Asymptotically accurate high-order space and time schemes for the Euler system in the low Mach regime Victor Michel-Dansac
SHARK-FV, 15-19 May 2017 Giacomo Dimarco, Univ. of Ferrara, Italy Raphaël Loubère, Univ. of Bordeaux, CNRS, France Marie-Hélène Vignal, Univ. of Toulouse, France Funding : ANR MOONRISE
Outline
1
General context : multi-scale models and principle of AP schemes
2
An order 1 AP scheme for the Euler system in the low Mach limit
3
High order schemes in time
4
High order schemes in time and space
5
Works in progress en perspectives
General context electron density (log scale) : t=0.1 0 2.4
Multiscale model : Mε depends on a parameter ε
−1
2.3 −2 2.2 −3 2.1 −4 2
In the (space-time) domain ε can
−5 1.9 −6
1.8
−7
1.7
be small compared to the reference scale be of same order as the reference scale
−8
1.6
1.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
−9
1
ε
0
10
take intermediate values −5
10
When ε is small : M0 = lim Mε asympt. model ε→0
Difficulties :
−10
10
0
0.2
0.4
0.6
0.8
1
x
Classical explicit schemes for Mε : stable and consistent if the mesh resolves all the scales of ε ⇒ huge cost Schemes for M0 with meshes independent of ε
But : ➠ M0 not valid everywhere, needs ε ≪ 1 ➠ location of the interface, moving interface
Principle of AP schemes A possible solution : AP schemes Use the multi-scale model Mε where you want. Discretize it with a scheme preserving the limit ε → 0 ➠ The mesh is independent of ε : Asymptotic stability.
➠ You recover an approximate solution of M0 when ε → 0 : Asymptotic consistency Asymptotically stable and consistent scheme ⇒ Asymptotic preserving scheme (AP) ([S.Jin] kinetic → hydro) You can use the AP scheme only to reconnect Mε and M0 Mε class. scheme
ε = O ( 1)
0000 1111 00000000000000 11111111111111 0000 1111 1111 0000 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 intermediate 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 zone 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111 0000 1111 00000000000000 11111111111111 0000 1111
M0 class. scheme ε≪1
Mε AP scheme
Outline
1
General context : multi-scale models and principle of AP schemes
2
An order 1 AP scheme for the Euler system in the low Mach limit
3
High order schemes in time
4
High order schemes in time and space
5
Works in progress en perspectives
The multi-scale model and its asymptotic limit ➠ Isentropic Euler system in scaled variables x ∈ Ω ⊂ IRd , t ≥ 0
(M ε )
(
∂t ρ + ∇ · (ρ u ) = 0,
(1)ε 1
∂t (ρ u ) + ∇ · (ρ u ⊗ u ) + ∇p (ρ) = 0, (2)ε ε Parameter : ε = M 2 = m |u |2 /(γ p (ρ)/ρ), M =Mach number Boundary and initial conditions : u · n = 0, on ∂Ω,
(
and
˜ 0(x ), ρ(x , 0) = ρ0 + ε ρ ˜0 (x ), with ∇ · u0 = 0. u (x , 0) = u0 (x ) + ε u
