetic energy (lower left), entropy (lower right).
33 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Two-stream instability
Consider the symmetric two stream instability, the VP systemwith initial condition
f (x , v , t = 0) =2
7√
2π(1 + 5v 2)(1 + α((cos(2kx)
+cos(3kx))/1.2 + cos(kx))exp(−v 2
2),
with α = 0.01, k = 0.5 and the length of domain inx−direction L = 2π
k .
34 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Figure: Two stream instability T = 53. The numerical solution ofmethod 1 (left) and Method 3 (right) with the numerical mesh64× 128.
35 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Figure: Third order semi-Lagrangian WENO method. Two streaminstability T = 53. The numerical mesh is 64× 128 (left) and256× 512 (right).
36 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Figure: Time development of L1 (upper left) and L2 (upper right)norm, discrete kinetic energy (lower left), entropy (lower right).
37 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
Ongoing and future work
• Algorithm:• Equations of variable coefficients• Truely multi-dimensional formulation of semi-Lagrangian
scheme evolving point values.
• Application• advection in incompressible flow• relativistic Vlasov equations
38 / 39
SL for linearadvection withapplications to
kineticEquations
Jingmei Qiu
Introduction
Proposedmethod
SL finitedifference I
SL finitedifference II
Comparison
Simulationresults
Summary
THANK YOU!
39 / 39