Second order Implicit-Explicit Total Variation Diminishing schemes for the Euler system in the low Mach regime

Second order Implicit-Explicit Total Variation Diminishing schemes for the Euler system in

the low Mach regime

Victor Michel-Dansac

Workhop bas Mach, 20-22 Nov. 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 Second-order schemes in time

4 Second-order schemes in time and space

5 Work in progress and perspectives

General context

Multiscale modelM_ε, depending on a parameterε In the (space-time) domain,εcan

be of same order as the reference scale;

be small compared to the reference scale;

take intermediate values.

0 0.10.20.30.40.50.60.70.80.9 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 epsilon

Whenεis small:M₀= lim

ε→⁰M_ε asympt. model Difficulties:

Classical explicit schemes forM_ε: they are stable and consistent if the mesh resolves all the scales ofε. =⇒ very costly whenε→⁰ Schemes forM₀ =⇒ the mesh is independent ofε

But: àM₀is not valid everywhere, it needsε¹ àthe interface may be moving: how to locate it?

Principle of AP schemes

A possible solution:Asymptotic Preserving (AP) schemes Use the multi-scale modelM_εeven for smallε.

DiscretizeM_εwith a scheme preserving the limitε→^0.

àThe mesh is independent ofε: Asymptotic stability.

àRecovery of an approximate solution ofM₀whenε→^0:

Asymptotic consistency.

àAsymptotically stable and consistent scheme

⇒Asymptotic preserving scheme (AP).

([Jin, ’99] kinetic→^hydro) The AP scheme may be used only to reconnectM_εandM₀.

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 reconnectMεandM0

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 Second-order schemes in time

4 Second-order schemes in time and space

5 Work in progress and perspectives

The multi-scale model and its asymptotic limit

àIsentropic Euler system in scaled variables: x∈Ω⊂IR^d,t ≥⁰

(M_ε)

( ∂tρ+∇·(ρu) =0 (1)_ε

∂t(ρu) +∇·(ρu⊗^u) +¹

ε∇p(ρ) =0 (2)_ε ^(withp(ρ) =ρ^γ) Parameter:ε=M²=m|^u|²/(γp(ρ)/ρ), M =Mach number Boundary and initial conditions:

u·ⁿ=0 on∂Ω and

( ρ(x,0) =ρ0+ε˜ρ0(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|ρ⁰(t)+ρ(t)

Z

∂Ωu·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=cst=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 equation 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 limit

Time discretization:

[Degond, Deluzet, Sangam & Vignal, ’09], [Degond & Tang, ’11], [Chalons, Girardin & Kokh, ’15]

Ifρⁿanduⁿare known at timetⁿ:











ρⁿ⁺¹−ρⁿ

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

∆t +∇·(ρu⊗^u)ⁿ+¹ε∇p(ρⁿ⁺¹) =0. (2) ^(AC) implicit treatment of the pressure wave eq.⇒uniform stability inε ε→^{0 gives} ∇p(ρⁿ⁺¹) =0 ⇒ consistency at the limit

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

∆t +∇·(ρu)ⁿ−∆t

ε ∆p(ρⁿ⁺¹)−∆t∇²: (ρu⊗^u)ⁿ=0

An order 1 AP scheme in the low Mach limit

The scheme proposed in [Dimarco, Loubère & Vignal, ’17]:

à 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.

àThe C.F.L. condition comes from the explicit fluxF_e(W):

∆t ≤ ∆x

λⁿ_j = ∆x 2|^u_jⁿ|,

whereλⁿ_j are the eigenvalues of the explicit Jacobian matrixDFe(W_jⁿ). àA linear stability analysis yields: if the implicit part is

•^centered⇒^L2stability;

•^upwind⇒^{TVD andL∞stability.}

SSP Strong Stability Preserving, [Gottlieb, Shu & Tadmor, ’01]

AP but diffusive results, 1D test case

ε=0.99, 300 cells Class: 273 loops

CPU time 0.07 AP: 510 loops

CPU time 1.46

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

1st-order AP 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

1st-order AP 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

1st-order AP Class. scheme

0 0.2 0.4 0.6 0.8 1

0.9999 1 1

Density

1st-order AP Class. scheme

x

ε=10⁻⁴, 300 cells Class: 4036 loops

CPU time 0.82 AP: 57 loops

CPU time 0.14

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

1st-order AP 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

1st-order AP 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

1st-order AP Class. scheme

0 0.2 0.4 0.6 0.8 1

0.9999 1 1

Density

1st-order AP Class. scheme

x

AP but diffusive results, 1D test case

ε=10⁻⁴ Underlying of the viscosity

AP but diffusive results,1-D test-case

ε = 10

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 use high order schemes But they must respect the AP properties

