the parareal algorithm applied to the korteweg-de vries-burgers’ equation

8/8/2019 The Parareal Algorithm Applied to the Korteweg-de Vries-Burgers Equation

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The Parareal Algorithm Applied to the Korteweg-de

Vries-Burgers Equation

Ma. Cristina Bargo12 Sidi Mahmoud Kaber1

1Laboratoire Jacques-Louis LionsUniversit Pierre et Marie Curie

Paris, France

2Institute of Mathematics

University of the Philippines Diliman

Quezon City, Philippines

Runion PITACJune 28, 2010

Bargo, Kaber (UPMC + UPD) Parareal Applied to KdVB Equation PITAC 1 / 70
http://find/http://goback/


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Outline

1 IntroductionThe Parareal AlgorithmProblem : The KdVB EquationAveraging/Renormalization

2 Implementation with Parareal AlgorithmThe Runge-Kutta (order 4) SchemeSpatial DiscretizationThe Lax-Wendroff Scheme

3 Results

4 Some Remarks



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The Parareal Algorithm

First presented by Lions, Maday and Turinici in 2001 (C. R. Acad. Sci.Paris Sr. I Math.) [6]

Current version by Baffico, Bernard, Maday, Turinici and Zrah in2002 (Phys. Rev. E) [1]



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First presented by Lions, Maday and Turinici in 2001 (C. R. Acad. Sci.Paris Sr. I Math.) [6]

Current version by Baffico, Bernard, Maday, Turinici and Zrah in2002 (Phys. Rev. E) [1]

The Problem

Find u such that

tu+ A (t,u) = 0, t > t0

u = u0, t = t0

(1)

where A : R V V (V a Hilbert space) and t0 0.



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We need :Fine solver F(t2; t1,u1) - fine approximation ofu(t2) to problem (1)with IC u(t1) = u1

Coarse solver G(t2; t1,u1) - coarse approximation ofu(t2), less

accurate than F(t2; t1,u1) but cheaper



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We need :Fine solver F(t2; t1,u1) - fine approximation ofu(t2) to problem (1)with IC u(t1) = u1

Coarse solver G(t2; t1,u1) - coarse approximation ofu(t2), less

accurate than F(t2; t1,u1) but cheaper

Parareal Algorithm

U00 = u0 and U

0n+1 = G(tn+1; tn,U

0n)

For k = 1, 2, , Uk0 = u0 and

Uk+1n+1 = G(tn+1; tn,U

k+1n ) + F(tn+1; tn,U

kn) G(tn+1; tn,U

kn) (2)



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Q : How do we choose F and G ?



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F : discretization scheme with timestep t



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G : discretization scheme with a larger timestep T (T t)



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G : discretization scheme with a larger timestep T (T t)based on a reduced model of the original problem (Baffico et al, [1])



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G : discretization scheme with a larger timestep T (T t)based on a reduced model of the original problem (Baffico et al, [1])large timestep T with coarse discretization in space (Fischer, Hechtand Maday, 2005, Domain Decomposition Methods in Science andEngineering [4])



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The KdVB Equation

The Korteweg-de Vries-Burgers (KdVB) equation

Find u(x, t), x R, t > 0 such that

tu + uxu =

1

R

2

xxu

3

xxxu, (3)

where :

IC : u(x, 0) = 1, x < 0 u(x, 0) = 0, x 0

BCs : u() = 1, u() = 0, ux() = 0

R is the Reynolds number



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The KdVB Equation

Click to play



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Averaging/Renormalization

Get the average solution u at each point x over the interval

(x /2, x + /2) ( is a predetermined length)

u(x, t) =1

x+/2x/2

u(s, t)ds

Q : What PDE will the average solution u satisfy ?

tu + uxu =1

R2xxu

3xxxu

Remark : Averaging operator and differentiation operator commute.

tu + uxu =1

R2xxu

3xxxu



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tu + uxu =1

R2xxu + uxu uxu

3xxxu

We rewrite the part in red as :

tu + uxu = x 1R

+ eff xu 3xxxueff is the effective viscosity, not necessarily constant

eff = bxu

, b = u2

2, u = u u


/


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Barenblatt, Ivanov and Shapiro, 1985 (Archive for Rational Mechanics and

Analysis) [2] :

Choose so that it is much larger than the length of a single hump,but much less than the thickness of the wave front.

