I3M-Montpellier - ANR MAIDESC
Bruno Koobus∗
(∗) I3M, Université Montpellier 2, France
Réunion MAIDESC - 9 avril 2014 - INRIA Roquencourt
1 Koobus I3M-Montpellier
I3M - current work
Test casesMultirate time-advancingA third order space-accurate scheme : CENO
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Test cases
Test case 1Circular cylinder at Reynolds 1M : three-dimensional flow with thin boundarylayers, unsteady separated shear layers and vortex shedding.
Circular cylinder (1.2M nodes), iso-contours of the vorticity magnitude.Reynolds number = 1M, Mach number = 0.1, RANS/VMS-LES hybrid model.
=⇒ Evaluation of the performance of the multirate scheme, and CENO (and turbulencemodeling...).
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Test cases
Test case 2Simulation of a moving contact discontinuity followed by the mesh (uniformpressure, uniform velocity, different density) with an ALE formulation :
time t1
time t2 > t1
=⇒ Evaluation of the efficiency of the multirate scheme.
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Multirate schemes
IntroductionMany physical phenomena show multiple scales which require locally-refinedmeshes (with possibly mesh adaption), but the time step required by the smallestdetails should not be applied to the larger details. The multiratetime-advancing approach, which allows to use different time steps in thecomputational domain, is a way to overcome this problem.
Some work has been made on these methods in the field of ODE (mostly) and PDE,but only few applications have been carried out in CFD...
The objective for Montpellier and INRIA Sophia-Antipolis is to develop andimplement a multirate scheme in the parallel scalable LES solver AIRONUM.Test cases 1 and 2 will be used to assess the effectiveness of the proposed multirateapproach.
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Multirate schemes
Several methods for integrating stiff ODE
To speed-up numerical integration of ODE (including those deriving from themethod of lines in PDE), three research directions have been followed in the lastdecades :
Multi-method schemes : different integration schemes used for the stiff andnon-stiff part of the solution.Multi-order schemes : same explicit method and same step size, but the orderof the method is chosen according to the stiffness level of the solution.Multirate schemes : the same explicit or implicit method, with the sameorder, is applied to the solution, but the step size is chosen according to thestiffness level of the solution.Remark : Multirate and multi-method schemes can be mixed.
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Multirate schemes
Works since 1960...Rice (1960), Andrus (1979), Osher-Sanders (1983), Gear-Wells (1984),Löhner-Morgan-Zienkiewicz (1984), Rentrop (1985), Byrne-Hindmarsh (1987),Skelboe (1989), Jorgen-Skelboe (1992), Andrus (1993), Günther-Rentrop (1993),Biesiadecki-Skeel (1993), Ven-Niemann-Tuitman-Veldman (1997),Engstler-Lubich (1997), Maurits-Ven-Veldman (1998), Günther-Kvaerno-Rentrop(1999), Kvaerno-Rentrop (1999), Kato-Kataoka (1999), Skelboe (2000),Günther-Kvaerno-Rentrop (2001), Dawson-Kirby (2001), Bartel-Günther (2002),Kirby (2002), Logg (2003, 2004), Guennouni-Verhoeven-Maten-Beelen (2004),Piperno (2005), Savcenco-Hundsdorfer-Verwer (2007), Savcenco (2007),Constantinescu-Sandu (2007, 2008), Savcenco-Mattheij (2008),Schlegel-Knoth-Arnold-Wolke (2008), Jansson-Log (2008), Ly (2008),Debreu-Blayo (2008), Faille-Nataf-Willien-Wolf (2009), Constantinescu-Sandu(2009, 2010), Mugg (2012), Fok-Rosales (2012),Seny-Lambrechts-Comblen-Legat-Remacle (2012),Dawson-Trahan-Kubatko-Westering (2013).
