A Parallel Method for Heat Equation with M

emory 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

• 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

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

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

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

• 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)

• 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:

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

• Space-time domain

t

t fdssAustKu0

)()( ),0[

)()(

1

ttK

• Space-frequency domain

0ˆˆˆ ufuAzuz

zzK )(ˆ

•Fourier-Laplace Transform

0

)()(ˆ dttfezf tz

1

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?

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

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)

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))(

)((),(

Space Discretization Error

nznx

5.1nznx

2nznx

• 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

Two dimensional case

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.