We can ignore the dispersion term 3xxxu and1R

The Average Equation (Barenblatt et. al., 1985 [2])

For a good choice of , the average solution u can be approximated byfinding v solution of

tv + vxv = x (effxv) , (4)

where v also satisfies the boundary conditions of problem (3)


A /R l


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Chorin, 2003 (Proc. Natl. Acad. Sci. USA) [3] :

eff is approximately constant : Find eff that minimizes

I() = +

|u(x) v(x + z)|2 dx

(Shift z is needed because the problem is translation invariant, andnumerical procedures can produce a shift with no intrinsicsignificance.)

For R = 60, = 20 and z = 0.4, we have eff 4.42


I l i i h P l Al i h


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Implementation with Parareal Algorithm

Idea :

Use the average equation (4) for G and the original equation (3) for F(in the spirit of Baffico et al, [1])

Involves 2 meshes in space, projection is needed (in the spirit of

Fischer, Hecht and Maday, [4])Numerical implementation : splitting scheme

4th order Runge-Kutta scheme for the linear partLax-Wendroff scheme for the nonlinear part

Spatial discretization : fourth-order finite difference schemes (Li andVisbal, 2006 [5])


Th R K ( d 4) S h


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The Runge-Kutta (order 4) Scheme

To approximate solutions to equations of the form :

y = f(t, y)y(t0) = y0

on nodes ti, i = 1, 2, , tN, it = ti ti1 = t (uniform)

Given yn (approximate solution at t = tn), solve for the following :k1 = tf(tn, yn)k2 = tf(tn +

t2

, yn +k12

)

k3 = tf(tn +t2

, yn +k22

)k4 = tf(tn + t, yn + k3)

Get approximate solution yn+1 at t = tn+1 :

yn+1 = yn +1

6(k1 + 2k2 + 2k3 + k4) (5)


S ti l Di ti ti


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Spatial Discretization

first derivative

fi = 1

3

fi+2 fi24h

+4

3

fi+1 fi12h

second derivative

fi = 13

fi+2 2fi + fi24h2

+ 43

fi+1 2fi + fi1h2

third derivative

fi = fi+3 3fi+1 + 3fi1 fi38h3

+ 2fi+2 2fi+1 + 2fi1 f i 2

2h3


Th L W d ff S h


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The Lax-Wendroff Scheme

(See [7], 8-1) To solve equations of the form :

t+ f() = 0 (6)

Taylor expansion of in t :

(x, tn+1) = (x, tn) + t

t(x, tn) +

1

2t2

2

t2(x, tn) + O(t

3)

Use (6) and change the partial derivatives in time in terms of partialderivatives in space


The La Wend off Sche e


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In one dimension :

t=

xf()

2

t2 = t fx = x ft = x f t=

x

A

f

x

where A = f


The Lax Wendroff Scheme


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Approximation :

n+1 = n tf

x+

1

2t2

xA

f

x(7)

with

f

x=

fi+1 fi12x

xA

f

x=

1

x Ai+ 1

2

(fi+1 fi)

x Ai1

2

(fi fi1)

x ,

Ai12

1

2(Ai + Ai1)


Simple parareal implementation


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Parameters :

x = x t T N 1 0,002 0,04 25

Convergence history :

Iteration kUk Uk1

1 0, 02914772 0, 00034015 8, 354 1011

10 2, 065 1014




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Figure: Solution using the fine scheme



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Figure: Convergence History


Reconstruction from the coarse mesh to the fine mesh


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Involves 2 meshes : x for the fine scheme, x >> x for the coarsescheme

Projection between the 2 meshes :

Uk+1n+1 = G(tn+1; tn, Uk+1n ) + F(tn+1; tn,Ukn)

G(tn+1; tn, Ukn ) (8)




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Uk+1n+1 = xx(G(tn+1; tn, xx (Uk+1n ))) + F(tn+1; tn,Ukn)

xx(G(tn+1; tn, xx (U

kn))) (8)




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kn))) (8)

xx : averaging matrix Aav (based on trapezoidal rule)




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kn))) (8)

xx : averaging matrix Aav (based on trapezoidal rule)

xx : ?



