VON NEUMANN STABILITY ANALYSIS FOR TIME- DEPENDENT DIFFUSION EQUATION by IRE Journals

Von Neumann Stability Analysis for Time- Dependent Diffusion Equation 1, 2

KHAING KHAING SOE WAI1, SAN SAN TINT2 Department of Engineering Mathematics, Technological University(Myitkyina)

Abstract -- In this paper, we describe two different finite difference schemes for solving the time fractional diffusion equation. And, we study the method of lines discretizations. Then, we use to check the stability of finite difference schemes by using Von Neumann analysis. Indexed Terms: Diffusion equation, explicit method, Crank-Nicolson method, Von-Neumann analysis

We have already studied the steady state version of

this equation and spatial discretizations of xx . We have also studied discretizations of the time derivatives. In practice we generally apply a set of finite difference equations on a discrete grid with grid

x  ih tn  nk (x , t ) points i n where i , . Here h  x is the mesh spacing on the x-axis and

INTRODUCTION

n k  t is the time step. Let Ui  u( xi , tn ) represent

We study finite difference methods for timedependent partial differential equations, where variations in space are related to variations in time. The heat equation (or diffusion equation), ut  k u xx . (1)

the numerical approximation at grid point

This is the classical example of a parabolic equation, and many of the general properties seen here carry over to the design of numerical methods for other

forward in time, determining the value

parabolic equations. We will assume k  1 for simplicity but some comments will be made about how the results scale to other values of k  0 . Along with this equation we need initial conditions at some time

, which we typically take to be

u ( x, 0)  ( x)

t0  0

(2)

and also boundary conditions if we are working on a bounded domain, for example, the Dirichlet conditions u (0, t )  g0 (t ) for t  0 for t  0 if 0  x  1.

u (1, t )  g1 (t )

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( 3)

( xi , tn )

Since the heat equation is an evolution equation that can be solved forward in time, we set up our difference equations in a form where we can march

U in 1 for all i

from the values U i at the previous time level, or perhaps using also values at earlier time levels with a multistep formula. As an example, one natural discretizations of Equation ( 1) would be U in 1  U in 1  2 (U in1  2U in  U in1 ) k h . (4) This uses our standard centered difference in space and a forward difference in time. This is an explicit n 1 method since we can compute each U i explicity in terms of the previous data k U in 1  U in  2 (U in1  2U in  U in1 ) h .

(5)

This is a one-step method in time, which is also called a two-level method in the context of partial differential equations since it involves the solution at two different time levels.

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© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 Another one-step method, which is much more useful in practice as we will see below, is the CrankNicolson method, U in 1  U in 1 2 n  ( D U i  D 2U in 1 ) 2 k (6)

1 (U in1  2U in  U in1  U in11  2U in 1  U in11 ) 2h 2



For example, we will discredited the heat Equation (1)

U i(t )  for

space

1 (U i 1 (t )  2U i (t )  U i 1 (t )) h2 ,

k (U in1  2U in  U in1  U in11  2U in 1  U in11 ) 2h 2

If i  1 , then

U1(t ) 



(7) or

rU

 (1  2r )Uin1  rUin11  rUin1  (1  2r )Uin  rUin1

, For i =1,….,m.

(8)

k r 2 2 h . This is an implicit method and gives where

a tridiagonal system of equations to solve for all the

U m (t ) 

1 (U m 1 (t )  2U m (t )  U m 1 (t )) h2





efficient per time step as an explicit method. The heat equation is “stiff”, and hence this implicit method, which allows much larger time steps to be taken than an explicit method, is a very efficient method for the heat equation. METHOD OF LINES DISCRETIZATIONS

The time-dependent partial differential equations relates to the stability theory we have already developed for time-dependent ordinary differential equations, it is easiest to first consider the so-called Method of lines discretization of the partial differential equation.

1  g0 (t )  2U1 (t )  U 2 (t ) h2 .

