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IMEX Schemes for Advection-Diffusion EquationsDoktorandenseminar WS 14/15
Serena Keller
Institute of Aerodynamics and Gas Dynamics, University Stuttgart
27. January 2015
Outline
Motivation
Stability Conditions for Advection-Diffusion EquationsExplicit RKIMEX RK
IMEX Runge-Kutta SchemeAlgorithm
Benchmark ProblemsSinus Transport in 2DSingular Perturbation Problem in 1D
Summary - Outlook
Motivation
I Explicit Scheme: Time step restriction due to stabilityconditions
I IMEX schemes: Weaker time step restriction(Implicit discretization of the diffusion part)
⇒ Usage of greater time steps⇒ Saving computing time
Linear Scalar Advection-Diffusion Equation
ut + a · ∇u = d∆u in [0,T ]× Ω,
u(t, x) = uB(t, x) in [0,T ]× ∂Ω,
u(0, x) = u0(x) in Ω,
a: advection velocity,d : diffusion parameter.
Stability Conditions of Explicit Schemes
For explicit time discretizations the following stability conditionshave to be satisfied:
|a|∆t
∆x< 1
d∆t
∆x2<
1
2.
Thus ∆t has to be chosen so that
∆t < min
∆x
|a|,
∆x2
2d
.
Stability Condition of IMEX Runge-Kutta Schemes
The idea of IMEX (Implicit-Explicit) schemes is to dispose of thediffusion stability condition
H
HHHHH
d∆t
∆x2<
1
2.
by discretizing the diffusion part F d(u) := d∆u implicitly and theadvective part F a(u) := −a · ∇u explicitly :
ut = F a(u) + F d(u).
So the convection stability condition remains:
|a|∆t
∆x< 1.
IMEX Algorithm
ut = F a(u) + F d(u).
Explicit
F a(u) := −a · ∇u
Butcher Tables:
A, b and c of an explicit(s + 1)-stage ERK
Implicit
F d(u) := d∆u
Butcher Tables:
A =
[0 0∗ A
], b =
[0b
]and
c =
[0c
].
A, b and c are the Butchertables of a s-stage DIRK
IMEX Algorithm
Let un be given and un+1 wanted.
1 k1 = Fd (tn, un)
2 k1 = F a(tn, un)
3 DO i=2,...σ
4 Solve for ki : ki = Fd (tn + ci∆t, ui )
with ui = un + ∆t (∑i
j=1 aij kj +∑i−1
j=1 aij kj )5 Evaluate ki : ki = F a(tn + ci∆t, ui )
6 END DO
7 un+1 = un + ∆t (∑σ
j=1 bj kj +∑σ
j=1 bj kj )
Implicit equation:
ki = F d(tn + ci∆t, un + ∆t
( i∑j=1
aij kj +i−1∑j=1
aij kj))
Linear Equation System
Implicit equation:
ki = F d(tn + ci∆t, un + ∆t
( i∑j=1
aij kj +i−1∑j=1
aij kj))
⇒ ki − F d(., dtaii ki ) = F d(., un + ∆t
( i−1∑j=1
aij kj +i−1∑j=1
aij kj))
⇒ (I − dtaiiAd)︸ ︷︷ ︸
=:A
ki︸︷︷︸=:x
= Ad(.,un + ∆t
( i−1∑j=1
aij kj +i−1∑j=1
aij kj))
︸ ︷︷ ︸=:b
→ Solving the linear equation system with GMRES
Comparison of explicit and IMEX schemes
I To see the advantages of IMEX schemes, choose ~a and d sothat
∆x2
2d= min
∆x
|~a|,
∆x2
2d
. (1)
I Explicit time step is diffusion dominated:
∆t <∆x2
2d<
∆x
|~a|. (2)
I Stability condition for IMEX:
∆t <∆x
|~a|.