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Multirate schemes
Base integration methods to solve y = f (t,y)
Linear multistep methods (including one-step methods as degenerate cases) :
yn =K1
∑i=1
αiyn−i +hK2
∑i=0
βiyn−i
where yn approximates y(tn), h = tn− tn−1 and yj = f (tj,yj)
BDF methods (K2 = 0,K1 = q) : yn =q
∑i=1
αiyn−i +hβ0yn
Adams methods :
- explicit of order q (K1 = 1,α1 = 1,K2 = q−1,β0 = 0) : yn = yn−1 +hq−1
∑i=1
βiyn−i
- implicit of order q (K1 = 1,α1 = 1,K2 = q−1) : yn = yn−1 +hq−1
∑i=0
βiyn−i
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Multirate schemes
Base integration methods to solve y = f (t,y)
Runge Kutta (RK) methods
r-stage explicit RK methods : yn = yn−1 +r
∑i=1
biki
with k1 = hf (tn−1,yn−1) and ki = hf (tn−1 +cih,yn−1 +i−1
∑j=1
aijkj) (i = 2 . . .r)
r-stage implicit RK methods : yn = yn−1 +r
∑i=1
biki
with ki = hf (tn−1 + cih,yn−1 +r
∑j=1
aijkj) (i = 1 . . .r)
r-stage Rosenbrock methods : yn = yn−1 +r
∑i=1
biki
ki = hf (tn−1 + cih,yn−1 +i−1
∑j=1
aijkj)+dih2 ∂ f∂ t
(tn−1,yn−1)+h∂ f∂y
(tn−1,yn−1)i
∑j=1
dijkj
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Multirate schemes
Base integration methods to solve y = f (t,y)r-stage explicit RK methods, representation in Butcher tableau :
yn+1 = yn +hr
∑i=1
biki with ki = f (tn + cih,yn +hi−1
∑j=1
aijkj)
c1 = 0 0c2 a21c3 a31 a32...
.
.
.
.
.
.. . .
cr ar1 ar2 · · · ar,r−1b1 b2 · · · br−1 br
or shorter c AbT or [A,b,c]
r-stage implicit RK methods, representation in Butcher tableau :
yn+1 = yn +hr
∑i=1
biki with ki = f (tn + cih,yn +hr
∑j=1
aijkj)
c1 a11 · · · a1r...
.
.
.
.
.
.cr ar1 · · · arr
b1 · · · br
or shorter c AbT or [A,b,c]
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Multirate schemes
Rice 1960, Multirate RK methods
x = F(t,x,y) x(t0) = x0, x latent componenty = G(t,x,y) y(t0) = y0, y active component
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Multirate schemes
Rice 1960, Multirate RK methodsEvaluation of x(m+1)K ,x(m+2)K , . . . :
x(m+1)K = xmK +3
∑i=1
biki
ki = hF(tmK + ciKh,xmK +i−1
∑j=1
aijkj,ymK +i−1
∑j=1
aijhj) (i = 1 . . .3)
hi = hG(tmK + ciKh,xmK +i−1
∑j=1
aijkj,ymK +i−1
∑j=1
aijhj) (i = 1 . . .2)
bi,ci,aij given by any RK3 method (work also done with RK4).
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Multirate schemes
Rice 1960, Multirate RK methodsEvaluation of ymK+j+1,ymK+j+2, . . . :
ymK+j+1 = ymK+j +3
∑i=1
αidi(j) for 0≤ j≤ K−1.
d1(j) = hG(tmK+j,xmK+j,ymK+j)
d2(j) = hG(tmK+j + µ2h,xmK+j +6
∑i=4
λi(j)ki−3,ymK+j + γ21d1(j))
d3(j) = hG(tmK+j + µ3h,xmK+j +9
∑i=7
λi(j)ki−6,ymK+j + γ31d1(j)+ γ32d2(j))
with extrapolation using previous ki : xmK+j = xmK +3
∑i=1
λi(j)ki 1≤ j≤ K−1,
where several sets of parameters “αi,µi,γik,λi(j)” are determined so that :option 1) local truncation error of integration formula for y(t) is in O(h4)option 2) extrapolation parameters λi(j) leads to an extrapolation truncation error in O(h4)
and integration parameters are determined independantly.
