Multi-Implicit Discontinuous Galerkin Method for Low Mach ... · Multi-Implicit Discontinuous Galerkin Method for Low Mach Number Combustion Will Pazner & Per-Olof Persson Division

Multi-Implicit Discontinuous Galerkin Method forLow Mach Number Combustion

Will Pazner & Per-Olof PerssonDivision of Applied Mathematics, Brown University

Department of Mathematics, University of California, Berkeley

Collaboration with Andy Nonaka, John Bell, Marc Day, Michael MinionCenter for Computational Sciences and Engineering,

Lawrence Berkeley National Laboratory

SIAM Conference on Computational Science and EngineeringFebruary 27, 2017

Outline

1 Introduction and Motivation

2 Spectral Deferred Correction (SDC) Method

3 Finite Volume Discretization

4 Extensions to DG

5 Preliminary Results

Outline




4 Extensions to DG


Introduction

Interested in modeling coupleddynamics

Reacting (low Mach number)fluid flow

Detailed chemical kinetics

Vastly different time scales forphysical processes:

Advection, diffusion,reaction

Low Mach Number Formulation

[Majda, Sethian, (1985)]

Acoustic propagation typically has negligible impact on thesystem state

Sound waves are analytically removed from the system

The set of conservation laws takes the form of a coupleddifferential-algebraic system

Governing Equations

Thermodynamic variables: ρ density, Yj mass fractions, h enthalpy

∂ρ

∂t= −∇ · (Uρ)

∂(ρYj)

∂t= −∇ · (UρYj) +∇ · ρDj∇Yj + ω̇j ,

∂(ρh)

∂t= −∇ · (Uρh) +∇ · λ

cp∇h+

∑j

∇ · hj(ρDj −

λ

cp

)∇Yj ,

ω̇j production rate, Dj diffusion coefficient, T temperate, cpspecific heat at constant pressure, U velocity

Velocity constraint

Equation of state:

p0 = ρRT∑j

YjWj

,

Taking Lagrangian derivative and enforcing constant pressureimplies

∇ · U =1

ρcpT

∇ · λ∇T +∑j

Γj · ∇hj

+

1

ρ

∑j

W

Wj∇ · Γj +

1

ρ

∑j

(W

Wj− hjcpT

)ω̇j =: S

Time integration

Want to integrate this system in time at advective time scale

For stability, need to treat diffusion and reaction implicitly

Multi-implicit splitting =⇒ weakly couple components of theequation

Outline




4 Extensions to DG


Spectral Deferred Correction (SDC) Method

Arbitrary order method for integrating ODEs, e.g.:

φt(t) = F (t, φ(t)), t ∈ [tn, tn + ∆t];

φ(tn) = φn,

Subdivide time step [tn, tn+1] into m time substeps, e.g.according to Gauss-Lobatto rule

tn

t0 t1 t2

tn + ∆t

t3

∆tm

SDC Method

Consider associated integral equation

φ(t) = φn +

∫ t

tnF (τ, φ(τ)) dτ.

Update equation:

φ(k+1)(t) = φn +

∫ t

tn

[F (φ(k+1))− F (φ(k))

]dτ +

∫ t

tnF (φ(k)) dτ,

φ(k+1)(t) = φn +

∫ t

tn

[F (φ(k+1))− F (φ(k))

]dτ +

∫ t

tnF (φ(k)) dτ,

Discretize two integrals on RHS with two quadrature rules:

First quadrature rule has order of accuracy p

Second quadrature rule has order of accuracy q > p

Each iteration increases order of accuracy of solution by p upto maximum of q

For example:

First term: forward or backward Euler (implicit or explicitmethod)

Second term: highly accurate Gauss-Lobatto rule

(Formally equivalent to certain RK/DIRK methods)

φ(k+1)(t) = φn +

∫ t

tn

[F (φ(k+1))− F (φ(k))

]dτ +

∫ t

tnF (φ(k)) dτ,





For example:




φ(k+1)(t) = φn +

∫ t

tn

[F (φ(k+1))− F (φ(k))

]dτ +

∫ t

tnF (φ(k)) dτ,





For example:




Multi-implicit SDC

φm+1,(k+1)A = φm,(k+1)

+

∫ tm+1

tm

[FA(φ

(k+1)A )− FA(φ(k))

]dt+

∫ tm+1

tmF (φ(k))dt

φm+1,(k+1)AD = φm,(k+1)

+

∫ tm+1

tm

[FA(φ

(k+1)A )− FA(φ(k)) + FD(φ

(k+1)AD )− FD(φ(k))

]dt

+

∫ tm+1

tmF (φ(k))dt,

φm+1,(k+1) = φm,(k+1)

+

∫ tm+1

tm

[FA(φ

(k+1)A )− FA(φ(k)) + FD(φ

(k+1)AD )− FD(φ(k))

+FR(φ(k+1))− FR(φ(k))]dt+

∫ tm+1

tmF (φ(k))dt.

