What is an AP scheme? I explain it through several examples!

1. Asymptotic Preserving Schemes
A conceptual journey
Hamed Zakerzadeh
Hamed Zakerzadeh AP schemes 1 / 4

2. Van der Pol oscillator
Balthasar van der Pol, around 1920,
introduced this -dependent equation for
triode oscillator:
v′′
− (1 − v2
)v′
+ v = 0
He even plotted the solution for different :
≪ 1 ∶ sinusoidal
oscillations
≫ 1 ∶ combination of
slow and fast dynamics
How to choose time step?
∆t = O(1/) → accurate
but too costly
∆t = O(1) → enough for
slow dynamics, useless or
unstable for fast
dynamics
Hamed Zakerzadeh AP schemes 2 / 4

3. Boundary layers and vanishing viscosity
Il’in, around 1969, investigated an
advection-diffusion equation over [0, 1]:
ν u′′
+ u′
= 0 u(0) = 0, u(1) = 1
whose exact solution is u(x) =
1 − e−x/ν
1 − e−1/ν
He found out that capturing the boundary
layer numerically is not easy!
uh
(kh) =
1 − (2ν−h
2ν+h
)
k
1 − (2ν−h
2ν+h
)
N
with h = 0.01 uh
(kh) =
1 − ( ν
ν+h
)
k
1 − ( ν
ν+h
)
N
with h = 0.01
Hamed Zakerzadeh AP schemes 3 / 4

4. Interaction of grid size and parameter may make numerical methods to fail → birth of
AP schemes as breakthrough in numerical resolution of multi-scale differential equations
Asymptotic Preserving (AP) schemes
Originated from the work of [Il’in,1969], but the term coined by Shi Jin (1995)
AP criteria:
-independent grid (in time and space)
stability for any
convergence to the correct asymptotic limit → 0
Hamed Zakerzadeh AP schemes 4 / 4