numerical solution of 2d heat equation

8/11/2019 Numerical Solution of 2D Heat Equation

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AMIRKABIRUNIVERSITYOFTECHNOLOGY

AEROSAPCEENGINEERINGDEPARTMENT

Numerical Solution of 2D Heat

Equation by ADI and SLOR methodsComputational Fluid Dynamics Course Assignment

Instructor: Dr. Jahangirian

Ata Ghasemi Esfahani

student ID: 92129071

W I N T E R 2 0 1 4


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Contents

1. Introduction ........................................................................................................................3

2. Problem definition ...............................................................................................................3

3. Discretization of 2D heat equation and MATLAB codes .............................................................4

3.1 ADI method ...................................................................................................................4

3.2 SLOR method .................................................................................................................8

4. Results and discussion .........................................................................................................12

4.1 ADI method ..................................................................................................................12

4.2 SLOR method ................................................................................................................21

4.3 Discussion .....................................................................................................................24

5. References .........................................................................................................................25


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1.Introduction

This report summarizes the results obtained from implementation of various implicit methods

for solving two-dimensional heat conduction equation with a specific set of boundary and

initial conditions.

The most crucial advantage associated with explicit method is its relative ease of

implementation. On the other hand, in order for the solution to be stable, for a fixed value of

x, one must consider limited values of t such that the stability criteria are met. This usually

results in long computer run times. Implicit methods often maintain stability even for much

larger values of t, but the massive matrix manipulations needed at each time step which

results in an increase in computation time. Also the implicit methods might not be as accurate

in yielding transient solutions since truncation error is large.

The methods investigated in this assignment, namely ADI and SLOR, have been devised to

take advantage of the so-called tri-diagonal matrix solution method that can be coded and

implemented easily. Instead of attempting to manipulate massive matrices, the ADI method

resorts to a method called two-level solution. This approach ensures that at each time step,

there are no more than three unknowns to solve for thus the tri-diagonal solution routine, also

known as Thomas algorithm, can be employed.

The SLOR (SOR by line, where SOR stands for successive over-relaxation) is a line-iterative

method. Although relaxation techniques can be formulated in explicit or implicit forms, the

specific discretization form presented here is implicit and again, relies on Thomas algorithm

for matrix manipulations. Within this framework, some quantities are assumed to be known

at initial iterative step and some other are updated as the iterative solution progresses.

Although both the ADI and SLOR methods are classified as implicit methods, care must be

taken in selecting grid size and time steps. As will be seen in what follows, careful selection

of the mentioned parameters will result in a physically sound solution and improper selection

of those yields inappropriate results.

2. Problem definition

Constant

Temperature

T=100 C

Constant

Temperature

T=100 C

Constant

Temperature

T=300 C

Constant

Temperature

T=0 C


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3. Discretization of 2D heat equation and MATLAB codes

3.1 ADI method

The unsteady two-dimensional heat conduction equation (parabolic form) has the following

form:

A forward time, central space scheme is employed to discretize the governing equation as

described in the next page.

The idea of the ADI-method (alternating direction implicit) is to alternate direction and thus

solve two one-dimensional problem at each time step. The first step keepsy-direction fixed:

In the second step we keepx-direction fixed:

The ADI calculation procedure is depicted in the figure below:

Both equations can be written in a triadiagonal form. If we define:


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Then we get:

Instead of five-band matrix in BTCS method, here each time step can be obtained in two

sweeps. Each sweep can be done by solving a tri-diagonal system of equations. The ADI

method is second order accurate with a T.E. of the order O[ t2, x2, y2]. The amplification

factor is as follows:

Where

It is obvious that this method is unconditionally stable, although the 3D ADI scheme is only

conditionally stable.

