Stabilization of explicit methods forconvection di¤usion equations by discrete

molli�cation ?

Carlos D. Acosta

Universidad Nacional de Colombia

Dept. of Mathematics and Statistics

Manizales Colombia

Carlos E. Mejía

Universidad Nacional de Colombia

Dept. of Mathematics

Medellín Colombia

Abstract

The main goal of this paper is to show that discrete molli�cation is a simple ande¤ective way to speed up explicit time-stepping schemes for partial di¤erential equa-tions. The second objective is to enhance the molli�cation method with a variety ofalternatives for the treatment of boundary conditions. The numerical experimentsindicate that stabilization by molli�cation is a technique that works well for a varietyof explicit schemes applied to linear and nonlinear di¤erential equations.

Key words: Convection-Di¤usion, Wave Propagation, Discrete Molli�cation,Stability Analysis

1 Introduction

Since the 80�s, the molli�cation method has been used by a considerable num-ber of authors as a regularization method for ill-posed problems ([9], [6], [8]).

? This work has been partially supported by COLCIENCIAS, project number1118-11-16705 and DIME, project number 30802867.Email addresses: cdacostam@unal.edu.co (Carlos D. Acosta),

cemejia@unalmed.edu.co (Carlos E. Mejía).

Preprint submitted to Elsevier Science 22 May 2006

More recently, in [10] and [7], the method was introduced as a stabilizer of theforward-time central-space explicit scheme for linear parabolic equations. Bystabilizer we mean a technique that provides a way to increase the stabilitybound of the explicit method.

The main goal of this paper is to show that discrete molli�cation is a sim-ple and e¤ective way to speed up explicit time-stepping schemes for partialdi¤erential equations. To ful�ll this objective, we concentrate on convection-di¤usion and advection equations. For the convection-di¤usion equations, wepresent a molli�ed forward-time central-space method. We establish stabilitybounds for linear equations and show through encouraging numerical experi-ments, that the same bounds are valid for some nonlinear cases.

The second goal of this paper is to enhance the molli�cation method with up-dated convergence results and a wide variety of alternatives for the treatmentof boundary conditions. This is a delicate matter when dealing with convolu-tions; our approach is inspired by the methods of digital image processing.

The outline of this paper is as follows: We begin with the de�nition and mainproperties of molli�cation in section 2. The study of boundary conditionsappears in section 2.3, which is followed by the results on stabilization insection 3. The last section presents illustrative numerical experiments.

2 Molli�cation

The molli�cation method is a �ltering procedure, based on convolution, thatis appropriate for the regularization of a variety of ill-posed problems, namely,inverse heat conduction problems, numerical di¤erentiation, coe¢ cient iden-ti�cation, problems related to digital signal processing, etc.

When molli�cation is applied to digital signal processing, it is convenient tohave the possibility of working with di¤erent molli�cation kernels. However,in this paper, we restrict our attention to a Gaussian kernel. For an overviewof the many applications of this method, we recommend [9], [10] and thereferences therein.

2

2.1 Abstract Setting

Let � > 0; p > 0 and

Ap� =

0B@ p=�Z�p=�

exp(�s2)ds

1CA�1

:

We work with the following truncated Gaussian kernel:

��p(t) =

8><>:Ap���1 exp(�t2=�2); jtj � p

0; jtj > p::

This kernel satis�es: ��p � 0; ��p 2 C1(�p; p); ��p is zero outside [�p; p] andRR ��p = 1:

Given f : R! R locally integrable, we de�ne its �p�mollification, denotedJ�pf; as the convolution of f with the kernel ��p: That is,

J�pf(t) = (��p � f) (t) (1)

=

1Z�1

��p(t� s)f(s)ds

=

t+pZt�p

��p(t� s)f(s)ds

=

pZ�p

��p(�s)f(t+ s)ds:

Roughly speaking, the parameters � and p are related to shape and supportof the molli�cation kernel respectively. Unless otherwise stated, we assumep = 3� for molli�cation in the abstract setting. Thus

Ap� =

0B@ p=�Z�p=�

exp(�s2)ds

1CA�1

=

0@ 3Z�3

exp(�s2)ds1A�1 (2)

is independent of �:

If f is de�ned on a bounded set Y; the computation of the convolution J�pfrequires either an extension of f to points out of Y or the restriction of f toa proper subset of Y: A useful approach for closed intervals was presented in[6]. Furthermore, in the last paragraph of section 5 of [10], Murio says: "Ingeneral, if the initial and/or boundary conditions are known, they should be

3

incorporated to the code to improve accuracy". Consequently, in this paperwe deal with several data extension procedures, based on the ideas presentedin [11] and [2]. We also include a scaling technique as an alternative for com-putations near the boundary. Following [11], we refer to all the extensions andto the scaling procedure as boundary conditions. The details on this topic arepresented in section 2.3.