The formal low Mach number limit ε → 0
(2)ε ⇒ ∇p (ρ) = 0, ⇒ ρ(x , t ) = ρ(t ). ′
Z
(1)ε ⇒ |Ω| ρ (t )+ρ(t )
u · n = 0, ⇒ ρ(t ) = ρ(0) = ρ0, ⇒ ∇· u = 0
∂Ω
The multi-scale model and its asymptotic limit The asymptotic model : Rigorous limit [Klainerman, Majda, 81]
where
ρ = cste = ρ0 , (M 0 ) ρ 0 ∇ · u = 0, ρ ∂ u + ρ ∇ · (u ⊗ u ) + ∇π = 0, 0 t 0 1 1 π1 = lim p (ρ) − p (ρ0) . ε→0 ε
(1)0 (2)0
Explicit eq. for π1 ∂t (1)0 − ∇ · (2)0 ⇒ −∆π1 = ρ0 ∇2 : (u ⊗ u ). The pressure wave eq. from Mε :
∂t (1)ε − ∇ · (2)ε
⇒
1
∂tt ρ − ∆p (ρ) = ∇2 : (ρ u ⊗ u ) ε
From a numerical point of view
√
(3)ε
Explicit treatment of (3)ε ⇒ conditional stability ∆t ≤ ε ∆x Implicit treatment of (3)ε ⇒ uniform stability with respect to ε
An order 1 AP scheme in the low Mach numb. limit Order 1 AP scheme in [Dimarco, Loubère, Vignal, SISC 2017] : If ρn and u n are known at time t n
n +1 ρ − ρn + ∇ · (ρ u )n+1 = 0, ∆t n +1 1 − (ρ u )n (ρ u ) + ∇ · (ρ u ⊗ u )n + ∇p (ρn+1 ) = 0. ∆t ε ∇ · (2) inserted into (1) : gives an uncoupled formulation
(1)
(AS)
(2)
(AC)
∆t ρn +1 − ρn + ∇ · (ρ u )n − ∆p (ρn+1 ) − ∆t ∇2 : (ρ u ⊗ u )n = 0, ∆t ε
➠ Results uniformly L∞ stable if the space discretization is well chosen ➠ Framework of IMEX (IMplicit-EXplicit) schemes :
ρu ρ 0 ∂t +∇ · +∇ · p(ρ) = 0. ρu ρu ⊗u Id ε | {z } | {z } | {z }
W
Fe (W )
Fi (W )
AP but diffusive results,1-D test-case ε = 0.99, 300 cells
x 10
1.8
Class. scheme AP scheme
5
Class. scheme AP scheme
1.6
Density
1.4
Time steps
Class : 273 loops CPU time 0.07 AP : 510 loops CPU time 1.46
−4
6
4
3
1.2 1 0.8
2
0.6
1
0.4
0 0
0.01
0.02
0.03
0.04
0.05
0.2
0.06
0
0.2
0.4
ε = 10−4 , 300 cells
1
−3
Density
−4
10
0.8
1
Class. scheme AP scheme
10
Time steps
Class : 4036 loops CPU time 0.82 AP : 57 loops CPU time 0.14
0.6
x
Time
Class. scheme
1
AP scheme
−5
10
0
0.01
0.02
0.03
Time
0.04
0.05
0.06
0.9999 0
0.2
0.4
0.6
x
0.8
1
AP but diffusive results,1-D test-case ε = 10−4
−4
10
1
Class. scheme AP scheme Density
Underlying of the viscosity
Time steps
Class. scheme AP scheme
−5
10
−6
10
0
0.01
0.02
0.03
0.04
0.05
0.06
1
0.9999 0
0.2
0.4
0.6
x
Time
⇓ It is necessary to use high order schemes But they must respect the AP properties
0.8
1
Outline
1
General context : multi-scale models and principle of AP schemes
2
An order 1 AP scheme for the Euler system in the low Mach limit
3
High order schemes in time
4
High order schemes in time and space
5
Works in progress en perspectives
Principle of IMEX schemes Biblio for stiff source terms or ode pb. : Asher, Boscarino, Cafflish, Dimarco, Filbet, Gottlieb, Le Floch, Pareschi, Russo, Ruuth, Shu, Spiteri, Tadmor... IMEX division : ∂t W + ∇ · Fe (W ) + ∇ · Fi (W ) = 0.