The first step of the ADI scheme is reduced to the following form that can be implemented as

a MATLAB code:

ATI-1,jn+1/2BTI,j

n+1/2+ATi+1,jn+1/2= Ki

Where:

A= (t/2x2)

B= 1+ (t/x2)

Ki= -Ti,jn- (t/2x2) [ Ti,j+1

n-2Ti,jn+Ti,j-1

n ]

The second step of the ADI scheme is implemented as follows:

CTI-1,jn+1/2

DTI,jn+1/2

+CTi+1,jn+1/2

= Ki


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%boundary conditions

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

fori=1:m+1

te(1,i)=300;

te(m+1,i)=0;

end

forj=2:m+1

te(j,1)=100;

te(j,m+1)=100;

end

te(:,:)

% formation of matrix A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

forj=1:m

fori=1:m

if(i==j)

ma(j,i)=-b;

end

if(i==j+1)

ma(j,i)=a;

end

if(i==j-1)

ma(j,i)=a;

end

end

end

fort=1:n

forj1=2:m

fori=2:m

k(i-1)=-te(j-1,i)-((alfa*dt)/(2*dy^2))*(te(j-1,i+1)-

2*te(j-1,i)+te(j-1,i-1));

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Thomas Algorithm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

md(1,1)=-b;

forj=1:m-1

fori=1:m-1

if(j~=1)

if(i==j)

md(j,i)=ma(j,i)-md(j-1,i)*a/md(j-1,j-1);

end

end

if(i==j+1)

md(j,i)=a;

endend

end

Boundary

conditions

Formation of coefficients

matrix


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fori=2:9

k(i)=k(i)-k(i-1)*a/md(i-1,i-1);

end

te(j1,m)=k(m-1)/md(m-1,m-1);

fori=1:m-2

i=m-i;

te(j1,i)=(k(i-1)-a*te(j1,i+1))/md(i-1,i-1);

end

end

fori=2:m

forj=2:m

l(j-1)=-te(j,i-1)-((alfa*dt)/(2*dx^2))*(te(j+1,i-1)-

2*te(j,i-1)+te(j-1,i-1));

end

fori=2:9

l(i)=l(i)-l(i-1)*a/md(i-1,i-1);

end

te(m,i)=l(m-1)/md(m-1,m-1);

forj=1:m-2

j=m-j;

te(j,i)=(l(j-1)-a*te(j+1,i))/md(j-1,j-1);

end

end

end

3.2 SLOR method

Assuming the 2D steady heat equation has the following form (Laplace's equation):


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We now have to describe the SLOR method. The general idea behind successive over-

relaxation method, also classified as an iterative method, is to introduce an arbitrary

correction to the intermediate values of the unknown considered in the following way:

where k denotes iteration level, ui,jk+1is the most recent value of ui,,jcalculated from the

Gauss-Seidel procedure, ui,jk'is the value from the previous iteration as adjusted by previous

application of this formula if the over-relaxation is being applied successively (at each

iteration), and ui,jk+1'is the newly adjusted or "better guess" for ui,,j, at the k + 1 iteration level.

That is, we expect ui,jk+1' to be closer to the final solution than the unaltered value ui,j

k+1from

the Gauss-Seidel calculation.

Here, w is the relaxation parameter, and when 1 < w < 2, overrelaxationis being employed.

For Laplace's equation on a rectangular domain with Dirichlet boundary conditions, theories

pioneered by Young (1954)and Frankel (1950)lead to an expression for the optimum w

(hereafter denoted by wop,). First, defining as:

and the optimum wis given by:

To illustrate the procedure, consider the solution to Laplacesequation on a square domain

with Dirichlet boundary conditions using the five-point scheme. If we agree to start at the

bottom of the square and sweep up by rows, we could write, for the general point:

Alternatively, the over-relaxation parameter w can be introduced prior to solution of the

simultaneous algebraic equations. This is done by substituting the over-relaxation scheme

introduced earlier into the above equation. The result is:

The above equation is then solved for each row by Thomas algorithm. The graphic depictionof the procedure can be seen in the figure below:


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For the specified conditions of our problem,is equal to 1 and wopt is calculated to be 1.989.

