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
1 Introduction and Motivation
2 Spectral Deferred Correction (SDC) Method
3 Finite Volume Discretization
4 Extensions to DG
5 Preliminary Results
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
1 Introduction and Motivation
2 Spectral Deferred Correction (SDC) Method
3 Finite Volume Discretization
4 Extensions to DG
5 Preliminary Results
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,
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,
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)
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
1 Introduction and Motivation
2 Spectral Deferred Correction (SDC) Method
3 Finite Volume Discretization
4 Extensions to DG
5 Preliminary Results
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)
Variable L1128 r128/256 L1
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)
Variable L1128 r128/256 L1
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
1 Introduction and Motivation
2 Spectral Deferred Correction (SDC) Method
3 Finite Volume Discretization
4 Extensions to DG
5 Preliminary Results
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
1 Introduction and Motivation
2 Spectral Deferred Correction (SDC) Method
3 Finite Volume Discretization
4 Extensions to DG
5 Preliminary Results
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