Asymptotically accurate high-order space and time schemes for the Euler system in the low Mach regime

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

Multiscale model :M_εdepends on a parameterε In the (space-time) domainεcan

be small compared to the reference scale be of same order as the reference scale take intermediate values

0 0.1 0.20.3 0.4 0.50.6 0.70.8 0.9 1

1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4

electron density (log scale) : t=0.1

−9

−8

−7

−6

−5

−4

−3

−2

−1 0

0 0.2 0.4 0.6 0.8 1

10⁻¹⁰ 10⁻⁵ 10⁰

x

ε

Whenεis small :M₀= lim

ε→⁰M_ε asympt. model Difficulties :

Classical explicit schemes forM_ε : stable and consistent if the mesh resolves all the scales ofε ⇒^{huge cost}

Schemes forM₀with meshes independent ofε But : ➠M₀not valid everywhere, needsε≪¹

➠location of the interface, moving interface

Principle of AP schemes

A possible solution :AP schemes

Use the multi-scale modelM_εwhere you want.

Discretize it with a scheme preserving the limitε→⁰

➠The mesh is independent ofε: Asymptotic stability.

➠You recover an approximate solution ofM₀whenε→^{0 :} Asymptotic consistency Asymptotically stable and consistent scheme

⇒Asymptotic preserving scheme (AP) ([S.Jin] kinetic→^hydro) You can use the AP scheme only to reconnectM_ε andM₀

00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000 00000000000000

11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111 11111111111111

00000000 00000000 00000000 0000 00000000 00000000 00000000 00000000 00000000 0000 00000000 00000000 00000000 00000000 0000 00000000 00000000 00000000 0000

11111111 11111111 11111111 1111 11111111 11111111 11111111 11111111 11111111 1111 11111111 11111111 11111111 11111111 1111 11111111 11111111 11111111 1111

00000000 00000000 00000000 0000 00000000 00000000 00000000 00000000 00000000 0000 00000000 00000000 00000000 00000000 0000 00000000 00000000 00000000 0000

11111111 11111111 11111111 1111 11111111 11111111 11111111 11111111 11111111 1111 11111111 11111111 11111111 11111111 1111 11111111 11111111 11111111 1111

ε=O(1) intermediate

zone ε≪1

M_ε class. scheme

M₀ class. scheme

M_ε 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 variablesx ∈Ω⊂IR^d,t ≥⁰ (^Mε)

( ∂_tρ+∇·(ρu) =0, (1)ε

∂_t(ρu) +∇·(ρu⊗^u) +¹

ε∇p(ρ) =0, (2)ε

Parameter :ε=^M2=^m|^u|²/(γ^p(ρ)/ρ)^{, M} =Mach number Boundary and initial conditions :

u·ⁿ=0, on∂Ω, and

( ρ(x,0) =ρ₀+ερ˜₀(x),

u(^x,⁰) =^u0(^x) +εu˜₀(^x), ^with∇·^u0=⁰. The formal low Mach number limitε→⁰

(2)ε ⇒ ∇p(ρ) =0, ⇒ ρ(x,t) =ρ(t).

(¹)ε ⇒ |Ω|ρ^′(^t)+ρ(^t) Z

∂Ω

u·ⁿ=⁰, ⇒ ρ(^t) =ρ(⁰) =ρ₀, ⇒ ∇·^u=⁰

The multi-scale model and its asymptotic limit

The asymptotic model :Rigorous limit [Klainerman, Majda, 81]

(^M0)







ρ=cste=ρ₀,

ρ₀∇·^u=⁰, (¹)0

ρ₀∂_tu+ρ₀∇·(u⊗^u) +∇π₁=0, (2)0

where

π₁= lim

ε→⁰

1 ε

p(ρ)−^p(ρ0) .

Explicit eq. forπ₁∂_t(¹)0−∇·(²)0 ⇒ −∆π1=ρ₀∇²: (^u⊗^u).

