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• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Parallel Numerical Solution of the 2D HeatEquation

16 January 2012

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

1 Introduction

2 Heat equation

3 Existence uniqueness

4 Numerical analysis

5 Numerical simulation

6 Conclusion

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Introduction

Aim: parallel solving of the heat equation with MPI.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Basic equation

Figure: = [0, Lx ] [0, Ly ]

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Basic equation

8>>>>>:

@u(x , y , t)

@tD4u(x , y , t) = f (x , y , t) in

u = g(x , y , t) on 0u = h(x , y , t) on 1

u(x , y , 0) = u0(x , y)

(1)

where @ = 0 [ 1, @ is the boundary of , t 2 [0,T ] R andx,y 2 . D > 0We assume that, u0 2 L2(), f 2 L2([0,T ],L2()),and we suppose g 2 H

12 (0), h 2 H

12 (1),

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Existence, uniqueness:

Existence and uniqueness of the continuous solution is provedusing the spectral method.

Proof.

For the proof, we refer to the report.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Implicit Euler scheme

Figure: Discretization of : particular case for nx=5 and ny=4

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Implicit Euler scheme

Spacial mesh points: xj

= jx and yi

= iy, 1 j nx and

1 i ny, where x = Lx(nx + 1)

, y =L

y

(ny + 1).

Temporal mesh points: t = nt. 1 n nt.Value of u at the point (x

j

, yi

) and at time tn

:unji

= u(xj

, xi

, tn

).

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Implicit Euler scheme

At the point (xj

, yi

) and at time tn

the original dierential equation(1) becomes:u

n

j,iun1j,i

t D

u

n

j1,i2un

j,i+un

j+1,i

x2 +u

n

j,i12un

j,i+un

j,i+1

y2

= f n

j ,i in

u = gnj ,0 on

s

0 u = gn

j ,ny+1 on n

0

u = hn0,i on w

1 u = hn

nx+1,i on e

1

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Implicit Euler scheme

We define the global notation l = (i 1)nx + j this is equivalent toi = [ l1

nx

+ 1] and j = mod(l 1, nx) + 1.

unl

un1l

tD

unl1 2unl + unl+1

x2+

unlnx 2unl + unl+nx

y2

= f n

l

in

(2)

l =

8>>>:

1, . . . , nx on s0nx(ny 1) + p p = 1, . . . , nx on n0(p 1)nx + 1 p = 1, . . . , ny on w1

pnx p = 1, . . . ny on e1

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Convergence

We define Lap

u:

unj ,i u

n1j ,i

tD

unj1,i 2unj ,i + unj+1,i

x2+

unj ,i1 2unj ,i + unj ,i+1

y2

= f n

j ,i

Using Taylor expansion one can show that the implicit Eulerscheme is of order 2 in space and of order 1 in time.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Linear System

We write the system (2) in matrix form AU=F.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Structure of the matrix A

A in the case where nx=5 and ny=4

A =

0

BB@

C B 0 0B C B 00 B C B0 0 B C

1

CCA

where 0 is the nx by nx 0 matrix,

C =

0

BBBB@

a b 0 0 0b a b 0 00 b a b 00 0 b a b0 0 0 b a

1

CCCCA

and B=cI5 with a = 1 +2tDx2 +

2tDy2 , b =

tDx2 and c =

tDy2 .

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Properties of the matrix A

The matrix A in (13) is real, symmetric, positive-defined and hasstrictly dominant diagonal since|A

i ,j | >P

j 6=i Ai ,j for all i. Therefore A is invertible, hence AU=Fhas a solution, that we will compute numerically using conjugategradient method.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

r0 := B AX0p0 := r0k := 0repeat

k

:=r

t

k

r

k

p

t

k

Ap

k

Xk+1 := Xk + kpk

rk+1 := rk kApkif r

k+1 is suciently small then exit loop end if

k

:=r

t

k+1rk+1

r

t

k

r

k

pk+1 := rk+1 + kpk

end repeatk := k + 1

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Parallelization

We are going focus on the parallelization of the operator A, wedivide the rows of the matrix A between the processors. Eachprocessors knows a peace of the vector of unknowns U.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Parallel Algorithm

1.Initializationstime loop2. construction of second member3. resolution of linear system : conjugate gradientmatrix vector productsscalar productcombinations of linear of vectors4. progression in timeFinalizations :destruction of MPI environmentsdeallocation of the matrix and vectorsThe big work here in this algorithm is the parallelization of thematrix vectors product.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Matrix vector products: Communication

In order to do the matrix vector products the processors mustcommunicate, their results two to two, so we use point-to-pointcommunication MPI SEND and MPI RECV. In fact the processorme must know the nx last values of the vector solution of theprocessor me +1 and the ns first values of the vector solution.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

scalar product:reduction

Each processor do the simple scalar product of the peaces of thevectors they have, we use MPI ALLREDUCE to make the sum ofelementary result, and transmit the result to all processors.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Illustration

To illustrate the problem we consider three dierent cases.For the first two cases we checked the correlation between thenumerical solution and exact one.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Illustration: stationary case 1

f (x , y) = 2(x x2 + y y2) g(x , y) = 0 h(x , y) = 0.Exact solution u(x , y) = xy(1 x)(1 y).

Figure: Numerical solution of stationary case 1 for nx=25 et ny=25 withfive processors.

Lydia Flore Mamoade Parallel Numerical Solution of the 2D Heat Equation

• IntroductionHeat equation

Existence uniquenessNumerical analysis

Numerical simulationConclusion

Illustration: stationary case 2

f (x , y) = sin(x) + cos(y) g(x , y) = sin(x) + cos(y)h(x , y) = sin(x) + cos(y).Exact solution u(x , y) = sin(x) + cos(y).

Figure: Numerical solution of stationary case 2 for nx=25 et ny=25 withfive processor