2.2 Discrete Molli�cation

De�nition 1 Let X = fxj : xj = x0 + jh; j 2 Zg be a discrete domain withx0 and h given real numbers and h > 0: Let G : X ! R be a function de�nedby G(xj) = yj: Set

Sj = (xj�1 + xj) =2; j 2 Z

Ij = [Sj; Sj+1) ; j 2 Z

f(t) =1X

j=�1yj�j(t); t 2 R;

(3)

with �j the characteristic function of Ij: Then for � > 0 and � a givennon-negative integer, we de�ne the �� � mollification of G as the �p �mollification of f with

p = (� + 1=2)h; (4)

that is,

J��G(x) = J�pf(x):

We are particularly interested in the value of J��G at the points in X: Let

tj = (j � 1=2)h; j 2 Z: (5)

Then we can write

J��G(xj) = J�pf(xj)

=

pZ�p

��p(�s)f(xj + s)ds

=�X

i=��

ti+1Zti

��p(�s)f(xj + s)ds:

Furthermore,

ti < s < ti+1 if and only if Si+j < xj + s < Si+j+1:

4

Thus,

J��G(xj) =�X

i=��wiyj+i; where wi =

ti+1Zti

��p(�s)ds; (6)

that is, the discrete molli�cation of G is the discrete convolution of the vectory with a kernel vector w of weights obtained from the kernel k�p. Notice thatw satis�es

�Xi=��

wi = 1:

For discrete molli�cation, the integer � is the parameter related to the supportof the discrete molli�cation kernel.

Theorem 1 (Convergence) Let g be a function de�ned on R with fourthderivative g(4) continuous and bounded in R: Let G be its discrete versionde�ned on X: If G" is another discrete function de�ned on X and such that

jG"(xj)�G(xj)j � "; for xj 2 X;

then, there exists a constant C such that

jJ��G"(xj)� J��G(xj)j � ";jJ��G(xj)� g(xj)j � Ch2:

Furthermore, if g is smooth enough, there exists a constant C such that

jD+J��G(xj)� g0(xj)j � Ch; (7)jD0J��G(xj)� g0(xj)j � Ch2;

jD�D+J��G(xj)� g00(xj)j � Ch2;

where D+; D� and D0 are the forward, backward and central �nite di¤erenceoperators respectively.

PROOF. (Stability) In fact,

jJ��G"(xj)� J��G(xj)j =�������X

i=��wi (G

"(xj+i)�G(xj+i))������

��X

i=��wi jG"(xj+i)�G(xj+i)j � ":

(Consistency) From Taylor´s theorem

g(xj+i) = g(xj) + (ih) g0(xj) +

(ih)2

2g00(xj) +

(ih)3

6g000(xj) +O(h

4):

5

Then

J��G(xj) =�X

i=��wig(xj+i) (8)

=�X

i=��wi

g(xj) + (ih) g

0(xj) +(ih)2

2g00(xj) +

(ih)3

6g000(xj)

!+O(h4)

= g(xj) +h2

2g00(xj)

�Xi=��

i2wi +O(h4):

Because, from symmetry in the kernel

�Xi=��

wi (ih) g0(xj) = hg

0(xj)�X

i=��iwi

= hg0(xj)�Xi=1

i (wi � w�i)

= 0:

And similarly,�X

i=��wi(ih)3

6g000(xj) = 0:

Hence,J��G(xj) = g(xj) +O

�h2�:

(Numerical Derivatives) From (8),

J��G(xj+1)� J��G(xj�1)2h

=g(xj+1)� g(xj�1)

2h

+h2

2

g00(xj+1)� g00(xj�1)2h

�Xi=��

i2wi +O(h4); (9)

J��G(xj+1)� J��G(xj�1)2h

= g0(xj) +h2

2g000(xj)

0@13+

�Xi=��

i2wi

1A+O(h4):The other bounds in (7) are obtained in the same way.