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Multirate schemes
Rice 1960, Multirate RK methodsApplications : 2 degrees of freedom problem of the type :
dxdt
= x/2 x(0) = 1
dydt
= x cos(25t) y(0) = 1/1250.5
Number of operations and functions evaluations :
Add. Mult. F evaluation G evaluationNormal RK3 22K 28K 3K 3KMultirate RK3 14(K+1) 17(K+1) 3 3K+2
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Multirate schemes
Skelboe 1989, Multirate BDF methods
y = f (t,y,z), y(t0) = y0, fast subsystem
z = g(t,y,z), z(t0) = y0, slow subsystem
Fast subsystem integrated by a k-step BDF formula (BDF-k) with step length h :
ym =k
∑i=1
αiym−i +hβ0f (tm,ym,zm)
Slow subsystem integrated by the same BDF-k formula with step length H = qh :
zn =k
∑i=1
αizn−qi +qhβ0f (tn,yn,zn)
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Multirate schemes
Skelboe 1989, Multirate BDF methods
Various strategies for the sequence of computation :
Fastest first algorithmstep 1) Integration of the fast subsystem from tn−q to tn (q steps) with extrapolated
values zm (n−q < m≤ n) based on zn−kq, . . . ,zn−q (Newton, zm =k
∑r=1
αr,m−(n−q)zn−rq).
step 2) Integration of the slow subsystem from tn−q to tn (one step).
Slowest first algorithmstep 1) Integration of the slow subsystem from tn−q to tn (one step) with extrapolatedvalue yn based on yn−q−k+1, . . . ,yn−q (Newton).step 2) Integration of the fast subsystem from tn−q to tn (q steps) with interpolated
values zm (n−q < m < n) based on zn−(k−1)q, . . . ,zn (Newton, zm =k−1
∑r=0
αr,m−(n−q)zn−rq).
Option : Waveform relaxation until convergence.16 Koobus I3M-Montpellier
Multirate schemes
Skelboe 1989, Multirate BDF methodsVarious strategies for the sequence of computation (continued) :
Implicit multirate algorithmInterpolated values zm (n−q < m < n) based on zn−(k−1)q, . . . ,zn where zn computedby BDF-k with yn computed by BDF-k.Integration from tn−q to tn ⇒ solution of one large system of algebraic equations :
K(
YnZn
)= L
(Yn−qZn−q
)where
K =(
Kyy KyzKzy Kzz
), L =
(Lyy LyzLzy Lzz
),
Yn = (yn,yn−1, . . . ,yn−k+1)T and Zn = (zn,zn−q, . . . ,zn−(k−1)q)T .
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Multirate schemes
Skelboe 1989, Multirate BDF methodsApplications : 2×2 test problems for investigating the stability properties of theprevious multirate algorithms (BDF-1 and BDF-2, interpolation of order 0 and 1).
Conclusion : These multirate methods not necessarily A-stable, even when basedon A-stable integration formulas.
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Multirate schemes
Günther-Rentrop 1993, Multirate ROW methods
Autonomous EDO (for the sake of clarity) :
y(t) = f (y), y(t0) = y0, y ∈ Rn
⇓
yS = fS(yS,yL), yS(t0) = yS0, yS ∈ RnS , active subsystem
yL = fL(yS,yL), yL(t0) = yL0, yL ∈ RnL , latent subsystem
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Multirate schemes
Günther-Rentrop 1993, Multirate ROW methods
yL integrated with ROW methods on one large time step H :
yHL (t0 +H) = yL0 +
s
∑i=1
ciki
ki = hfL(yS(t0 +αiH),yL0 +i−1
∑j=1
αijkj)+HJL
i
∑j=1
γijkj , JL =∂ fL∂yL
(yS0,yL0)
where αi =i−1
∑j=1
αij and yS(t) is an extrapolated value for yS(t).
yS integrated with ROW methods and m time steps h = H/m :
yHS (t0 +(λ +1)h) = yS0(t0 +λh)+
s
∑i=1
cili
li = hfS(yS(t0 +λh)+i−1
∑j=1
αijlj, yL(t0 +λh+αi))+hJS
i
∑j=1
γijlj ,
JS =∂ fS∂yS
(yS(t0 +λh), yL(t0 +λh)), for λ = 0,1, . . . ,m−1
where yL(t) is an extrapolated value for yL(t).