The MATLAB code can be seen in the following pages:

%Heat equation solution by SLOR method

%by Ata Ghasemi Esfahani 92129071

clear all

clc

%Constants%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

omega=1.989;

a=-omega/4;

m=10;

iteration=1000;

%Initial conditions

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

fort=1:iteration

forj=1:m

fori=1:m

te(t,j,i)=0;end

end

end

%Boundary conditions

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

fort=1:iteration+1

fori=1:m+1

te(t,1,i)=300;

te(t,m+1,i)=0;

endend

Solution domain

increments


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fort=1:iteration+1

forj=2:m+1

te(t,j,1)=100;

te(t,j,m+1)=100;

end

end

te2(:,:)=te(iteration,:,:);

te2(:,:)

%formation of matrix A

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

forj=1:m

fori=1:m

if(i==j)

ma(j,i)=1;

end

if(i==j+1)

ma(j,i)=a;

end

if(i==j-1)

ma(j,i)=a;

end

end

end

fort=1:iteration

forj=2:m

fori=2:m

k(i-1)=(1-omega)*te(t,j,i-1)+te(t,j+1,i-1)+te(t+1,j-1,i-

1);

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Thomas Algorithm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

md(1,1)=1;

forj1=1:m-1

fori=1:m-1

if(j1~=1)

if(i==j1)

md(j1,i)=ma(j1,i)-(a^2)/md(j1-1,i);

end

end

if(i==j1+1)

md(j1,i)=a;

end

end

end

fori=2:9

k(i)=k(i)-k(i-1)*a/md(i-1,i-1);

endte(t,j,m)=k(m-1)/md(m-1,m-1);

fori=1:m-1

Formation of coefficients

matrix


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i=m-i;

te(t,j,i)=(k(i)-a*te(t,j,i+1))/md(i,i);

end

end

end

te1(:,:)=te(iteration,:,:);

4. Results and discussion

4.1 ADI method

The results of the numerical procedure are presented in this section. Temperature

distributions in the domain are presented so that the reader is able to compare the

consequences of introducing various changes to the problem. In what follows ndenotes the

time level.

Fig. 1 Temperature distribution, ADI method,x= 0.1, n = 5,t = 0.1


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300 300 300 300 300 300 300 300 300 300 300

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

100 0 0 0 0 0 0 0 0 0 100

Fig. 9 Temperature distribution, ADI method,x= 0.1, n = 1,t = 0.1, solution matrix


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Columns 1 through 7

300.0000 300.0000 300.0000 300.0000 300.0000 300.0000 300.0000

100.0000 274.8239 297.8872 299.8227 299.9850 299.9975 299.9850

100.0000 49.8107 49.6214 49.4322 49.2429 49.0536 48.8643

100.0000 49.8107 49.6214 49.4322 49.2429 49.0536 48.8643

100.0000 49.8107 49.6214 49.4322 49.2429 49.0536 48.8643

100.0000 49.8107 49.6214 49.4322 49.2429 49.0536 48.8643

100.0000 49.8107 49.6214 49.4322 49.2429 49.0536 48.8643

100.0000 49.8107 49.6214 49.4322 49.2429 49.0536 48.8643

100.0000 49.8107 49.6214 49.4322 49.2429 49.0538 48.8666

100.0000 49.8107 49.6214 49.4322 49.2429 49.0538 48.8666

100.0000 0 0 0 0 0 0

Columns 8 through 11

300.0000 300.0000 300.0000 300.0000

299.8227 198.0254 274.8239 100.0000

48.6750 102.3387 49.2429 100.0000

48.6750 58.3989 49.2429 100.0000

48.6750 49.9288 49.2429 100.0000

48.6750 48.8138 49.2429 100.0000

48.6750 48.6585 49.2429 100.0000

48.6750 48.3911 49.2429 100.0000

48.7019 46.3765 49.2695 100.0000

48.7019 34.8840 49.2695 100.0000

0 0 0 100.0000



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Columns 1 through 7

300.0000 300.0000 300.0000 300.0000 300.0000 300.0000 300.0000

100.0000 292.8531 299.8297 299.9959 299.9999 300.0000 299.9999

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 49.9931 49.9863 49.9794 49.9726 49.9657 49.9588