2.3 Boundary Conditions

In this section we present the numerical procedure for the computation ofthe ��� discrete molli�cation of a data vector y 2 Rm: This molli�cation isdenoted y��:

The procedure makes use of extensions of the data outside the domain and,in this sense, follows the ideas in [10] and other references on molli�cation,

6

in which two constant extensions of the data are computed by solving twooptimization problems. However, we take a di¤erent approach, based on thetechniques for image reconstruction and digital signal processing described in[11], Section 7 of [2] and Chapter 24 of [12].

Let us assume, following De�nition 1, that

yj = G (xj) ;

where a = x1 < x2 < � � � < xm = b and xj � xj�1 = h > 0 for j = 2; :::m: By(6),

[y��]j = J��G(xj) =�X

i=��wiyj+i; where wi =

ti+1Zti

��p(�s)ds

and the t0is are given by (5), that is,

ti = (i� 1=2)h; i2 Z:

This yieldsy�� = Tlyl + Ty + Tryr; (10)

where

Tl =

26666666666664

w�� � � � w�1. . .

...

w��

0

37777777777775m��

; yl =

26666666666664

y��+1

y��+2...

y�1

y0

37777777777775��1

;

T =

26666666666664

w0 � � � w� 0.... . . . . .

w��. . . w�

. . . . . ....

0 w�� � � � w0

37777777777775m�m

; y =

26666666666664

y1

y2...

ym�1

ym

37777777777775m�1

; (11)

Tr =

26666666666664

0

w�.... . .

w1 � � � w�

37777777777775m��

; yr =

26666666666664

ym+1

ym+2...

ym+��1

ym+�

37777777777775��1

:

7

The vectors yl and yr are our concern now. How do we de�ne them? Someadditional hypotheses are in order. They are called Boundary Conditions andwe describe them according to [11].

2.3.1 The zero (Dirichlet) boundary condition

In this case, yl and yr are null and y�� is computed from y by the Toeplitzmatrix T . More precisely,

y�� = Ty; (12)where T and y are given by (11).

This boundary condition is useful when it is known that y decays rapidlyto zero outside the domain [a; b] : Of course, it becomes a liability if y doesnot have this property, as is clearly stated in [2] and [12] for digital imageprocessing.

2.3.2 Scaled boundary condition

In this case, yl and yr are null and y�� is computed from y by a scaled versionT�� of matrix T given by

T�� = D�1T;

where D = diag [d1; d2; :::; dm] and dk is the sum of the elements in the kthrow of T: Thus,

y�� = T��y: (13)This scaling is recommended when there is no clue on the behavior of thefunction G outside the domain.

2.3.3 The periodic boundary condition

In this case,

yl =

26666666666664

y��+1

y��+2...

y�1

y0

37777777777775=

26666666666664

ym��+1

ym��+2...

ym�1

ym

37777777777775; yr =

26666666666664

ym+1

ym+2...

ym+��1

ym+�

37777777777775=

26666666666664

y1

y2...

y��1

y�

37777777777775and it is possible to compute discrete molli�cation in a straightforward way.The associated linear operator is the circulant matrix

C�� =h0m�(m��)jTl

i+ T +

hTrj0m�(m��)

i

8

andy�� = C��y:

Since C�� is circulant, it can be diagonalized by the discrete Fourier matrix.This is an important advantage of using periodic boundary conditions.

Another common approach is to extend the data by re�ection of the informa-tion in the domain. This re�ection can be even or odd. In the even case, themolli�cation operator is a Toeplitz plus Hankel matrix. Section 3 of [11] is anappropriate reference at this point.

2.4 Parameter Selection

When working with molli�cation as a regularization procedure, the parameter� may be automatically selected by GCV, Generalized Cross Validation ([5]),and the integer � is obtained from � as

� =

$3�

h� 12

%;

which is consistent with equation (4). In this paper we establish that, evenwhen boundary conditions are assumed, discrete molli�cation is de�ned by alinear operator. Therefore GCV can be applied in all cases. For the applicationof GCV to molli�cation, we recommend [10].