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Multirate schemes
Günther-Rentrop 1993, Multirate ROW methods
Rational (1,1)-extrapolation scheme (Padé approximant) :
ySi(t0 + h) = ySi(t0)+2h fSi(t0)2
2fSi(t0)− hnS
∑j=1
∂ fSi
∂ySj(y(t0))fSj(y(t0))− h
n
∑j=nS+1
∂ fSi
∂yLj(y(t0))fLj(y(t0))
yLi(t0 + h) = yLi(t0)+2h fLi(t0)2
2fLi(t0)− hnS
∑j=1
∂ fLi
∂ySj(y(t0))fSj(y(t0))− h
n
∑j=nS+1
∂ fLi
∂yLj(y(t0))fLj(y(t0))
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Multirate schemes
Günther-Rentrop 1993, Multirate ROW methods
Applications : simulation of electric circuits (inverter chain)⇒ stiff EDO (system of 250-4000 differential equations).
Results : implementation of a multirate 4-steps ROW method, A-stable, speedup upto 2.8 compared to a RK4 method.
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Multirate schemes
Löhner-Morgan-Zienkiewicz 1984, Explicit multirate for hyperbolic problems∂U∂ t
+∇ ·F(U) = 0 in Ω = Ω1 ∪Ω2,
with, for a given explicit scheme, an allowable time step ∆t1 in Ω1 and ∆t2 = ∆t1/n in Ω2.
1D model and a splitting into 2 subdomains
One global time step of the proposed multirate explicit scheme (for 2 subdomains) :
Add to Ω2 two grid points of Ω1 → new subdomain Ω′2.
Specify a BC for U (free or fixed) at point C and advance one global time step ∆t1 in Ω1.
Specify a BC for U (free or fixed) at point A and advance n small time steps ∆t2 = ∆t1/n in Ω′2.
UA is obtained from Ω1, UC from Ω′2, UB = mean values obtained from Ω1 and Ω′2.
The same procedure can be performed with more than 2 subdomains splitting, and in the multidimensional case.
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Multirate schemes
Löhner-Morgan-Zienkiewicz 1984, Explicit multirate for hyperbolic problemsApplications :
Implementation of the proposed multirate scheme with a second order explicit FE scheme (Taylor-Galerkinmethod of Donea).
Transient solution of a 1D shock tube problem (Sod).
Transient solution of a 2D supersonic inviscid flow around a circular cylinder.
Steady-state solution of a 2D supersonic inviscid flow past a wedge.
Speedup of 2 between the multirate and single-rate scheme (2D supersonic wedge).
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Multirate schemes
Kirby 2002, Multirate forward Euler for hyperbolic conservation laws
∂y(t,x)∂ t
+∂F(y(t,x))
∂x= 0
⇓ Semi-discretization
yi(t) = fi(y1(t), . . . ,yn(t)), i = 1, . . . ,n
⇓ Partitioning in slow/fast components
yF = fF(yF,yS) (fast solution subsystem, explicit Euler time step ∆t/m)
yS = fS(yF,yS) (slow solution subsystem, explicit Euler time step ∆t)
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Multirate schemes
Kirby 2002, Multirate forward Euler for hyperbolic conservation laws
A multirate scheme based on forward Euler steps
yF : m steps integration from tn to tn+1
yn+ηkF = yn+ηk−1
F +σk∆t fF(yn+ηk−1F ,yn
S), k = 1, . . . ,m−1yn+1
F = yn+ηm−1F +σm∆t fF(yn+ηm−1
F ,ynS)
yS : 1 step integration from tn to tn+1
yn+1S = yn
S +∆t fS(ynF,yn
S)
wherem
∑k=1
σk = 1 with 0 < σk ≤ 1,
ηl =l
∑k=1
σk, η0 = 0 and tn+ηk = tn +ηk∆t.
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Multirate schemes
Kirby 2002, Multirate forward Euler for hyperbolic conservation laws
Results : the proposed multirate scheme satisfies the TVD property and amaximum principle under local CFL conditions, but only first order timeaccurate.
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Multirate schemes
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation laws
∂y(t,x)∂ t
+∂F(y(t,x))
∂x= 0
⇓ Semi-discretization
yi(t) = fi(y1(t), . . . ,yn(t)), i = 1, . . . ,n
⇓ Partitioning in slow/fast subsystems
yF = fF(yF,yS), fast subsystem (6= fast solution subsystem)
yS = fS(yF,yS), slow subsystem (6= slow solution subsystem )
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Multirate schemes
Sandu-Constantinescu 2007, Multirate for hyperbolic conservation laws
Fast solution : solution with fast characteristic time (6= yF)Slow solution : solution with slow characteristic time (6= yS)ΩF : fast characteristic time , small time step ∆t/m used in the multirate schemeΩFB : slow characteristic time , but small time step ∆t/m used in the multirate schemeΩS : slow characteristic time , large time step ∆t used in the multirate schemeyF = fast solution ∪ fast buffer solution→ small time step ∆t/myS = slow solution \ fast buffer solution→ large time step ∆t
ΩFB : important for the TVB property of the multirate scheme under local CFL conditions(size of fast buffer = half of stencil size).