Temporal schemes - Spatial Schemes
I Temporal:In the following we will consider schemes of 3rd order:
I 3-stage ERKI 3-stage IMEX (ERK + SDIRK): Ascher3I 4-stage IMEX (ERK + ESDIRK): Kennedy4
I Spatial:I Discontinuous Galerkin Spectral Element Method (DG-SEM),
Code Flexi
Sinus Transport in 2DI Exact solution of the linear scalar advection-diffusion equation
in 2D
u(x , y , t) = sin(ω(x − a1 − t)) sin(ω(y − a2 − t))e−2dω2t .
where ~a = (a1, a2) = (1, 1).
CoordinateX
Co
ord
inat
eY
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Solution
0.90.80.70.60.50.40.30.20.10
-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9
t=0
CoordinateX
Co
ord
inat
eY
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Solution
0.90.80.70.60.50.40.30.20.10
-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9
t=2
CoordinateX
Co
ord
inat
eY
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Solution
0.90.80.70.60.50.40.30.20.10
-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9
t=4
CoordinateX
Co
ord
inat
eY
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1
Solution
0.90.80.70.60.50.40.30.20.10
-0.1-0.2-0.3-0.4-0.5-0.6-0.7-0.8-0.9
t=6
Parameter Standard ValuesType of nodes Gauss
Polynomial Order 8Mesh size 8x8
εGMRES truncation 10−4
d
Comparison of Explicit and IMEX schemes
d
tim
e st
ep
10-2 10-1 100 101 10210-7
10-6
10-5
10-4
10-3
10-2
Explicit ERK3IMEX Ascher3IMEX Kennedy4
d
L2
erro
r
10-2 10-1 100 101 10210-18
10-16
10-14
10-12
10-10
10-8
10-6
10-4
Explicit ERK3IMEX Ascher3IMEX Kennedy4
d
CP
U t
ime
10-2 10-1 100 101 10210-1
100
101
102
103
Explicit ERK3IMEX Ascher3IMEX Kennedy4
Singular Perturbation Problem in 1D
I Singular Perturbation Problem (SPP) can be seen as a modelfor the formation of a boundary layer
I Consider the linear scalar advection diffusion equation in 1Dwith a singular perturbation of the boundary values:
ut + a ux = d uxx in [0,T ]× [0, 1],
u(0, t) = 0 in [0,T ],
u(1, t) = 1 in [0,T ],
u(x , 0) = u0(x) in [0, 1].
I Consider the stationary limit of the problem with a = −1.∣∣∣‖uh(tn+1)− uex‖L2([0,1]) − ‖uh(tn)− uex‖L2([0,1])
∣∣∣ < c .
x
u
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
d=0.075d=0.1d=0.115
Parameter Standard ValuesType of nodes Gauss
Polynomial Order 8Mesh size 8
εGMRES truncation 10−2
Stationary truncation c 10−12
Comparison of Explicit and IMEX schemes
d
tim
e st
ep d
t
0.025 0.05 0.075 0.110-4
10-3
10-2
Explicit ERK3IMEX Ascher3IMEX Kennedy4
d
L2
erro
r
0.01 0.03 0.05 0.07 0.09 0.11 0.1310-8
10-7
10-6
10-5
10-4
10-3
10-2
Explicit ERK3IMEX Ascher3IMEX Kennedy4
d
CP
U t
ime
0.01 0.03 0.05 0.07 0.09 0.11 0.13
5
10
15
20 Explicit ERK3IMEX Ascher3IMEX Kennedy4
Summary
I For the Sinus transport: Kennedy4 and Ascher3 are fasterthan the explicit scheme, but less accurate.
I For the convergence to the stationary limit: Kennedy4 is thefastest scheme
I Although Kennedy4 has 4 stages, it needs less or the sametime than Ascher3 with the same L2 accuracy for theconsidered test cases
Outlook
I Preconditioning of the implicit part
I Implementing IMEX schemes for Navier Stokes with a nonlinear solver (Newton)