However, for stabilization, it is convenient to keep � as the main parameter.This means, we know in advance the amount of points to take into account assupport for the discrete molli�cation kernel. Given �; the value of � is foundfrom (4) as

� =

�� + 1

2

�h

3:

3 Stabilization

Explicit methods for partial di¤erential equations are subject to restrictionson the time step. By stabilization of an explicit scheme, we mean a procedurethat speeds up computations by allowing greater time steps. Additionally, itis desirable that the method prevents the appearance of spurious oscillations.

To comply with the �rst task, we show that our stabilized schemes allowgreater time steps than the established by regular stability restrictions of theCFL type, this is obtained by using the next theorem 2. For references on the

9

subject, we recommend [13], [1], [4], [7] and [10]. The second task is relevantwhen dealing with non-linear equations like viscous Burgers�equation.

Our approach is stabilization via discrete molli�cation applied to explicitschemes for convection-di¤usion. Our results generalize those in [7] and section5 of [10].

Theorem 2 Let

vn+1m =�+1X

j=���1Wjv

nm+j;

be a time-stepping numerical scheme for solving the convection-di¤usion equa-tion

ut + aux = buxx;

on a equally spaced discrete domain

x1 < x2 < � � � < xN :

If the entries�modulus of the Fourier transform of the RN vector�W0 W1 � � � W�+1 0 � � � 0 W���1 � � � W�1

�(14)

are all less than or equal to one, then the scheme is stable under periodicboundary conditions.

PROOF. Under periodic boundary conditions, the numerical solution at tn+1can be computed from the solution at tn; by multiplying the latter by thecirculant matrix with �rst row (14). But, the eigenvalues of such a matrix arethose entries in the Fourier transform of (14). Then the spectral radius of theiteration matrix is the in�nity norm of the Fourier transform of (14). So, fromthe hypothesis the iteration matrix has spectral radius less than or equal toone, thus the scheme is stable.

This simple result gives a su¢ cient condition for stability. Furthermore, thecondition of the theorem is very easy to check.

3.1 Convection-di¤usion equation

Consider the Convection-Di¤usion Equation

ut + aux = buxx (15)

10

and the forward-time central-space �nite di¤erence scheme (FTCS) for thisequation,

vn+1m � vnmk

+ avnm+1 � vnm�1

2h= b

vnm+1 � 2vnm + vnm�1h2

: (16)

Here, vnm is the discrete approximation to the value of u at the nodal point(xm; tn); h and k are the uniformmesh sizes for the space and the time variablesx and t respectively. Notice that (16) is equivalent to

vn+1m = vnm + b�(1 + �)vnm�1 � 2b�vnm + b�(1� �)vnm+1; (17)

where

� =k

h2; � =

k

hand � =

ha

2b=a�

2b�:

The stability bounds for this scheme are

b� � 1=2; � � 1:

Let znm = J��vn (xm) be the �� �mollification (in space) of vn evaluated at

xm: We consider two molli�ed versions for (17), the �rst one

vn+1m = vnm + b�(1 + �)znm�1 � 2b�znm + b�(1� �)znm+1; (18)

results from replacing the data coming from spatial discrete di¤erentiation fortheir molli�ed versions. The second molli�ed scheme is obtained from (17) byreplacing the right hand side terms by their molli�ed versions. This yields

vn+1m = znm + b�(1 + �)znm�1 � 2b�znm + b�(1� �)znm+1: (19)

To analyze the convergence of (18), we rewrite it in terms of the de�nition ofdiscrete molli�cation. We begin by setting wj = 0 for jjj > �; so that

znm =�X

j=��wjv

nm+j =

�+1Xj=���1

wjvnm+j;

znm�1 =�X

j=��wjv

nm�1+j =

�+1Xj=���1

wj+1vnm+j;

znm+1 =�X

j=��wjv

nm+1+j =

�+1Xj=���1

wj�1vnm+j:

This yields

vn+1m = vnm + b��+1X

j=���1f(1 + �)wj+1 � 2wj + (1� �)wj�1g vnm+j;

11

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

α

Max

imum

bµ

η = 6η = 5η = 4η = 3η = 2η = 1η = 0

Fig. 1. CFL Condition for the molli�ed FTCS Scheme (18)

or equivalently

vn+1m =�+1X

j=���1[�0;j + b� f(1 + �)wj+1 � 2wj + (1� �)wj�1g] vnm+j; (20)

where �0;j represents the Kronecker´s delta. Similarly, we can write (19) in theform

vn+1m =�+1X

j=���1[wj + b� f(1 + �)wj+1 � 2wj + (1� �)wj�1g] vnm+j: (21)

3.1.1 Stability

Theorem 2 can be applied to schemes (20) and (21). Figure 1 shows the max-imum b� allowed for the scheme (20).