The pressure wave eq. fromM_ε :

∂_t(1)ε−∇·(2)ε ⇒ ∂_ttρ−¹ε∆p(ρ) =∇²: (ρu⊗^u) (3)ε

From a numerical point of view

Explicit treatment of(3)ε⇒conditional stability∆t ≤√ε∆x Implicit treatment of(³)ε⇒uniform stability with respect toε

An order 1 AP scheme in the low Mach numb. limit

Order1AP scheme in[Dimarco, Loubère, Vignal, SISC 2017]: Ifρⁿanduⁿare known at timetⁿ











ρⁿ⁺¹−ρⁿ

∆^t +∇·(ρu)ⁿ⁺¹=0, (1) ^(AS) (ρ^u)ⁿ⁺¹−(ρ^u)ⁿ

∆t +∇·(ρ^u⊗^u)ⁿ+¹

ε∇p(ρⁿ⁺¹) =⁰. (²) ^(AC)

∇·(2)inserted into(1):gives an uncoupled formulation ρⁿ⁺¹−ρⁿ

∆t +∇·(ρ^u)ⁿ−∆^t

ε ∆^p(ρⁿ⁺¹)−∆^t∇²: (ρ^u⊗^u)ⁿ=⁰,

➠ Results uniformlyL^∞stable if the space discretization is well chosen

➠ Framework of IMEX (IMplicit-EXplicit) schemes :

∂_t ρ

ρu

| {z }

W

+∇· 0

ρu⊗^u

| {z }

Fe(W)

+∇· ρu

p(ρ) ε Id

| {z }

Fi(W)

=⁰.

AP but diffusive results,1-D test-case

ε=0.99, 300 cells Class: 273 loops

CPU time 0.07 AP: 510 loops

CPU time 1.46

0 0.01 0.02 0.03 0.04 0.05 0.06

0 1 2 3 4 5 6x 10⁻⁴

Time

Timesteps

AP scheme Class. scheme

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Density AP scheme Class. scheme

x

ε=10⁻⁴, 300 cells Class: 4036 loops

CPU time 0.82 AP: 57 loops

CPU time 0.14

0 0.01 0.02 0.03 0.04 0.05 0.06

10⁻⁵ 10⁻⁴ 10⁻³

Time

Timesteps

AP scheme Class. scheme

0 0.2 0.4 0.6 0.8 1

0.9999 1 1

Density

AP scheme Class. scheme

x

AP but diffusive results,1-D test-case

ε=¹⁰⁻⁴ Underlying of the viscosity

0 0.01 0.02 0.03 0.04 0.05 0.06

10⁻⁶ 10⁻⁵ 10⁻⁴

Time

Timesteps

AP scheme Class. scheme

0 0.2 0.4 0.6 0.8 1

0.9999 1 1

Density

AP scheme Class. scheme

x

⇓

It is necessary to usehigh order schemes But they must respect the AP properties

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 :∂_tW+∇·^Fe(^W) +∇·^Fi(^W) =⁰. General principle :Step n :Wⁿis known

Quadrature formula with intermediate values W(tⁿ⁺¹) =W(tⁿ)−

Z tⁿ⁺¹ tⁿ

∇·^Fe(W(t))dt

| {z }

− Z tⁿ⁺¹

tⁿ

∇·^Fi(W(t))dt

| {z }

Wⁿ⁺¹=Wⁿ −∆t

s

∑

j=1

˜b_j ∇·^Fe(W^n,j) −∆t

s

∑

j=1

b_j ∇·^Fi(W^n,j)

Quadratures exact on the constants :∑^sj=1˜b_j =∑^sj=1b_j=¹

Intermediate valuest^n,j=tⁿ+c_j∆t W^n,j ≈^W(tⁿ) +

Z t^n,j tⁿ

∂_tW(t)dt =Wⁿ+ ∆t Z cj

0

∂_tW(tⁿ+s∆t)ds

Principle of IMEX schemes

Quadrature formula for intermediate values :i=¹,···,^s W^n,j=Wⁿ−∆t

∑

k<j

˜

a_j,k∇·^Fe(W^n,k)−∆t

∑

k≥^j

a_j,k∇·^Fi(W^n,k),

Quadratures exact on the constants :

s k

∑

˜ aj,k = ˜^cj,

s k

∑

aj,k =^cj

Wⁿ⁺¹=Wⁿ −∆t

s

∑

j=1

˜b_j ∇·^Fe(W^n,j) −∆t

s

∑

j=1

b_j ∇·^Fi(W^n,j) Butcher tableaux :

Expl. part Imp. part

0 0 0 ··· ⁰

c₂ a˜_2,1 0 . . . 0

... ... . .. . .. ... c_s ˜a_s,1 . . . ˜a_s,s−1 0

˜b₁ . . . ˜_bs

c₁ a_1,1 0 ··· ⁰

c₂ a_2,1 a_2,2 . . . 0

... ... . .. . .. ... c_s a_s,1 . . . a_s,s−1 a_s,s

b₁ . . . ^bs

Conditions for 2nd order :