General principle : Step n : W n is known Quadrature formula with intermediate values Z t n+1 Z t n+1 ∇ · Fi (W (t )) dt ∇ · Fe (W (t )) dt − W (t n + 1 ) = W (t n ) − W n +1 = W n
|t
n
s
{z
} |t
n
s
{z
−∆t ∑ b˜j ∇ · Fe (W n,j ) −∆t ∑ bj ∇ · Fi (W n,j ) j =1
j =1
}
˜j = ∑sj=1 bj = 1 Quadratures exact on the constants : ∑sj=1 b Intermediate values t n,j = t n + cj ∆t Z Z t n,j n n ,j n ∂t W (t ) dt = W + ∆t W ≈ W (t ) + tn
0
cj
∂t W (t n + s ∆t ) ds
Principle of IMEX schemes Quadrature formula for intermediate values : i = 1, · · · , s W n,j = W n − ∆t
∑ a˜j ,k ∇ · Fe (W n,k ) − ∆t ∑ aj ,k ∇ · Fi (W n,k ),
k 0) ci n+1,O1 n+1,O1 n+1,O1 wj = wjn − √ (wj − wj −1 ) − ce (wjn − wjn−1 )
ε
√
Order 2 AP scheme ARS with the parameter β = 1 − 1/ 2 Lemma (VMD,Loubère,Vignal) Under the CFL condition ∆t ≤ ∆x /ce
β ≃ 0.41 θ≤ 1−β
⇒
TV (w n+1 ) ≤ TV (w n ) kw n + 1 k∞ ≤ kw n k∞
A MOOD procedure Limited AP scheme : w n+1,lim = θ w n+1,O2 + (1 − θ) w n+1,O1
with
θ=
Problem : More accurate than order 1 but not order 2
β 1−β
Solution : MOOD procedure (Clain, Diot, Loubère, 11) On the toy equation w n+1,HO MOOD AP scheme, CFL ∆t ≤ ∆x /ce Compute the order 2 approximation w n+1,O2 Detect if the max. principle is satisfied : kw n+1,O2 k∞ ≤ kw n k∞ ? If not, compute the limited AP approximation w n+1,lim 1.5
0.5
Exact AP Order 1 AP Order 2 AP limited MOOD AP
0
w for
ε = 10−4
1
-0.5
-1
-1.5
1 0.5 0
w for
ε = 10−2
1.5
Exact AP Order 1 AP Order 2 AP limited MOOD AP
-0.5 -1 -1.5
0
0.2
0.4
0.6
x
0.8
1
0
0.2
0.4
0.6
x
0.8
1
Outline
1
General context : multi-scale models and principle of AP schemes
2
An order 1 AP scheme for the Euler system in the low Mach limit
3
High order schemes in time
4
High order schemes in time and space
5
Works in progress en perspectives
Error curves for the model problem Order 2 in space with MUSCL but explicit slopes for implicit fluxes Error curves on a smooth solution for the toy model ε = 10−2 ε=1 0
-1
10
-2
10
-3
10-4
10
L∞ error
10
10
AP Order 1 AP Order 2 AP limited MOOD AP
-5
100
200
400
number of cells
800
10
-1
10
-2
10
-3
10-2
-3
10
-4
10
-5
AP Order 1 AP Order 2 AP limited MOOD AP
400
ε = 10−4
-1
10
0
10
1600
L∞ error
L∞ error
10
AP Order 1 AP Order 2 AP limited MOOD AP 400
800
1600
number of cells
3200
6400
800
1600
number of cells
3200
6400
Second-order scheme for the Euler equations Recall the first-order IMEX scheme for the Euler system :
n +1 ρ − ρn + ∇ · (ρ u )n+1 = 0, ∆t n +1 − (ρ u )n 1 (ρ u ) + ∇ · (ρ u ⊗ u )n + ∇p (ρn+1 ) = 0. ∆t ε
(1) (2)
We apply the same convex combination procedure :
W n+1,lim = θ W n+1,O2 + (1 − θ) W n+1,O1 , with θ = β/(1 − β). we use the value of θ given by the study of the model problem But we need we need a criterion to detect oscillations in the MOOD procedure !