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Multirate schemes
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation lawsMultirate partitioned RK scheme (2nd order accurate, conservative, nonlinearly stable) :
c AbT Base method (RKB )
1m c 1
m A1m 1+ 1
m c 1m 1bT 1
m A
.
.
.
.
.
.. . .
m−1m 1+ 1
m c 1m 1bT · · · 1
m 1bT 1m A
1m bT 1
m bT · · · 1m bT
c Ac A
.
.
.. . .
c A1m bT 1
m bT · · · 1m bT
Fast method (RKF ) : yF = fF (yF ,yS) Slow method (RKS ) : yS = fS(yF ,yS)
Same weight coefficients for RKF and RKS
(bFi
= bSi=
bim
): important for second order accuracy and conservation properties of the multirate scheme.
30 Koobus I3M-Montpellier
Multirate schemes
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation laws
Case RK2 and m=2 :0 0 01 1 0
1/2 1/2
0 01/2 1/2 01/2 1/4 1/4 01 1/4 1/4 1/2 0
1/4 1/4 1/4 1/4
0 01 1 00 0 0 01 0 0 1 0
1/4 1/4 1/4 1/4
Base method (RKB) Fast method (RKF ) Slow method (RKS )
RKB , RKF and RKS stages :
RKB (y = f (y)) : RKF (yF = fF (yF ,yS)) : RKS (yS = fS(yF ,yS)) :
k1 = f (yn) k1F = fF (yn
F ,ynS) k1
S = fS(ynF ,yn
S)
y(1) = yn +∆t k1 y(1)F = yn
F + ∆t2 k1
F y(1)S = yn
S +∆t k1S
k2 = f (y(1)) k2F = fF (y(1)
F ,y(1)S ) k2
S = fS(y(1)F ,y(1)
S )
yn+1 = yn + ∆t2 (k1 + k2) y(2)
F = ynF + ∆t
4 k1F + ∆t
4 k2F y(2)
S = ynS
k3F = fF (y(2)
F ,ynS) k3
S = fS(y(2)F ,yn
S)
y(3)F = y(2)
F + ∆t2 k3
F y(3)S = yn
S +∆t k3S
k4F = fF (y(3)
F ,y(3)S ) k4
S = fS(y(3)F ,y(3)
S )
yn+1F = yn
F + ∆t4 (k1
F + k2F + k3
F + k4F ) yn+1
S = ynS + ∆t
4 (k1S + k2
S + k3S + k4
S )
At each stage of the multirate formula, evaluation of the flux functions at the same argument values : important for the conservation properties of the multirate scheme.
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Multirate schemes
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation laws
ΩSB = slow buffer, fS depends on yF and yS (size = m × half of stencil size).