∑

∑

∑

∑

AP Order 2 scheme for Euler

ARS scheme (Asher, Ruuth, Spiteri, 97) :“only 1” intermediate step

0 0 0 0

β β 0 0

1 β−^{1 2}−β 0 β−^{1 2}−β 0

0 0 0 0

β 0 β 0

1 0 1−β β 0 1−β β

β=¹−^√1₂

W^n,1=^Wn

W^n,2=W^⋆=Wⁿ−∆tβ∇·^Fe(Wⁿ)−∆tβ∇·^Fi(W^⋆),

W^n,3=Wⁿ⁺¹=Wⁿ−∆t(β−¹)∇·^Fe(Wⁿ)−∆t(2−β)∇·^Fe(W^⋆)

−∆t(1−β)∇·^Fi(W^⋆)−∆tβ∇·^Fi(Wⁿ⁺¹).

0 0.2 0.4 0.6 0.8 1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Densityε=1

Exact AP Order 1 AP Order 2

x

0 0.2 0.4 0.6 0.8 1

1 1.00001 1.00002 1.00003 1.00004 1.00005 1.00006 1.00007 1.00008 1.00009 1.0001

Densityε=10−4

Exact AP Order 1 AP Order 2

x

Better understand the oscillations

For a scalar hyperbolic eq. ∂_tw+∂_xf(w) =0

Oscillations measured by the Total Variation and theL^∞norm TV(^wn) =

∑

j

|^wjⁿ+1−^wjⁿ| k^wnk^∞= max|^wjⁿ|

TVD (Total Variation Diminishing) property andL^∞stability TV(wⁿ⁺¹)≤^TV(wⁿ)

k^wn+1k^∞≤ k^wnk^∞ ⇔no oscillations

First idea :Find an AP order 2 scheme which satisfies these properties Impossible

Theorem(Gottlieb,Shu,Tadmor, 01)There are no implicit Runge-Kutta schemes of order higher than one which preserves the TVD property.

A limiting procedure

Another idea :use a limited scheme

Wⁿ⁺¹=θW^n+1,O2+ (1−θ)W^n+1,O1, W^n+1,Oj =orderj AP approximation

θ∈[⁰,¹]largest value such thatWⁿ⁺¹does not oscillate Toy scalar equation ∂_tw+√^ci

ε∂_xw+^ce∂_xw=⁰

Order 1 AP scheme with upwind space discretizations (c_e,^ci >⁰⁾ w_j^n+1,O1=^w_jⁿ−√^ci

ε(^w_j^n+1,O1−^wjⁿ−^+11,O1)−^ce(^w_jⁿ−^wjⁿ−¹) Order 2 AP scheme ARS with the parameterβ=¹−¹/√

2 Lemma(VMD,Loubère,Vignal)Under the CFL condition∆t ≤∆x/c_e

θ≤ β

1−β≃⁰.41 ⇒

TV(wⁿ⁺¹)≤^TV(wⁿ) k^wn+1k^∞≤ k^wnk^∞

A MOOD procedure

Limited AP scheme :

w^n+1,lim=θw^n+1,O2+ (¹−θ)^{wn+1,O1 with} θ=₁₋^β_β Problem :More accurate than order 1 but not order 2 Solution :MOOD procedure(Clain, Diot, Loubère, 11)

On the toy equationw^n+1,HO MOOD AP scheme,CFL∆^t≤∆^x/ce

Compute the order 2 approximationw^n+1,O2

Detect if the max. principle is satisfied :k^wn+1,O2k^∞≤ k^wnk^∞? If not, compute the limited AP approximationw^n+1,lim