Euler equations : MOOD procedure The previous detector (L∞ criterion on the solution) is irrelevant for the Euler equations, since ρ and u do not satisfy a maximum principle. we need another detection criterion We pick the Riemann invariants Φ± = u ∓
2
γ−1
s
1 ∂p (ρ)
ε ∂ρ
, since they
satisfy an advection equation for smooth solutions. On the Euler equations W n+1,HO MOOD AP scheme, CFL ∆t ≤ ∆x /λ Compute the order 2 approximation W n+1,O2
Detect if both Riemann invariants break the maximum principle at the same time If so, compute the limited AP approximation W n+1,lim
Euler equations : Numerical results Riemann problem : left rarefaction wave, right shock ; top curves : ε = 1 ; bottom curves : ε = 10−4 Density ρ Momentum q = ρu 1. 8 1. 6 1. 4 1. 2 1
2 1. 8 1. 6 1. 4 1. 2 1
0. 2 0. 4 0. 6 0. 8
0 1.0001
1
0
0. 2 0. 4 0. 6 0. 8
1 1.009 1.006
1.00005 1
1.003 0
0. 2 0. 4 0. 6 0. 8
exact solution
first-order
1
0
0. 2 0. 4 0. 6 0. 8
second-order
1
1
MOOD-AP scheme
Euler equations : Numerical results ε=1
ε = 10−2
10−1
10−4
10−2
10−5
10−3
10−6
10−4
10−7 102
103
10−8
103
ε = 10−4 first-order second-order limited AP MOOD AP
10−7 10−8 10−9 10−10 103
104
Euler equations : Numerical results Initial data for the explosion : density cylinder (left) ; outwards velocity field (right). 1 2
0. 5
1. 5
0
1 −1
1 0
0
−1 −1 −0.5
1 −1 1
1. 2 1. 4 1. 6 1. 8
−0.5
2
0
0. 1
0
0. 5
1
0. 2
0. 3
0. 4
Euler equations : Numerical results reference solution
first-order scheme 1. 6 1. 4
1. 5
1. 2
1
−1
1 0
1 −1
0
1
0. 8
1. 5 1
−1
1. 5
1. 5
1
1
−1
0
0
1
1 −1 second-order scheme
−1
0
0
1 −1
0
0
1
1
1 −1 MOOD-AP scheme
Outline
1
General context : multi-scale models and principle of AP schemes
2
An order 1 AP scheme for the Euler system in the low Mach limit
3
High order schemes in time
4
High order schemes in time and space
5
Works in progress en perspectives
Works in progress and perspectives Choose the order 2 time discretization to get a θ as close as possible to 1 for the stability Study a local value of θ, depending on the presence of oscillations in a given cell Extension to full Euler (order 1 scheme exists but we have trouble decomposing between an explicit and an implicit flux) Simulations of physically relevant phenomena Domain decomposition with respect to ε
M0 class. scheme
000 111 0000000000000 1111111111111 000 111 111 000 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111
Mε class. scheme
ε = O ( 1)
000 111 0000000000000 1111111111111 000 111 111 000 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 000 111 0000000000000 1111111111111 000 111 intermediate111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 zone 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111 000 111 0000000000000 1111111111111 000 111
ε≪1
Mε AP scheme
Thanks !
Bibliography All speed schemes Preconditioning methods : [Chorin 65], [Choi, Merkle 85], [Turkel, 87], [Van Leer, Lee, Roe, 91], [Li,Gu 08,10], · · · Splitting and pressure correction :
[Harlow, Amsden, 68,71],
[Karki, Patankar, 89], [Bijl, Wesseling, 98], [Sewall, Tafti, 08], [Klein, Botta, Schneider, Munz, Roller 08], [Guillard, Murrone, Viozat 99, 04, 06] [Herbin, Kheriji, Latché 12,13], · · ·
➠ Asymptotic preserving schemes [Degond, Deluzet, Sangam, Vignal, 09], [Degond, Tang 11], [Cordier, Degond, Kumbaro 12], [Grenier, Vila, Villedieu 13] [Dellacherie, Omnes, Raviart,13], [Noelle, Bispen, Arun, Lukacova, Munz,14], [Chalons, Girardin, Kokh,15] [Dimarco, Loubère, Vignal, 17]