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Multirate schemes
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation laws
Case RK2 and m=2 :0 0 01 1 0
1/2 1/2
0 01/2 1/2 01/2 1/4 1/4 01 1/4 1/4 1/2 0
1/4 1/4 1/4 1/4
0 01 1 00 0 0 01 0 0 1 0
1/4 1/4 1/4 1/4
Base method (RKB) Fast method (RKF ) Slow method (RKS )
RKB (y = f (y)) : ΩF ∪ΩFB , RKF (yF = fF (yF ,yS)) : ΩSB , RKS (yS = fS(yF ,yS)) : ΩS\ΩSB , RKS → RKB (yS = fS(yS))
k1 = f (yn) k1F = fF (yn
F ,ynS) k1
S = fS(ynF ,yn
S) k1S = fS(yn
S)
y(1) = yn +∆t k1 y(1)F = yn
F + ∆t2 k1
F y(1)S = yn
S +∆t k1S y(1)
S = ynS +∆t k1
Sk2 = f (y(1)) k2
F = fF (y(1)F ,y(1)
S ) k2S = fS(y(1)
F ,y(1)S ) k2
S = fS(y(1)S )
yn+1 = yn + ∆t2 (k1 + k2) y(2)
F = ynF + ∆t
4 k1F + ∆t
4 k2F y(2)
S = ynS
y(2)S = yn
S
k3F = fF (y(2)
F ,ynS) k3
S = fS(y(2)F ,yn
S)
k3S = fS(yn
S) = k1S
y(3)F = y(2)
F + ∆t2 k3
F y(3)S = yn
S +∆t k3S
y(3)S = yn
S +∆t k3S = y(1)
S
k4F = fF (y(3)
F ,y(3)S ) k4
S = fS(y(3)F ,y(3)
S )
k4S = fS(y(3)
S ) = k2S
yn+1F = yn
F + ∆t4 (k1
F + k2F + k3
F + k4F ) yn+1
S = ynS + ∆t
4 (k1S + k2
S + k3S + k4
S ) yn+1S = yn
S + ∆t2 (k1
S + k2S )
33 Koobus I3M-Montpellier
Multirate schemes
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation lawsProperties of the proposed multirate partitioned RK scheme :
second order accurate
conservative
nonlinear stable (positivity, maximum principle preserving, TVB)
theoritical speedup (single rate/multirate) :
Speedup =m(NΩF +NΩFB +NΩS )
m(NΩF +NΩFB +NΩSB )+NΩS−NΩSB=
m(NΩF +NΩFB +NΩS )m(NΩF +NΩFB +Nintm∆)+NΩS−Nintm∆
where NΩX = number of nodes in ΩX , ∆= half of stencil size, and Nint= number of interface nodes between NΩFB
and NΩS .
for large m, decrease of speedup⇒ nested partitioning.in practice, NΩSB << min(NΩF +NΩFB ,NΩS)
⇒ Speedup ' m(NΩF +NΩFB +NΩS )m(NΩF +NΩFB )+NΩS
⇒ Speedup close to the ideal value of m if NΩF +NΩFB << NΩS .
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Multirate time scheme
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation lawsNested partitioning :
Nested partitioning, example with 3 levels
N +1 levels of partitioning, with time step requirement ∆tj = ∆tmj , j = 0, . . . ,N.
⇒ Speedup =
(mN
N
∑j=0
Lj
)/( N
∑j=0
mjLj
)where Lj = number of grid points associated to ∆tj (level j)
If Lj+1 << Lj, then Speedup ' mN .
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Multirate time scheme
Sandu-Constantinescu 2007, Multirate RK for hyperbolic conservation laws
Applications :multirate RK2 scheme with m = 2 (∆t/2) and m = 3 (∆t/3), 2 levels ofpartitioning.1D advection equation (initial solutions : step, triangular and exponentialshape), fixed and moving grids, 2nd order limited FV scheme.1D burger equation (initial solutions : step and exponential shape), fixed grids,3rd order TVD FV scheme.numerical solutions : 2nd order accurate, positive, obey the maximumprinciple, TVD, wiggle free; conservative time steps.Speedup (single rate/multirate, burger Eq.), fast region ' 10 % entire domain :
Time Single rate Multirate Experimental Theoriticalratio time (sec) time (sec) Speedup Speedup
m = 2 25.28 13.71 1.84 1.80m = 3 36.73 15.07 2.43 2.45
36 Koobus I3M-Montpellier
Multirate time scheme
Sandu-Constantinescu 2009, Multirate Adams for hyperbolic conservation lawsMultirate explicit Adams : same solution component partitioning as for multirate RKschemes.
Semi-discretization of hyperbolic PDE’s⇓
Partitioning in slow/fast subsystems
yF = fF(yF,yS), fast subsystem
yS = fS(yF,yS), slow subsystem
yF integrated with small time step h at times . . . , tn−m, tn−m+1, tn−m+2, . . . , tn−1, tn, . . . :. . . ,yn−m
F ,yn−m+1F ,yn−m+2
F , . . . ,yn−1F ,yn
F, . . .yS integrated with large time step mh at times . . . , tn−3m, tn−2m, tn−m, tn, . . . :
. . . ,yn−3mS ,yn−2m
S ,yn−mS ,yn
S, . . .