0 0.2 0.4 0.6 0.8 1

-1.5 -1 -0.5 0 0.5 1 1.5

wforε=10−2

Exact AP Order 1 AP Order 2 AP limited MOOD AP

x

0 0.2 0.4 0.6 0.8 1

-1.5 -1 -0.5 0 0.5 1 1.5

Exact AP Order 1 AP Order 2 AP limited MOOD AP

x

wforε=10−4

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

100 200 400 800 1600

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

AP Order 1 AP Order 2 AP limited MOOD AP L∞error

ε=¹

number of cells ^{400 800 1600 3200 6400}

10^-3 10^-2 10^-1 10⁰

AP Order 1 AP Order 2 AP limited MOOD AP

L∞error

ε=¹⁰⁻²

number of cells

400 800 1600 3200 6400

10^-5 10^-4 10^-3 10^-2 10^-1

AP Order 1 AP Order 2 AP limited MOOD AP L∞error

ε=¹⁰⁻⁴

number of cells

Second-order scheme for the Euler equations

Recall the first-order IMEX scheme for the Euler system :











ρⁿ⁺¹−ρⁿ

∆t +∇·(ρu)ⁿ⁺¹=0, (1) (ρ^u)ⁿ⁺¹−(ρ^u)ⁿ

∆t +∇·(ρ^u⊗^u)ⁿ+¹

ε∇p(ρⁿ⁺¹) =⁰. (²) We apply the same convex combination procedure :

W^n+1,lim =θW^n+1,O2+ (¹−θ)^{Wn+1,O1, with}θ=β/(¹−β).

we use the value ofθgiven 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, sinceρandudo not satisfy a maximum principle.

we needanother detection criterion

We pick theRiemann invariantsΦ_± =^u∓γ−²¹ s

1 ε

∂p(ρ)

∂ρ , since they satisfy an advection equation for smooth solutions.

On the Euler equationsW^n+1,HO MOOD AP scheme,CFL∆^t≤∆^x/λ Compute the order 2 approximationW^n+1,O2

Detect if both Riemann invariants break the maximum principle at the same time

If so, compute the limited AP approximationW^n+1,lim

Euler equations : Numerical results

Riemann problem : left rarefaction wave, right shock ; top curves :ε=1 ; bottom curves :ε=¹⁰⁻⁴

0 0.^{2 0}.^{4 0}.^{6 0}.^{8 1} 1

1.² 1.⁴ 1.⁶ 1.8 2

Densityρ

0 0.2 0.4 0.6 0.8 1 1 1.² 1.4 1.⁶ 1.8 Momentumq=ρu

0 0.^{2 0}.^{4 0}.^{6 0}.^{8 1} 1

1.00005 1.0001

exact solution first-order second-order MOOD-AP scheme 0 0.^{2 0}.^{4 0}.^{6 0}.^{8 11}

1.⁰⁰³ 1.⁰⁰⁶ 1.⁰⁰⁹

Euler equations : Numerical results

10² 10³

10⁻⁴ 10⁻³ 10⁻² 10⁻¹

ε=1

10³ 10⁻⁸

10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴

ε=¹⁰⁻²

10³ 10⁴

10⁻¹⁰ 10⁻⁹ 10⁻⁸ 10⁻⁷

ε=10⁻⁴ first-order

second-order limited AP MOOD AP

Euler equations : Numerical results

Initial data for the explosion : density cylinder (left) ;

outwards velocity field (right).

−¹ 0

1−¹ 0

1 1

1.5 2

1 1.^{2 1}.^{4 1}.^{6 1}.^{8 2}

−¹ −⁰.^{5 0 0}.^{5 1}

−¹

−⁰.⁵ 0 0.5 1

0 0.^{1 0}.^{2 0}.^{3 0}.⁴

Euler equations : Numerical results

−¹ ₀

1−^{1 0} 1 1

1.5

reference solution

0.8 1 1.² 1.4 1.⁶

−¹ ₀

1−^{1 0} 1 1

1.5

first-order scheme

−¹ ₀

1−^{1 0} 1 1

1.5

second-order scheme

−¹ ₀

1−^{1 0} 1 1

1.5

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 aθas close as possible to 1 for the stability

Study a local value ofθ, 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 toε

0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000

1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111 1111111111111

000000 000000 000000 000 000000 000000 000000 000000 000000 000 000000 000000 000000 000000 000 000000 000000 000000

111111 111111 111111 111 111111 111111 111111 111111 111111 111 111111 111111 111111 111111 111 111111 111111 111111

000000 000000 000000 000 000000 000000 000000 000000 000000 000 000000 000000 000000 000000 000 000000 000000 000000

111111 111111 111111 111 111111 111111 111111 111111 111111 111 111111 111111 111111 111111 111 111111 111111 111111

ε=O(1) intermediate

zone ε≪1

M_ε class. scheme

M₀ class. scheme

M_ε 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]