Multi-implicit SDC

Explicit advection =⇒ discretize update with forward Euler

Implicit diffusion, reaction =⇒ discretize update withbackward Euler

φm+1,(k+1)AD = φm,(k+1) + ∆tm

[FA(φm,(k+1))− FA(φm,(k))

+FD(φm+1,(k+1)AD )− FD(φm+1,(k))

]+Im+1

m

[F (φ(k))

]φm+1,(k+1) = φm,(k+1) + ∆tm

[FA(φm,(k+1))− FA(φm,(k))

+FD(φm+1,(k+1)AD )− FD(φm+1,(k))

+ FR(φm+1,(k+1))− FR(φm+1,(k))]

+Im+1m

[F (φ(k))

]

Volume Discrepancy Constrained Evolution

Recall velocity constraint: ∇ · U = S,Linearization =⇒ pressure no longer constantContinuity equation =⇒

∇ · U =1

ρpρ

(−DpDt

)+ S

Define correction δχ = DpDt

, discretize as

δχ ≈1

p0

(p0 − pEOS

∆t

)Solve corrected constraint for velocity:

∇ · U = S + δχ

975000

980000

985000

990000

995000

1e+06

1.005e+06

1.01e+06

1.015e+06

1.02e+06

0 0.2 0.4 0.6 0.8 1 1.2 200

400

600

800

1000

1200

1400

1600

pE

OS [

g/(

cm

-s2)]

Te

mp

era

ture

[K

]

x [cm]

pEOS, No δχ CorrectionpEOS, Using δχ Correction

Temperature

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2

pE

OS -

p0 [

g/(

cm

-s2)]

x [cm]

n=128n=256n=512

n=1024

Each MISDC iteration drives the solution to EOS

Outline




4 Extensions to DG


1D Finite Volume Discretization

Solution represented by cell-averages:

〈φ〉i ≡1

∆x

∫ xi+1

2

xi− 1

2

φ(x) dx

Reconstruct polynomial to 4th order obtain gradients, point values,etc.

φ̂i = 〈φ〉i −1

24(〈φ〉i−1 − 2〈φ〉i + 〈φ〉i+1)

φ̃i+ 12

=−φ̂i−1 + 9φ̂i + 9φ̂i+1 − φ̂i+2

16

∇̃φi+ 12

=〈φ〉i−1 − 15〈φ〉i + 15〈φ〉i+1 − 〈φ〉i+2

12∆x

Numerical Results (Finite Volume)

Premixed Hydrogen Flame

(9 chemical species, 27 reactions)

Variable L1128 r128/256 L1

256 r256/512 L1512

Y (H2) 5.91E-08 4.01 3.67E-09 3.98 2.33E-10Y (O2) 1.10E-06 4.00 6.83E-08 4.05 4.14E-09Y (H2O) 1.01E-06 4.01 6.25E-08 4.05 3.76E-09Y (H) 1.17E-09 3.70 9.00E-11 3.91 5.97E-12Y (O) 2.70E-08 3.93 1.77E-09 4.01 1.10E-10Y (OH) 3.17E-08 4.01 1.97E-09 4.06 1.18E-10Y (HO2) 3.56E-08 3.71 2.72E-09 3.88 1.86E-10Y (H2O2) 1.41E-08 3.70 1.09E-09 3.84 7.58E-11Y (N2) 1.77E-07 3.95 1.15E-08 4.07 6.85E-10ρ 5.00E-09 4.01 3.10E-10 4.09 1.82E-11T 1.21E-02 4.02 7.44E-04 4.05 4.48E-05ρh 6.77E+00 3.99 4.26E-01 4.07 2.54E-02

Premixed Methane Flame

(53 species, 325-step chemical reaction network)


256 r256/512 L1512

Y (CH4) 1.11E-06 4.00 6.97E-08 3.98 4.42E-09Y (O2) 3.77E-06 3.96 2.42E-07 4.07 1.44E-08Y (H2O) 2.30E-06 4.02 1.42E-07 4.05 8.53E-09Y (CO2) 1.87E-06 4.02 1.15E-07 4.07 6.87E-09Y (CH3) 3.11E-08 2.48 5.59E-09 3.75 4.16E-10Y (CH2(S)) 8.01E-11 4.14 4.54E-12 3.85 3.15E-13Y (O) 1.05E-07 4.08 6.20E-09 3.90 4.16E-10Y (H) 3.48E-09 3.83 2.45E-10 3.81 1.75E-11Y (N2) 3.58E-07 3.74 2.68E-08 4.00 1.67E-09ρ 1.25E-08 4.03 7.64E-10 4.05 4.61E-11T 3.52E-02 4.01 2.18E-03 4.06 1.31E-04ρh 4.09E+01 3.97 2.60E+00 4.00 1.62E-01