100.0000 0 0 0 0 0 0


300.0000 300.0000 300.0000 300.0000

299.9959 264.7318 292.8531 100.0000

49.9520 70.9772 49.9726 100.0000

49.9520 50.9738 49.9726 100.0000

49.9520 49.9887 49.9726 100.0000

49.9520 49.9531 49.9726 100.0000

49.9520 49.9518 49.9726 100.0000

49.9520 49.9442 49.9726 100.0000

49.9520 49.7358 49.9726 100.0000

49.9520 45.4135 49.9726 100.0000

0 0 0 100.0000


Columns 1 through 7

300.0000 300.0000 300.0000 300.0000 300.0000 300.0000 300.0000


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100.0000 219.6133 278.4530 294.1989 298.3425 299.1713 298.3425

100.0000 48.7815 47.5629 46.3444 45.1258 43.9073 42.6887

100.0000 48.7815 47.5629 46.3444 45.1258 43.9073 42.6887

100.0000 48.7815 47.5629 46.3444 45.1258 43.9073 42.6887

100.0000 48.7815 47.5629 46.3444 45.1258 43.9073 42.6887

100.0000 48.7815 47.5629 46.3444 45.1258 43.9073 42.6887

100.0000 48.7815 47.5629 46.3444 45.1258 43.9073 42.6887

100.0000 48.7821 47.5655 46.3540 45.1619 44.0419 43.1912

100.0000 48.7821 47.5655 46.3540 45.1619 44.0419 43.1912

100.0000 0 0 0 0 0 0


300.0000 300.0000 300.0000 300.0000

294.1989 132.3841 219.6133 100.0000

41.4702 99.9029 45.1258 100.0000

41.4702 84.3623 45.1258 100.0000

41.4702 60.1294 45.1258 100.0000

41.4702 47.9626 45.1258 100.0000

41.4702 42.2473 45.1258 100.0000

41.4702 37.2052 45.1258 100.0000

43.3455 28.5179 46.8756 100.0000

43.3455 16.2208 46.8756 100.0000

0 0 0 100.0000



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Fig. 14 Temperature distribution, SLOR method,x= 0.0025, iteration level = 1000, w=1.989



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4.3 Discussion

Although both methods are stable, careful examination of the ADI method results indicates

that:

1. Increasing the time steps would speed up the convergence to final state

2. Increasing the size of the grid prevents us from seeing the accurate temperature

distributions that we would see otherwise.

3. With the same grid size and time and same time level, bigger time steps yield different

results.

4. The code is biased from left to right, in other words the boundary conditions from the left

side of the domain propagate through the domain faster than the boundary conditions on the

right.

5. The computation time does notincrease significantly with time steps that are in the order

of convergence time step.

It must be mentioned that the time step nwas changed manually to observe temperature

variations but the code converged at 976 steps.

The SLOR method, on the other hand, presented physically unacceptable results. It can be

observed that:

1. At the same iteration level, using smaller mesh net results in the diminishment of the

region that has a higher temperature than that of the boundaries.

2. Although the optimum relaxation factor calculated according to [1] is 1.989, reducing the

relaxation factor results in more acceptable results in terms of the physical reality of the

problem.

3. Increasing the iteration levels exponentially increases the computation time, although

even at 1000 iterations, the code has not converged yet.

Therefore, by comparing the results obtained from ADI and SLOR methods it is found that

the ADI method, although a little bit harder to code,

1. Yields realistic results

2. Significantly reduces the computation time

3. Converges much more quickly


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5. References

[1] Computational Fluid Mechanics and Heat Transfer, John C. Tannehil, Dale A. Anderson,

Richard H. Pletcher, Seond Edition, Taylor and Francis, 1997

[2] Computational Fluid Dynamics: The Basics with Applications, John D. Anderson,

McGraw-Hill, 1995

numerical solution of 2d heat equation

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