1 (U 2 (t )  2U 3 (t )  U 4 (t )) h2 .

u (0, t )  g0 (t ) u (1, t )  g1 (t ) , come into

Since a tridiagonal system of m equations can be solved with O( m) work, this method is essentially as

1 (U 0 (t )  2U1 (t )  U 2 (t )) h2

U 3 (t ) 

boundary

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(9)

1 U1 (t )  2U 2 (t )  U 3 (t )  h2 .

If i  2 , then

conditions these equations.

II.

U 2 (t ) 

If i  2 , then

If i  m , then

U in 1 simultaneously. The values

i  1, 2,  , m .

U in 1  U in

n 1 i 1

point

where prime now means differentiation with respect to time.

which can be rewritten as for i =1,….,m.



grid

1 (U m 1 (t )  2U m (t )  g1 (t )) h2 .

We can find this as a coupled system of m ordinary differential equations for the variables U i (t ) , which vary continuously in time along the lines . This

 system can be written as, U (t )  AU (t )  g (t ) .

g (t )

(10) where the tridiagonal matrix A and includes the terms needed for the boundary conditions,  2 1   1 2 1     1 2 1 1  A 2      h    1 2 1    1 2  

U 0 (t )  g 0 (t )

and

U m1 (t )  g1 (t )

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The eigenvector u components

u jp for

corresponding to

(11)

j th component of the vector Au p is 1 ( Au p ) j  2 (u jp1  2u jp  u jp1 ) h

The

DEFINITION

equations of the form A U  g where h is the mesh h

( A h ) 1

width. We say that the method is stable if h  h0 exists for all h sufficiently small (for , say) and if there is a constant C, independent of h, such that fo all

h  h0

(12)



1  sin( p( j  1)h)  2sin( pjh)  sin( p( j  1)h) h2



1  sin( pjh) cos( ph)  cos( pjh) sin( ph) h2

2sin( pjh)  sin( pjh) cos( ph)  cos( pjh)sin( ph) 

1  sin( pjh) cos( ph)  2sin( pjh)  sin( pjh) cos( ph) h2

  p u jp . For j  1 and

j  m , the j th

looks slightly different (the STABILITY IN THE 2 –Norm

If the matrix A from Equation (11 ) is symmetric, the 2-norm of A is equal to its spectral radius is

 ( A)  max  p ,  p is the p th eigen 1 p  m

A 1

value of the matrix. The matrix symmetric and the eigen values of the inverses of the eigen values of A.

A1

So,



1 p  m

1

is also

are simply

 ( A1 )  max ( p )1

 min  p



( 14 )

We can be verified by checking that Au p   p u p .

Suppose a finite difference method for a linear boundary value problem gives a sequence of matrix

IV.

 p has

j  1, 2,  , m given by

u jp  sin( pjh)

III.

( Ah )1  C

component of

Au p

u jp1 or u jp1 term is

missing) but that the above form and trigonometric manipulations are still valid provided that we define

u0p  ump1  0 , as is consistent with Equation ( 14 ). From Equation (13 ) we see that the smallest eigenvalue of A (in magnitude) is 1  22 (cos(h)  1) h 2 1 2 2 1 4 4 2 2   2    h   h  O(h6 )    O(h ) .   h 2 24

1 p  m

This is clearly bounded away from zero as h  0 , and so we see that the method is stable in the 2-norm.

1

So all we need to do is compute the eigen values of A and show that they are bounded away from zero as h0. Then

the m eigen values of A are given 2 by  p  2 (cos( ph )  1) , for p  1, 2,  , m . ( 13 ) h

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VON NEUMANN ANALYSIS

The Von Neumann approach to stability analysis is based on Fourier analysis and hence is generally limited to constant coefficient linear partial differential equations. For simplicity it is usually applied to the Cauchy problem, which is the partial differential equation on all space with no boundaries,

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© AUG 2019 | IRE Journals | Volume 3 Issue 2 | ISSN: 2456-8880 in the one-dimensional case. Von Neumann analysis can also be used to study the stability of problems with periodic boundary conditions. The Cauchy problem for linear partial differential equations can be solved using Fourier transforms. The basic reason this works is that the functions e

i x

with wave number   constant are eigen functions i x

v( x )

j operator. For example, if we approximate by 1 D0 V j  (V j 1  V j 1 ) 2h , then in general the grid D V function 0 is not just a scalar multiple of V.