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Multirate time scheme
Sandu-Constantinescu 2009, Multirate Adams for hyperbolic conservation laws
Time integration from tn−m to tn by the multirate explicit k-steps Adams(∗) scheme (fastestfirst strategy):
Step 1, for l = 1, . . . ,m : yn−m+lF = yn−m+l−1
F +hk
∑i=1
βifF(yn−m+l−iF ,yn−im
S )
Step 2, ynS = yn−m
S +hk
∑i=1
βi
(m
∑l=1
fS(yn−m+l−iF ,yn−im
S )
)
Remark : on ΩS\ΩSB, yS = fS(yS) ⇒ ynS = yn−m
S +mhk
∑i=1
βifS(yn−imS )
(base explicit k-steps Adams scheme retrieved with an integration step of mh).
Evaluation of the flux functions with the same argument values and same weights βi for yFand yS : important for the conservation properties and 2nd order accuracy of the multiratescheme.(∗) Base explicit k-steps Adams scheme applied to y = f (y) : yn = yn−1 +h
k∑
i=1βi f (yn−i)
38 Koobus I3M-Montpellier
Multirate time scheme
Sandu-Constantinescu 2009, Multirate Adams for hyperbolic conservation lawsCase of explicit 2-steps Adams and m = 2 :
yn−1F = yn−2
f + 3h2 fF(yn−2
F ,yn−2S )− h
2 fF(yn−3F ,yn−4
S ),
ynF = yn−1
f + 3h2 fF(yn−1
F ,yn−2S )− h
2 fF(yn−2F ,yn−4
S )
ynS = yn−2
S + 3h2
(fS(yn−1
F ,yn−2S )+ fS(yn−2
F ,yn−2S )
)− h
2
(fS(yn−2
F ,yn−4S )+ fS(yn−3
F ,yn−4S )
)Remark : on ΩS\ΩSB, yS = fS(yS) ⇒ yn
S = yn−2S +
3×2h2
fS(yn−2S )−2h
2fS(yn−4
S )(base explicit k-steps Adams scheme retrieved with an integration step of 2h)
39 Koobus I3M-Montpellier
Multirate time scheme
Sandu-Constantinescu 2009, Multirate Adams for hyperbolic conservation lawsProperties of the proposed multirate explicit k-steps Adams scheme :
second order accurate
conservative
nonlinear stable (positivity, maximum principle preserving, TVB)
theoritical speedup (single rate/multirate) :
Speedup =m(NΩF +NΩFB +NΩS )
m(NΩF +NΩFB +NΩSB )+NΩS−NΩSB=
m(NΩF +NΩFB +NΩS )m(NΩF +NΩFB +Nintm∆)+NΩS−Nintm∆
where NΩX = number of nodes in ΩX , ∆= half of stencil size, and Nint= number of interface nodes between NΩFB
and NΩS .
for large m, decrease of speedup⇒ nested partitioning.in practice, NΩSB << min(NΩF +NΩFB ,NΩS)
⇒ Speedup ' m(NΩF +NΩFB +NΩS )m(NΩF +NΩFB )+NΩS
⇒ Speedup close to the ideal value of m if NΩF +NΩFB << NΩS .
