Download Semi-Lagrangian Formulations for Linear Advection

SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Semi-Lagrangian Formulations for Linear
Advection Equations and Applications to
Kinetic Equations
Jingmei Qiu
Department of Mathematical and Computer Science
Colorado School of Mines
joint work w/ Chi-Wang Shu
Supported by NSF and AFOSR.
CSCAMM, University of Maryland College Park
1 / 39
SL for linear
advection with
applications to
kinetic
Equations
Outline
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
• Background: numerical methods for kinetic equations
• Semi-Lagragian finite difference methods for linear
advection equations
• Simulation results
• Ongoing and future work
2 / 39
SL for linear
advection with
applications to
kinetic
Equations
VP system
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
The Vlasov-Poisson (VP) system,
∂f
+ p · ∇x f + E(t, x) · ∇p f = 0,
∂t
E(t, x) = −∇x φ(t, x),
−4x φ(t, x) = ρ(t, x).
(1)
(2)
Summary
f (t, x, p): the probability of finding a particle with velocity p at
position xRat time t.
ρ(t, x) = f (t, x, p)dp - 1: charge density
3 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Numerical approach:
Lagrangian vs. Eulerian vs.
semi-Lagrangian
• Lagrangian: tracking a finite number of macro-particles.
– e.g., PIC (Particle In Cell)
dx
dv
= v,
=E
dt
dt
• Eulerian: fixed numerical mesh
– e.g., finite difference WENO, finite volume, finite
element, spectral method.
(3)
• Semi-Lagrangian:
–e.g., finite difference, finite volume, finite element,
spectral method.
4 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Strang splitting for solving the
Vlasov equation
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
∂f
+ v · ∇x f + E(t, x) · ∇v f = 0,
∂t
Nonlinear Vlasov eqation ⇒ a sequence of linear equations.
• 1-D in x and 1-D in v :
∂f
∂f
+v
=0,
∂t
∂x
∂f
∂f
+ E (t, x)
=0.
∂t
∂v
(4)
(5)
5 / 39
SL for linear
advection with
applications to
kinetic
Equations
SL schemes
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
1
Solution space: point values, or cell averages, or piecewise
polynomials 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 tracking
characteristics backward in time. The scheme is not subject to
CFL time step restriction.
6 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Various formulations
of SL finite difference schemes
SL finite difference (point values)
• Scheme I: advective scheme
the solution is evolved along characteristics (in Lagrangian
spirit) approximating advective form of equation
ft + afx = 0.
• Scheme II: convective scheme
the 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 linear
advection with
applications to
kinetic
Equations
Scheme I: advective scheme
Jingmei Qiu
Introduction
Proposed
method
fi n+1 = f (xi , t n+1 ) = f (xi? , t n ) = f (xi − ξ0 ∆x, t n ),
SL finite
difference I
SL finite
difference II
Comparison
When ξ0 ∈ [0, 12 ]
Simulation
results
u rrr u
u
u
u r r r u t n+1
u u
u
u r r r u tn
u
Summary
ξ0 =
a∆t
∆x
u rrr u
x0
xi−2 xi−11 xi
ξ0 ∈ [0, 2 ]
xi+1 xi+2
xN
8 / 39
SL for linear
advection with
applications to
kinetic
Equations
Third order example
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
1 n
1
1 n
n
fi n+1 = fi n + (− fi−2
+ fi−1
− fi n − fi+1
)ξ0
6
2
3
1 n
1 n
+ ( fi−1
− fi n + fi+1
)ξ02
2
2
1 n
1 n
1
1 n
+ ( fi−2
− fi−1
+ fi n − fi+1
)ξ03 ,
6
2
2
6
(6)
? Method 1: apply WENO interpolation on (6).
? No mass conservation.
9 / 39
SL for linear
advection with
applications to
kinetic
Equations
Matrix vector form
Jingmei Qiu
n
n
n
fi n+1 = fi n + ξ0 (fi−2
, fi−1
, fi n , fi+1
) · AL3 · (1, ξ0 , ξ02 )0 ,
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
(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
1/6
0
0
0
 5/6
  −1/6
1/2
−1/3
0
1/6
 
= 
 1/3 −1/2 1/6  −  5/6
1/2 −1/3
0
0
0
1/3 −1/2 1/6


.