⇓

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

we also wish to retain theL^∞stability

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 Second-order schemes in time

4 Second-order schemes in time and space

5 Work in progress and perspectives

Principle of IMEX schemes

Bibliography for stiff source terms or ODE problems: Ascher, Boscarino, Cafflish, Dimarco, Filbet, Gottlieb, Happenhofer, Higueras, Jin, Koch, Kupka, 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)−∆tZ _tⁿ⁺¹

tⁿ ∇·^Fe(W(t))dt

| {z }

−∆tZ _tⁿ⁺¹

tⁿ ∇·^Fi(W(t))dt

| {z } Wⁿ⁺¹=Wⁿ −∆t

∑

˜bj ∇·^Fe(W^n,j) −∆t

∑

bj ∇·^Fi(W^n,j)

Quadratures exact on the constants:∑^sj=1˜_bj =∑^sj=1b_j=1 Intermediate values at timest^n,j=tⁿ+c_j∆t:

W^n,j ≈^W(tⁿ) + Z _t^n,j

tⁿ ∂tW(t)dt =Wⁿ+ ∆tZ _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

˜aj,k∇·^Fe(W^n,k)−∆t

∑

k≤^j

aj,k∇·^Fi(W^n,k),

Quadratures exact on the constants:

∑

k=1

˜a_j,k = ˜c_j,

∑

k=1

a_j,k =c_j

Wⁿ⁺¹=Wⁿ−∆t

∑

j=1

˜bj ∇·^Fe(W^n,j)−∆t

∑

j=1

bj ∇·^Fi(W^n,j) Butcher tableaux:

Explicit part Implicit part

0 0 0 ··· ⁰

c2 ˜a2,1 0 . . . 0

... ... ... ... ...

cs ˜as,1 . . . ˜as,s−¹ 0

˜b₁ . . . ˜_bs

c₁ a₁,1 0 ··· ⁰

c2 a2,1 a2,2 . . . 0

... ... ... ... ...

cs as,1 . . . as,s−¹ as,s

b₁ . . . b_s

Conditions for 2nd order:

∑

∑

∑

∑

AP Order 2 scheme for Euler

ARS scheme [Ascher, Ruuth & Spiteri, ’97]:

“only one” 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−√¹ 2

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)

AP Order 2 scheme for Euler

Densityρfor the ARS time discretization:

0 0.2 0.4 0.6 0.8 1 1

1.05 1.1

ε=10⁻¹

0 0.2 0.4 0.6 0.8 11 1.005 1.01 ε=10⁻²

0 0.2 0.4 0.6 0.8 1 1

1.0005 1.001

ε=10⁻³

0 0.2 0.4 0.6 0.8 11 1.00005 1.0001

ε=10⁻⁴ exact 1st order 2nd order

Better understand the oscillations

Consider the scalar hyperbolic equation ∂tw+∂xf(w) =0.

Oscillations measured by the Total Variation and theL^∞norm:

TV(wⁿ) =

∑

j |^wjⁿ+1−^wjⁿ| ^and k^wnk∞= max

j |^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+ce∂xw+√^ciε∂xw=0

Order 1 AP scheme with upwind space discretizations (ce,ci >0):

w_j^n+1,O1=w_jⁿ−^ce(w_jⁿ−^wjⁿ−¹)−√^ciε(w_j^n+1,O1−^w_jⁿ₋⁺₁^1,O1).

Order 2 AP scheme: ARS with the parameterβ=1−¹/√ 2.

Lemma(Dimarco, Loubère, M.-D., Vignal): Under the CFL condition∆t ≤∆x/ce,

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)w^n+1,O1 with θ= β 1−β Problem: More accurate than order 1 but not order 2

Solution: MOOD procedure: see [Clain, Diot & Loubère, ’11]

On the toy equation:w^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 0

1 ε=10⁻²

0 0.2 0.4 0.6 0.8 1

−¹ 0 ε=10⁻⁴ 1

exact 1st order 2nd order TVD-AP TVD-AP-MOOD

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 Second-order schemes in time

4 Second-order schemes in time and space

5 Work in progress and perspectives

Error curves for the toy scalar equation

Order 2 in space: MUSCL (with the MC limiter) with explicit slopes for implicit fluxes.