Dimethyl Ether Flame

(39 species, 175 reactions)


256 r256/512 L1512

Y (CH3OCH3) 2.29E-06 3.83 1.62E-07 3.93 1.06E-08Y (O2) 2.99E-06 3.63 2.42E-07 4.02 1.49E-08Y (CO2) 2.51E-06 3.83 1.76E-07 4.04 1.07E-08Y (H2O) 1.62E-06 3.51 1.42E-07 4.01 8.85E-09Y (CH3OCH2O2) 1.55E-10 4.51 6.76E-12 3.88 4.61E-13Y (OH) 3.24E-07 3.80 2.32E-08 4.02 1.43E-09Y (HO2) 1.46E-07 3.80 1.05E-08 3.95 6.77E-10Y (O) 1.70E-07 3.55 1.46E-08 3.92 9.66E-10Y (H) 8.35E-09 3.68 6.52E-10 3.96 4.20E-11Y (N2) 1.09E-06 3.76 8.01E-08 3.93 5.25E-09ρ 9.44E-09 3.67 7.42E-10 4.02 4.58E-11T 2.54E-02 3.59 2.11E-03 4.01 1.31E-04ρh 5.89E+01 3.83 4.15E+00 4.02 2.56E-01

Outline




4 Extensions to DG


Advantages of DG

Arbitrary order of accuracy

Unstructured and complex geometries

Straightforward generalization to multiple dimensions

Amenable to parallelization

Weak Form

∫K

∂ρ

∂tv dx =

∫KρU · ∇v dx−

∫∂K

ρ̂U(n)v dA∫K

∂(ρYj)

∂tv dx =

∫K

(ρYjU + Γj) · ∇v dx+

∫Kω̇jv dx

−∫∂K

(ρ̂YjU(n) + Γ̂j(n)

)v dA∫

K

∂(ρh)

∂tv dx =

∫KρhU · v dx−

∫∂K

ρ̂hU(n)v dA

−∫K

λ

cp∇h · ∇v dx+

∫∂K

λ̂

cp∇h+ . . .

General approach

Transform second-order equations into system of first-orderequations and use LDG method (Cf. Cockburn and Shu)

Solve weak form of reaction equations =⇒ reaction solvecouples all nodes within each element (expensive!)∫K

u(k+1),m+1 − u(k+1),m

∆tmdx =

∫Kr(k+1),m+1AD +ω̇(u(k+1),m+1) dx

Oscillatory second derivatives =⇒ filter S for stability

Outline




4 Extensions to DG


Numerical Results (Hydrogen Flame)

101 102 103 10410-11

10-10

10-9

10-8

10-7

10-6

10-5 H2

101 102 103 10410-13

10-12

10-11

10-10

10-9

10-8

10-7 O2

101 102 103 10410-12

10-11

10-10

10-9

10-8

10-7

10-6 H2O

101 102 103 10410-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4 H

101 102 103 10410-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6 ρ

101 102 103 10410-5

10-4

10-3

10-2

10-1

100

101

102 ρh

p= 1

p= 2

p= 3

Numerical Results (Hydrogen Flame)

Error in total concentration of O2 after 1 ms simulation

nx p = 1 Rate p = 2 Rate p = 3 Rate

64 1.80× 10−8 - 6.40× 10−10 - 7.58× 10−10 -128 4.48× 10−9 2.01 3.56× 10−11 4.17 2.64× 10−11 4.84256 1.14× 10−9 1.97 2.65× 10−12 3.74 1.64× 10−12 4.01512 2.89× 10−10 1.98 2.85× 10−13 3.221024 7.28× 10−11 1.99

Thanks!

Volume Discrepancy

975000

980000

985000

990000

995000

1e+06

1.005e+06

1.01e+06

1.015e+06

1.02e+06

0 0.2 0.4 0.6 0.8 1 1.2 200

400

600

800

1000

1200

1400

1600

pE

OS [g/(

cm

-s2)]

Tem

pera

ture

[K

]

x [cm]

pEOS, No δχ CorrectionpEOS, Using δχ Correction

Temperature

Volume Discrepancy (Refinement in Space)

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1 1.2

pE

OS -

p0 [g/(

cm

-s2)]

x [cm]

n=128n=256n=512

n=1024

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Multi-Implicit Discontinuous Galerkin Method for Low Mach ... · Multi-Implicit Discontinuous Galerkin Method for Low Mach Number Combustion Will Pazner & Per-Olof Persson Division