10 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Conservative formulation
Rewrite the advective scheme into a conservative form
Introduction
n
n
fi n+1 = fi n − ξ0 ((fi−1
, fi n , fi+1
) · C3L · (1, ξ0 , ξ02 )0
Proposed
method
n
n
−(fi−2
, fi−1
, fi n ) · C3L · (1, ξ0 , ξ02 )0 )
n
n
= fi n − ξ0 (fˆi+1/2
− fˆi−1/2
)
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
with

C3L = 
− 61
5
6
1
3
0
1
2
− 12
1
6

− 13  .
1
6
? Method 2: apply WENO reconstructions on flux functions
n
fˆi+1/2
? The scheme is locally conservative. However, such splitting
can NOT be generalized to equations with variable coefficients.
11 / 39
SL for linear
advection with
applications to
kinetic
Equations
Scheme II: conservative scheme
Jingmei Qiu
Integrating the conservative form of PDE
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
ft + (af )x = 0,
over [t n , t n+1 ] at xi gives
Z
fi n+1 = fi n − (
t n+1
tn
af (x, τ )dτ )x |x=xi
.
= fi n − Fx |x=xi
1
?
= fi n −
(F̂ 1 − F̂i− 1 ) + O(∆x k )
2
∆x i+ 2
R t n+1
where F(x) = t n af (x, τ )dτ , and F̂i± 1 are numerical fluxes.
2
12 / 39
SL for linear
advection with
applications to
kinetic
Equations
F(x)
Jingmei Qiu
Z
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
t n+1
F(xi ) =
Z
af (xi , τ )dτ =
xi
tn
xi?
by applying the Divergence Theorem on
R
u rrr u
u
u
u rrr u
u
u
u
f (ξ, t n )dξ,
Ω (ft
+ (af )x )dx = 0.
u r r r u t n+1
Summary
x0
xi−2 xi−1
xi
u
u r r r u tn
xi+1 xi+2
xN
13 / 39
SL for linear
advection with
applications to
kinetic
Equations
F(x)
Jingmei Qiu
Z
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
t n+1
F(xi ) =
Z
af (xi , τ )dτ =
xi
tn
xi?
by applying the Divergence Theorem on
R
u rrr u
u
u
u
f (ξ, t n )dξ,
Ω (ft
+ (af )x )dx = 0.
u r r r u t n+1
Ω
u r r
r u
u
x0 6 xi−2 xi−1
xi?
u
xi
u
u r r r u tn
xi+1 xi+2
xN
13 / 39
SL for linear
advection with
applications to
kinetic
Equations
F̂i± 12
Jingmei Qiu
Introduction
Proposed
method
Let H(x) to be a function s.t.
SL finite
difference I
SL finite
difference II
Comparison
1
F(x) =
∆x
Simulation
results
Summary
then
Fx |x=xi =
Z
x+ ∆x
2
H(ξ)dξ
x− ∆x
2
1
(H(xi+ 1 ) − H(xi− 1 )).
2
2
∆x
Let numerical fluxes F̂i± 1 = H(xi± 1 ).
2
2
14 / 39
SL for linear
advection with
applications to
kinetic
Equations
A first order scheme
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Z
fi n+1 = fi n − (
t n+1
af (x, τ )dτ )x
tn
.
= fi n − Fx
1
= fi n −
(F(xi ) − F(xi−1 )), if a > 0
∆x Z
Z xi−1
xi
1
n
n
= fi −
(
f (ξ, t )dξ −
f (ξ, t n )dξ)
?
∆x xi?
xi−1
Remark. When a is a constant and cfl < 1, the scheme
reduces to
fi n+1 = fi n −
a∆t n
n
(f − fi−1
)
∆x i
which is the first order upwind scheme.
15 / 39
SL for linear
advection with
applications to
kinetic
Equations
High order WENO reconstructions
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
High order WENO reconstructions are applied in the following
reconstructions
(Z
)N
n
oN
xi
WENO
n N WENO
n
{f (xi , t )}i=1 −→
−→ F̂i+ 1
f (ξ, t )dξ
xi?
Simulation
results
2
i=1
i=1
Summary
• Computational expensive: two weno reconstruction
procedures
• The reconstruction stencil is widely spread (not compact),
leading to instability of algorithm.
16 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
{f (xi , t
n
WENO
)}N
i=1 −→
(Z
xi
)N
WENO
n
−→
f (ξ, t )dξ
xi?
n
oN
F̂i+ 1
i=1
2
i=1
n
oN
WENO/ENO
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
→
F̂
{f (xi , t n )}N
1
i=1
i+
2
i=1
17 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Third order WENO reconstruction
when a = 1, dt < dx.
Consider two substencils
S1 = {xi−1 , xi },
S2 = {xi , xi+1 }.
dt
Let ξ = dx
, the third order reconstruction from the three point
stencil {xi−1 , xi , xi+1 }
Summary
(1)
(2)
F̂i+ 1 = γ1 F̂i+ 1 + γ2 F̂i+ 1
2
2
2
where the linear weights
γ1 =
1−ξ
,
3
γ2 =
2+ξ
,
3
18 / 39
SL for linear
advection with
applications to
kinetic
Equations
and
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
3
1
1
1
(1)
n
F̂i+ 1 = (−( ξ + ξ 2 )fi n + ( ξ + ξ 2 )fi−1
)dx
2
2
2
2
2
1
1
1
1
(2)
n
F̂i+ 1 = ((− ξ + ξ 2 )fi n − ( ξ + ξ 2 )fi+1
)dx
2
2
2
2
2
Simulation
results
Summary
Idea of WENO: adjust the linear weighting γi to a nonlinear
weighting 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 linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
• Smoothness indicator:
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
n
β1 = (fi−1
− fi n )2 ,
n 2
β2 = (fi n − fi+1
)
• Nonlinear weights
w̃1 = γ1 /( + β1 )2 ,
w̃2 = γ2 /( + β2 )2
• Normalized nonlinear weights wi :
w1 = w̃1 /(w̃1 + w̃2 ),
w2 = w̃2 /(w̃1 + w̃2 )
20 / 39
SL for linear
advection with
applications to
kinetic
Equations
Summary of scheme II
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Method 3
1
