A Parallel Method for Heat Equation with Memory
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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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