40 Koobus I3M-Montpellier
Multirate time scheme
Sandu-Constantinescu 2009, Multirate Adams for hyperbolic conservation laws
Applications :multirate explicit 2-steps Adams scheme with m = 2 (2h) and m = 3 (3h), 2levels of partitioning.1D advection equation (initial solution : step shape), fixed grids, 3rd orderlimited FV scheme.1D burger equation (initial solution : step ashape), fixed grids, 3rd order TVDFV scheme.numerical solutions : 2nd order accurate, positive, obey the maximumprinciple, TVD, wiggle free; conservative time steps.Speedup (single rate/multirate, advection Eq.), fast region ' 10 % entire domain :
Time Single rate Multirate Experimental Theoriticalratio time (sec) time (sec) Speedup Speedup
m = 2 39.81 19.44 2.04 1.81m = 3 39.81 14.22 2.79 2.50
41 Koobus I3M-Montpellier
Transition to order 3
Definition of the scheme (CENO)
Finite Volume approach (2D) :
ddt
∫Ci
u(x,y, t)dxdy+∫
∂Ci
~f (u(x,y, t)).~nds = 0
ddt
∫Ci
u(x,y, t)dxdy+ ∑k∈V(i)
∫∂Ci∩∂Ck
~f (u(x,y, t)).~nds = 0
42 Koobus I3M-Montpellier
Transition to order 3 (2)
Definition of the scheme (CENO)
Polynomial reconstruction :Average of a function g over cell Ck : gk = 1
area(Ck)∫
Ckg(x,y)dxdy
We define Pni = uni + ∑
α∈Icn
i,α
[(X−X0,i)α − (X−X0,i)α
i]
Pni
i = uni is satisfied.cn
i,α chosen to minimize Hi = ∑k∈N(i)
(Pni
k−unk)2
⇒ Linear system with unknowns cni,α (5 in 2D)
43 Koobus I3M-Montpellier
Transition to order 3 (3)
Definition of the scheme (CENO)
Flux evaluation :
Interfaces Ci∩∂Ck between Ci and Ck, (1) : ∂C(1)ik and (2) : ∂C(2)
ik∫∂Ci∩∂Ck
~f (u(x,y, t)).~nds = ∑l=1,2
∫∂C(l)
ik
~f (u(x,y, t)).~nds
= ∑l=1,2
∫∂C(l)
ik
~f (Pi(x,y, t)).~nds
= ∑m=1,2
ωm~f (Pi(x(l)gm,ik,y
(l)gm,ik, t))~νik
(l)
44 Koobus I3M-Montpellier
Transition to order 3 (4)
Definition of the scheme (CENO)
Flux evaluation (2) :
~f (Pi(x(l)gm,ik,y
(l)gm,ik, t)). ~νik = Φ(Pi(x
(l)gm,ik,y
(l)gm,ik, t),Pk(x
(l)gm,ik,y
(l)gm,ik, t), ~νik)
where Roe’s scheme is used as approximate Riemann solver :
Φ(u1,u2,~ν) =~f (u1)+~f (u2)
2.~ν− γ
2
∣∣∣∣∣ ∂~f∂u
(u1 +u2
2
).~ν
∣∣∣∣∣(u2−u1)
45 Koobus I3M-Montpellier
Application with mesh adaption
Scramjet, thesis of A. Carabias (INRIA Sophia and Rocquencourt)
Figure: 2D anisotropic mesh adaption (31460 nodes), iso-contours of Mach number. InletMach number = 3, CENO scheme.
46 Koobus I3M-Montpellier
Thank you for your attention.
47 Koobus I3M-Montpellier
Appendix : Turbulence modeling
VMS-LES approach
Main features :Approach based on variational projections of the Navier-Stokes equations ⇒equations governing different scales of the solution (large resolved scales,small resoved scales, unresolved scales),Effects of the unresolved scales only modeled in the equations governing thesmall resolved scales :
petites echelles resoluesgrandes echelles resolues
modelisation
echelles non resolues
echelles resolues
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Appendix : Turbulence modeling
VMS-LES approach (2)
The VMS-LES option chosen allows to take into account :
the 3D compressible Navier-Stokes equations,
unstructured meshes,
a finite element/finite volume formulation,
the scales separation with a simple and efficient procedure obtained fromaveraging on macro-cells,
bluff body flows with vortex sheding.
49 Koobus I3M-Montpellier
Appendix : Turbulence modeling
RANS/VMS-LES hybrid model
Central idea of this hybrid approach :
Solve the RANS equations in the whole domain,Correct the mean flow field by adding fluctuations provided by a VMS-LESmodel in regions where the grid resolution is fine enough for VMS-LES.
Basic ingredients of this hybrid approach :a RANS model,a VMS-LES model,a blending function.
50 Koobus I3M-Montpellier
Appendix : Turbulence modeling
RANS/VMS-LES hybrid model (2)(∂Wh
∂ t ,Xi
)+(∇ ·F(Wh),Xi,Φi) =−θ(τRANS (Wh) ,Φi)
−(1−θ)(τLES(W ′h),Φ′i)
where θ = tanh[(
∆
lRANS
)2]
with lRANS =k3/2
εet ∆ = local mesh size.
⇓
∆ lRANS : θ → 0 (VMS-LES mode)∆ lRANS : θ → 1 (RANS mode)
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