Error curves on a smooth solution for the toy scalar equation:

10³ 10⁻⁶

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

ε=1

10³ 10⁻⁴

10⁻³ 10⁻² 10⁻¹ 10⁰

ε=10⁻²

10³ 10⁻⁴

10⁻³

ε=10⁻⁴ first-order

second-order TVD-AP

TVD-AP-MOOD

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) (ρu)ⁿ⁺¹−(ρu)ⁿ

∆t +∇·(ρu⊗^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)W^n+1,O1, withθ= β 1−β. We use the value ofθgiven by the study of the toy scalar equation.

But how can we detect oscillations for 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 s1

ε

∂p(ρ)

∂ρ ^{: in a} Riemann problem, at least one of them satisfies a maximum principle.

[Conway & Smoller, ’73]

On the Euler equations:

W^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: 1D 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 1.12

1.4 1.6 1.82

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

0 0.2 0.4 0.6 0.8 11 1.003 1.006 1.009

exact 1st order 2nd order TVD-AP-MOOD

Euler equations: 1D Numerical results

Error curves inL^∞norm, smooth 1D solution

10² 10⁻⁴

10⁻³ 10⁻² 10⁻¹

ε=1

10² 10³

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

ε=10⁻²

10⁴ 10⁻⁹

10⁻⁸ 10⁻⁷ 10⁻⁶ 10⁻⁵

ε=10⁻⁴

first-order second-order TVD-AP

TVD-AP-MOOD

Euler equations: 2D Numerical results

Error curves inL^∞norm, smooth 2D traveling vortex

10³ 10⁴

10⁻⁴ 10⁻³ 10⁻²

ε=1

10³ 10⁴

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³

ε=10⁻²

10³ 10⁴

10⁻⁷ 10⁻⁶ 10⁻⁵

ε=10⁻⁴

first-order second-order TVD-AP

TVD-AP-MOOD

Euler equations: 2D Numerical results

reference solution obtained solving the vorticity formulation

∂tω+U·∇ω=0, withω=∂xv−∂yu

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 Second-order schemes in time

4 Second-order schemes in time and space

5 Work in progress and perspectives

Work in progress and perspectives

Pick an order≥^{2 andL2}-stable time discretization to get aθas close as possible to 1 for the stability of the limited scheme.

Study a local value ofθ, depending on the presence of oscillations in a given cell.

Extension to full Euler (order 1 scheme exists).

Domain decomposition with respect toε:

Principle of AP schemes

A possible solution :AP schemes

Use the multi-scale modelMεwhere 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_εandM0

ε=O(1) intermediate

zone ε≪¹

Mε class. scheme

M0

class. scheme M_ε AP scheme

Thanks!

Euler equations: 2D Numerical results

To obtain a 2D reference incompressible solution, setω=∂xv−∂yuand consider thevorticity formulationof the incompressible Euler equations:

∂_tω+U·∇ω=0,

∇·^U=0 =⇒ ∃stream functionΨsuch that

U = ^t(∂_yΨ,−∂_xΨ),

−∆Ψ = ω.

To get the time evolution of the vorticity fromωⁿ:

1 solve−∆Ψⁿ=ωⁿ forΨⁿ(with periodic BC and assuming that the average ofΨvanishes);

2 getUⁿ fromUⁿ=^t(∂yΨⁿ,−∂xΨⁿ);

3 solve∂tω+Uⁿ·∇ωⁿ=0 to getωⁿ⁺¹.

We get a reference incompressible vorticity ω(x,t), to be compared to the vorticity of the solution given by the compressible scheme with smallε(we takeε=M²=10⁻⁵).

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],. . . à Asymptotic preserving schemes

[Degond, Deluzet, Sangam & Vignal, ’09], [Degond & Tang, ’11], [Cordier, Degond & Kumbaro, ’12], [Grenier, Vila & Villedieu, ’13]

[Dellacherie, Omnès & Raviart, ’13],

[Noelle, Bispen, Arun, Lukáˇcová & Munz, ’14],

[Chalons, Girardin & Kokh, ’15] [Dimarco, Loubère & Vignal, ’17]