a parallel method for heat equation with memory
DESCRIPTION
A Parallel Method for Heat Equation with Memory. Kwon, Kiwoon and Sheen,Dongwoo. Dept. of Math. Seoul National University. Naturally Parallel Algorithms. Highly massive computing needs parallel computation. One of major naturally parallel algorithm is Domain decomposition method. - PowerPoint PPT PresentationTRANSCRIPT
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A Parallel Method for Heat Equation with M
emory Kwon, Kiwoon and Sheen,Dongwoo
Dept. of Math.
Seoul National
University
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Naturally Parallel Algorithms
• Highly massive computing needs parallel computation
• One of major naturally parallel algorithm is Domain decomposition method
• Evolution equation is classically solved by time marching( stepping ) method, but It is not parallelizable.
• But Frequency domain method is Naturally parallelizable algorithm for evolutionequation
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Frequency domain method
1.Douglas Jr.,J E Santos,D Sheen : Wave with absorbing boundary condition 2.C-O Lee,J Lee, D Sheen, Y Yeom :heat equation 3.D Sheen,I H Sloan,V Thomee : heat equation(Fourier-Laplace transform) 4.C-O Lee,J Lee,D Sheen : linearized Navier-Stokes 5.K Kwon,D Sheen
: heat equation with memory
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2. Unable to account formemory effects,
which is prevalent in some materials
1. conservation law of energy qhet
div
ukq Cuee 0
2. Fourier’s law
Ch
uCk
ut
Classical heat equation
1. A thermal disturbance at one point propagated instantly to everywhere of the body ( wave – inite speed)Classical heat equation : drawbacks
Classical heat equation
Drawbacks
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Heat equation with memory
• Coleman(64), Gurtin and Pipkin(68) : Replace Fourier’s law with equation with memory term
dsstusKtqt
)()(~)(0
fdssustKut
t )()(0
Integro-differential equation:
Applications
1.The transmission of heat pulses
observed in liquid helium
2.Some dielectrics at low temperature
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• K(s) is a constant a wave equation• K(s) is a Dirac delta function a heat equation
• K(0) is finite The speed of propagation is finite(wave)• K’(0) is divergent The speed of propagation is infinite :The discontinuity is smoothed out(heat)
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• Original Problem
t
t fdssAustKu0
),,0[ )()(),[0, 0),( txu
0},{t )0,( 0 uxu
where A is a symmetric positive definite operator
Weak formulation
),( ),()),(()(),(0
10
t
t HvvfdsvsuAstKvu
. )0,( 0 uxu),,0[ 0),( txu
The weak formulation:
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Positive Memory and Regularity
• The memory )) ,0([)( 1 LtK
Is called a positive memory if it satisfies
T t
dsdtsystKty0 0
0)()()(
)) ,0([ Cy for each• [Regularity] If
K is a positive Memory, then the solution )(tu
satisfies
t
dsfutu0
0 ||||2||||||)(||
• 10 , )(
)(1
ttK
is a positive memory
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• Space-time domain
t
t fdssAustKu0
)()( ),0[
)()(
1
ttK
• Space-frequency domain
0ˆˆˆ ufuAzuz
zzK )(ˆ
•Fourier-Laplace Transform
0
)()(ˆ dttfezf tz
1
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Contour at a frequency domain
• Is it possible to take a Fourier-Laplace transform at each point of a contour?
• Is there a Space-Frequency domain solution at this frequency?
(Avoid singular point!) • Is it possible to take a inverse Fourier-Laplace
transform along the contour?• When any quadrature scheme is used, in which contour the order of convergence is g
ood?
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Discretization in the space domain
• (k-1)th degree finite element space and Ritz projection is used
))||||||(||||(||||||0000 dsuuhuuCuu
k
t
tk
k
hh
))||||||(||||(||||||00
1
1001 dsuuhuuCuuk
t
tk
k
hh
)( 2hOWhen piecewise linear element is used
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Discretization in the frequency domain
))(1(||)()(|| |cos|,,
rrtrtrnz e
srts
eeCtutU
||))(ˆ||supmax||(|| )(0 zfunz k
zrk
r
where
,
For It holds
,
)( rnzO
•Euler-MacLaurin formula
•Spectral analysis
•Semi group theory
•Suitable choice of contour
Point of the proof(SST)
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Fully discretization||)()(||||)()(||||)()(|| ,,,, tutututUtutU hhhnzhnz
||))(ˆ||supmax||(|| )(0,,, zfunzC j
zrj
rtr
)||||||(||00,, t
ktkk
kt dsuuhC
)( rk nzhO approximation
Numerical Test(1D),] ,0[ ,A ,5.0xu sin0
Then the unique solution is
xetxu t sin),( x
zztxf sin
11
)1(),(ˆ
tst xdse
stetxf
0
1
.sin))(
)((),(
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Space Discretization Error
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nznx
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5.1nznx
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2nznx
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• Backward Euler:
)( 12 ntnxO
)( )1(2 ntnxO
• Crank-Nicolson:
)( 2 rnznxO • Frequency domain method:
r:the regularity of right hand side
2ntnx
1ntnx
2/rnxnz
Nx: space domain division numberNt: time domain division numberNz: frequency domain division number
1.0,9.0 p
t
2/rnznx )( rnzO
trs / 1/
0 trs
p
Order of convergence
•Strategy:
•Choice of parameter
•Approximation is bad if T is too small or beta is too close to 0 or 1
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Two dimensional case
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ReferencesA study on inverse problems and numerical methods for partial differential equations,Ph.D thesis, Kiwoon Kwon, Dept. of math.
Seoul National University, 2001,2.
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