Reconstruction numerical fluxes F̂i± 1 from WENO/ENO
reconstruction of {fi n }N
i=1
2
2
1
(F̂ 1 − F̂i− 1 ),
2
∆x i+ 2
? Compare with the method of line approach (Runge Kutta
time discretization), the time integration of the PDE here is
exact.
? The scheme can be extended to accommodate extra large
time step evolution and to the case of variable coefficient
a(x, t).
fi n+1 = fi n −
21 / 39
SL for linear
advection with
applications to
kinetic
Equations
Comparison
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
• 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 linear
advection with
applications to
kinetic
Equations
Numerical Simulations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Method 1, 2, 3 with fifth order WENO reconstruction
• Linear advection equation
Simulation
results
• Rigid body rotation
Summary
• Vlasov Poisson system
23 / 39
SL for linear
advection with
applications to
kinetic
Equations
Linear transport: ut + ux + uy = 0
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Table: Order of accuracy with u(x, y , t = 0) = sin(x + y ) at T = 20.
dt = 1.1dx = 1.1dy .
–
mesh
20×20
40×40
60×60
80×80
method 1
error
order
6.03E-4
–
2.24E-5 4.75
3.10E-6 4.88
7.50E-7 4.93
method 2
error
order
8.28E-4
–
2.62E-5 4.97
3.44E-6 5.00
8.16E-7 5.00
method 3
error
order
7.94E-4
–
2.51E-5 4.98
3.29E-6 5.01
7.80E-7 5.00
24 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Rigid body rotation:
ut − yux + xuy = 0
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Initial profile and numerical solution of method 1 from the
numerical mesh 100 × 100, dt = 1.1dx = 1.1dy at T = 12π
25 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Rigid body rotation:
ut − yux + xuy = 0
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Method 2 and 3 with the numerical mesh 100 × 100,
dt = 1.1dx = 1.1dy at T = 12π.
26 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Rigid body rotation:
ut − yux + xuy = 0
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Method 1 (left) and 3 (right).
27 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Rigid body rotation:
ut − yux + xuy = 0
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Method 1 (left) and 3 (right).
28 / 39
SL for linear
advection with
applications to
kinetic
Equations
Landau damping
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Consider the VP system with initial condition,
1
v2
f (x, v , t = 0) = √ (1 + αcos(kx))exp(− ),
2
2π
(8)
29 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Weak Landau damping:
α = 0.01, k = 2
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Time evolution of L2 norm of electric field.
30 / 39
SL for linear
advection with
applications to
kinetic
Equations
Weak Landau damping (cont.)
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Time evolution of L1 (upper left) and L2 (upper right) norm,
discrete kinetic energy (lower left), entropy (lower right).
31 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Strong Landau damping:
α = 0.5, k = 2
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Time evolution of the L2 norm of the electric field.
32 / 39
SL for linear
advection with
applications to
kinetic
Equations
Strong Landau damping (cont.)
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Time evolution of L1 (upper left) and L2 (upper right) norm,
discrete kinetic energy (lower left), entropy (lower right).
33 / 39
SL for linear
advection with
applications to
kinetic
Equations
Two-stream instability
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Consider the symmetric two stream instability, the VP system
with initial condition
f (x, v , t = 0) =
2
√ (1 + 5v 2 )(1 + α((cos(2kx)
7 2π
+cos(3kx))/1.2 + cos(kx))exp(−
v2
),
2
with α = 0.01, k = 0.5 and the length of domain in
x−direction L = 2π
k .
34 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Two stream instability T = 53. The numerical solution of
method 1 (left) and Method 3 (right) with the numerical mesh
64 × 128.
35 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
Figure: Third order semi-Lagrangian WENO method. Two stream
instability T = 53. The numerical mesh is 64 × 128 (left) and
256 × 512 (right).
36 / 39
SL for linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
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 linear
advection with
applications to
kinetic
Equations
Ongoing and future work
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
Summary
• 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 linear
advection with
applications to
kinetic
Equations
Jingmei Qiu
Introduction
Proposed
method
SL finite
difference I
SL finite
difference II
Comparison
Simulation
results
THANK YOU!
Summary
39 / 39