This document presents a stochastic sharpening approach for improving the pinning and facetting of sharp phase boundaries in lattice Boltzmann simulations. It summarizes that multiphase lattice Boltzmann models are prone to unphysical pinning and facetting of interfaces. By replacing the deterministic sharpening threshold with a random variable, the approach delays the onset of these issues and better predicts propagation speeds, even with very sharp boundaries. The random projection method preserves the shape of a propagating circular patch, unlike the standard LeVeque model which results in facetting and non-circular shapes at high sharpness ratios.

1. A stochastic sharpening approach for the
pinning and facetting of sharp phase
boundaries in lattice Boltzmann simulations
T. Reis and P. J. Dellar
Department of Mathematical Sciences
University of Greenwich

2. Motivation
• Multiphase LBEs are prone to lattice pinning and facetting
(Latva-Kokko & Rothman (2005), Halliday et al. (2006)).
• Similar difficulties arise in combustion (Colella (1986), Pember
(1993)).
• Spurious phenomena associated with stiff source terms
investigated by LeVeque & Yee (1990).
• Unphysical behaviour can be corrected with a random projection
method (Bao & Jin (2000)).
• LB formulation of these models can shed light on pinning and
facetting.

3. Phase fields, pinning and facetting
Two types of particles: red & blue.
Scalar field φ, the “colour”
φ ≈ 1 in red phase (oil)
φ ≈ 0 in blue phase (water)
Diffuse interfaces marked by
transitions from φ ≈ 1 to φ ≈ 0.
∂t φ + u · φ = S(φ).

4. Model problem for propagating sharp interfaces
∂φ
∂t
+ u
∂φ
∂x
= S(φ) = −
1
T
φ 1 − φ 1
2 − φ
TS(φ) = − φ 1 − φ 1
2 − φ dφ
dt = S(φ)
“ The numerical results presented above indicate a disturbing feature
of this problem – it is possible to compute perfectly reasonable results
that are stable and free from oscillations and yet are completely
incorrect. Needless to say, this can be misleading.”
LeVeque & Yee (1990)

5. Lattice Boltzmann Implementation
Suppose φ = i fi and postulate the discrete Boltzmann equation:
∂t fi + ci ∂x fi = −
1
τ
fi − f
(0)
i + Ri .
With the following equilibrium function and discrete source term :
f
(0)
i = Wi φ(1 + λci u),
Ri = −1
2 S(φ)(1 + λci u);
the Chapman-Enskog expansion yields the
advection-reaction-diffusion equation:
∂φ
∂t
+ u
∂φ
∂x
= S(φ) +
τ
λ
(1 − u2
)
∂2
φ
∂x2
.

6. Reduction to Fully Discrete Form
The previous discrete Boltzmann equation can be written as
∂t fi + ci ∂x fi = Ωi
Integrate along a characteristic for time ∆t:
fi (x + ci ∆t, t + ∆t) − fi (x, t) =
∆t
0
Ωi (x + ci s, t + s) ds,
and approximate the integral by the trapezium rule:
fi (x +ci ∆t, t+∆t)−fi (x, t) =
∆t
2
Ωi (x +ci ∆t, t+∆t)
+ Ωi (x, t) +O ∆t3
.
This is an implicit system.

7. Change of Variables
To obtain a second order explicit LBE at time t + ∆t define
fi (x , t ) = fi (x , t ) +
∆t
2τ
fi (x , t) − f
(0)
i (x , t) −
∆t
2
Ri (x , t)
The new algorithm is
fi (x + ci ∆t , t + ∆t) − fi (x, t)
= −
∆t
τ + ∆t/2
fi (x, t) − f
(0)
i (x, t) +
τ∆t
τ + ∆t/2
Ri (x, t),
However, we need to find φ in order to compute f
(0)
i and Ri .

8. Reconstruction of φ
The source term does not conserve φ:
¯φ =
i
fi =
i
fi +
∆t
2τ
fi − f
(0)
i −
∆t
2
Ri
= φ +
∆t
2T
φ 1 − φ 1
2 − φ = N(φ).
We can reconstruct φ by solving the nonlinear equation N(φ) = ¯φ
using Newton’s method:
φ → φ −
N(φ) − ¯φ
N (φ)
,
Note that N(φ) = φ + O(∆t/T).

9. Sharpening without Advection
Initial condition: φ = sin2
(πx)
T = τ = 0.01

10. Pinning as a result of sharpening
Numerical solutions for τ/T = 1 (left) and τ/T = 50 (right).

11. Excessive Sharpening: τ/T = 100
For large timescale ratios, the interface can become completely
pinned to the lattice . . .

12. Conservation Trouble
In the 2D case the circular region shrinks and eventually disappears:
This is due to the curvature of the interface:
We find φc that satisfies
M =
Ω
S(φ, φc)dΩ.

13. Facetting with a LeVeque D2Q9 Model
Propagating circular patch when τ/T = 10. Notice the beginnings of
a facet on the leading edge . . .
When τ/T = 50 the facetting is severe . . .

14. Recap: Model problem for propagating sharp
interfaces
∂φ
∂t
+ u
∂φ
∂x
= S(φ) = −
1
T
φ 1 − φ 1
2 − φ
Usually implemented as separate advection and sharpening steps.
∂φ
∂t + u ∂φ
∂x = 0
∂φ
∂t
= S(φ)

15. The Random Projection Method
The LeVeque & Yee source term is effectively the deterministic
projection
SD(φ) : φ(x, t + ∆t) =
1 if φ (x, t) > 1/2,
0 if φ (x, t) ≤ 1/2.
Bao & Jin (2000) replace the critical φc = 1/2 with a uniformly
distributed random variable φ
(n)
c :
SR(φ) : φ(x, t + ∆t) =
1 if φ (x, t) > φ
(n)
c ,
0 if φ (x, t) ≤ φ
(n)
c .
∂φ
∂t
+ u · φ = −
1
T
φ 1 − φ φ(n)
c − φ .

16. Comparison of Propagation Speed (1D)
The speed of the interface is dictated by the ratio of timescales, τ
T .

17. Facetting with a RPM D2Q9 Model
Propagating circular patch when τ/T = 10. The blob retains its initial
shape well . . .
. . . even when τ/T = 50:

18. Comparing the Radii of the Patches at t = 0.4
The Random Projection model preserves the circular shape at
τ/T = 10. . .

19. Comparing the Radii of the Patches at t = 0.4
. . . and is clearly superior to the LeVeque model when τ/T = 50

20. Conclusion
• We have studied the combination of advection and sharpening of
the phase field that commonly arises in lattice Boltzmann models
for multiphase flows.
• Overly sharp phase boundaries can become pinned to the
lattice. This has been shown to be a purely kinematic effect.
• Replacing the threshold with a uniformly distributed
pseudo-random variable allows us to predict the correct
propagation speed at very sharp boundaries (τ/T > 100) and
delays the onset of facetting.
• This offers an explanation and improvement of the spurious
phenomena of pinning and facetting observed in multiphase
lattice Boltzmann simulations.