Multi-Implicit Discontinuous Galerkin Method for Low Mach Number Combustion Will Pazner & Per-Olof Persson Division of Applied Mathematics, Brown University Department of Mathematics, University of California, Berkeley Collaboration with Andy Nonaka, John Bell, Marc Day, Michael Minion Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory SIAM Conference on Computational Science and Engineering February 27, 2017
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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