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Piecewise Constant Function

xx : piecewise constant

The value of the function at the interval [xj4, xj] will be the same asthe value of the function at Xi1, and so on...




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Figure: Test Function




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Figure: Solution using the parareal algorithm : Iteration 1




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Reconstruction from the coarse mesh to the fine meshPi i C F i


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Reconstruction from the coarse mesh to the fine meshPi i C F i


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Reconstruction from the coarse mesh to the fine meshPse doin erse


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Pseudoinverse



Reconstruction from the coarse mesh to the fine meshPseudoinverse


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Pseudoinverse





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Pseudoinverse





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Pseudoinverse

Convergence history :

Iteration kU

k Uk1

1 1, 7889744

2 1, 89755895 2, 263787710 2, 749808415 2, 931551720 3, 2399287

25 3, 4933626




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Pseudoinverse





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Pseudoinverse





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seudo e se





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Other reconsruction matrices tried :

Piecewise linear (discontinuous)

Piecewise linear (continuous)xx = A

av (AavA

av)

1

right inverse


Transformation from average space to real space


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Same mesh in space, only a transformation of the function from the

real space xj to the average space xj : suppose u contains thevalues at xj and u contains the values at xj

u = Bavu

where Bav is the transformation matrix (square, invertible)



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G : solved on the average spaceF : solved on the real space

Uk+1n+1 = (Bav)1G(tn+1; tn, BavUk+1n ) + F(tn+1; tn,Ukn)

(Bav)1G(tn+1; tn, BavU

kn) (9)




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S l ti ( ?) Filt i i l l d iti f B


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Solution ( ?) : Filter using singular value decomposition of BavBav = U SV

, where

S diagonal matrix, with nonnegative entries arranged in decreasingorderU, V unitary matrices

B1av = V S1U

S1 : remove the last d diagonal entries of S1 and replace them with0

B1av = VS1U

Use B1av instead of B1av in equation (9)

Uk+1n+1 = (Bav)

1G(tn+1; tn, BavUk+1n ) + F(tn+1; tn,U

kn)

(Bav)1G(tn+1; tn, BavU

kn)




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Some Remarks


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Average equation of Chorin is not a good model that can be used inthe coarse scheme

Problems with reconstruction from the coarse mesh to the fine mesh(in space)


L. Baffico, S. Bernard, Y. Maday, G. Turinici, and G. Zrah.Parallel-in-time molecular-dynamics simulations.Phys. Rev. E, 66(5) :057701, Nov 2002.


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G. I. Barenblatt, M. Ya. Ivanov, and G. I. Shapiro.

On the structure of wave fronts in nonlinear dissipative media.87(4) :293303, 1985.

Alexandre J. Chorin.Averaging and renormalization for the Korteveg-de Vries-Burgers

equation.Proc. Natl. Acad. Sci. USA, 100(17) :96749679 (electronic), 2003.

Paul F. Fischer, Frdric Hecht, and Yvon Maday.A parareal in time semi-implicit approximation of the Navier-Stokesequations.

In Domain Decomposition Methods in Science and Engineering,volume 40 of Lecture Notes in Computational Science and Engineering,pages 433440. Springer, Berlin, 2005.

Jichun Li and Miguel R. Visbal.


High-order compact schemes for nonlinear dispersive waves.26(1) :123, 2006.

Jacques-Louis Lions, Yvon Maday, and Gabriel Turinici.Rsolution dEDP par un schma en temps pararel


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Rsolution d EDP par un schma en temps pararel .C. R. Acad. Sci. Paris Sr. I Math., 332(7) :661668, 2001.

Elaine S. Oran and Jay P. Boris.Numerical Simulation of Reactive Flow.Elsevier Science Publishing Co., Inc., 1987.

Bargo Kaber (UPMC + UPD) Parareal Applied to KdVB Equation PITAC 70 / 70

the parareal algorithm applied to the korteweg-de vries-burgers’ equation

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