But for the special case of W, we obtain 1 i ( j 1) h  ei ( j 1) h ) Do Wj  2h (e 1 ih  ih ijh i ijh  ( e  e )e  sin(h )e h 2h

is simply the eigen function

evaluated at the point

 ( x ) of  x

, and for small

approximate the eigenvalue of

i i 1  sin(h )   h  h33  O(h55 )   h 6 h i  i  h 2  3   6 .

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g ()

where number.

h

we can

( 15 )

U nj 1  g () eijh

( 16 )

is the amplification factor for this wave

Inserting these expressions into Equation ( 5) give

g () eijh  eijh 

k i ( j 1) h (e  2eijh  ei ( j 1) h ) h2

k    1  2 (eih  2  eih ) eijh  h   1

functions and eigen values of the operator x found

x

,

k  i h ( e  2  e i h ) h2

k (2  2 cos(h )) h2 ,and

1 4 we see that

Then,

hence

2k (cos(h)  1) h2 .

Since 1  cos(h)  1 for any value of

Note the relation between these and the eigen

Then we expect that,

g ( )  1 

So W is an “eigen gridfunciton” of the operator i sin( h) h with eigen value .

earlier:

works on a single wave number U nj  eijh we set .

 1

We will consider the Equation ( 5). To apply Von Neumann analysis we consider how this method

g ()

i sin(h ) W j h .

i

O(h 2 3 ) .

i x

of the differential operator x , x e  i e , and hence of any constant coefficient linear differential operator. Von Neumann analysis is based W  eijh on the fact that the related grid function j is an eigen function of any standard finite difference



This corresponds with the eigenvalue

,

k  g ( )  1  h2 , for all .

g ()  1

equivalent

2k (cos(h)  1) h2 1 2k h 1  2 (2sin 2 ( )) h 2  1,

g ( )  1 

4k 2 h sin ( ) h2 2  1. 4k h 1  2 sin 2 ( ) h 2  1, So, 1  1

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REFERENCES



[1]

Evans, L.C., "Partial Differential Equations", American Mathematical Society, Providence, New York, 1998.

[2]

This is exactly the stability restriction Now we will prove that the Crank-Nicolson method is stable for all k and h can also be shown using Von Neumann analysis. Substituting Equation (15) and Equation (16) into the difference Equation ( 7) and

Griffits, D.F., "An Introduction to Matlab Version 2.2", University of Dundee, Ireland, 1996.

[3]

Jungel, A., "Numerical Methods for Partial Differential Equations I": Elliptic and Parabolic Equations, (Lecture Notes), 2001.

eijh gives the

[4]

Mathews, J.H., "Numerical Methods for Mathematics", Science and Engineering, Prentice-Hall International Education, 1992..

[5]

Smith, G.D., "Numerical Solution of Partial Differential Equations, Finite Difference Methods", 3rd Ed., Clarendon Press, Oxford, 1985.

[6]

Thomas, J.W., "Numerical Partial Differential Equations, Conservation Laws and Elliptic Equations", Springer-Verlag, New York, 1998.

r

cancelling the common factor following relation,

k 1  h2 2 .

g () eijh  eijh k  2  ei ( j 1) h  2eijh  ei ( j 1) h  g ()ei ( j 1) h 2 g ()eijh  g ()ei ( j 1) h  2h

g  1 Then,

k (e  ih  2  eih )(1  g ) 2h 2 ,

z zg z  1  (1  g ) g   1 g 2 2 2. and hence , z 1 g (1  )  1  z 2 2 , Thus,

g So,

z where

1  12 z 1  12 z ,

k  ih ( e  2  e ih ) h2



2k (cos(h)  1) h2 .Since z  0 for all

 Therefore, g  1

and the method is stable for

any k and h.

VI.

CONCLUSION

This paper has presented two finite difference methods for solving diffusion equation. Next, we proved that finite difference Crank- Nicolson method is stable by using Von Neumann analysis

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