From this �gure some additional remarks have to be made:

1. For values of � close to 1; there is no essential improvement in the stabilityof the original scheme.

2. For values of � close or equal to 0; the stability bound is greater than 12

whenever molli�cation is used (� > 0). As a consequence, when solving theconvection-di¤usion equation with � close to 1; one can increase the spaceresolution by reducing h without having to select a very small k:

3. The case � = 0 corresponds to the heat equation, for which stabilization

12

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

α

Max

imum

bµ

η = 7η = 6η = 5η = 4η = 3η = 2η = 0

Fig. 2. CFL Condition for the molli�ed FTCS Scheme (19)

by molli�cation was considered by Murio in [10]. There is agreement betweenour results and Murio�s.

The CFL condition for the molli�ed scheme (21) is shown in �gure 2. Themain feature here is stabilization for � close to 1, which is missing from theprevious scheme.

3.1.2 Consistency

For consistency with the problem

ut + aux = buxx;

we rewrite (18) and (19) as

vn+1m � vnmk

+ aD0znm = bD2z

nm; (22)

vn+1m � znmk

+ aD0znm = bD2z

nm; (23)

13

respectively, and note that

ut �vn+1m � vnm

k= O (k)

ut �vn+1m � znm

k= O(k + h

2

k)

ux(xm; tn)�D0znm = O

�h2�

uxx(xm; tn)�D2znm = O

�h2�

From here we obtain consistency of the schemes as k; h! 0 as long as h2=k !0 as well.

4 Numerical Examples

In this section we illustrate the stabilization property of the molli�ed explicitschemes by testing them with a variety of examples, based on the convection-di¤usion equation

ut + aux = buxx: (24)

Some of the examples are illustrations of the theory presented above andthe others are preliminary encouraging experiments that show the potentialof discrete molli�cation as stabilizer of explicit marching schemes for partialdi¤erential equations. All of the examples were implemented using MATLAB7.0 SP1 under Windows XP SP2 on a PC with an Intel Pentium 4 processorof 3.02 GHz and 512Mb of RAM .

Example 1 (Di¤usion-dominated) Molli�ed Forward-time Central-spacescheme (20) for the homogeneous heat equation

ut = uxx; 0 < x < 1; 0 < t

u (x; 0) = sin (�x) ; 0 < x < 1;

u (0; t) = u (1; t) = 0; 0 < t

The exact solution is known. For the table we use h = 1=64; the Neumann(odd extension) boundary condition and the maximum � allowed. The error

14

norms are computed for all x and a set of t-values on the discrete grid.

� b� Inf Error L2 Error T ime (seg)

0 0:5 1:4563e� 4 2:3612e� 4 1:532

1 0:6475 2:5998e� 3 7:3468e� 4 1:234

2 1:48 6:4994e� 3 1:8757e� 3 0:547

3 2:3125 8:3427e� 3 2:5916e� 3 0:344

4 3:4225 8:6844e� 3 3:1920e� 3 0:234

5 4:68 8:7602e� 3 3:8941e� 3 0:172

6 6:2125 7:9251e� 3 4:8629e� 3 0:125

Example 2 Molli�ed Forward-time Central-space scheme for the convection-di¤usion equation (24) with

u(x; t) =1p1 + t

exp

�(x� a(1 + t))

2

4b(1 + t)

!:

This example is presented at [3] as a test problem for studying the e¤ect ofdissipation and dispersion in two particular schemes. Here we use this exampleto compare schemes (20) and (21). Actually, this problem requires to choosea proper value for the parameter b� inside the stability region in order to getgood accuracy. The original scheme FTCS (17) yields acceptable results forb� = 0:125: Scheme (20) provides good results for b� = 0:25 with � � 0:5but for a greater � it is required a smaller b�: In contrast, the scheme (21) iscapable of solving the problem with good accuracy and using b� values greaterthan 0:5: So, only in this case we can talk of an e¤ective stabilization. Thefollowing tables show the maximum absolute error (") at t = 2 for scheme(21) with several values of �; b� and �: The experiments were run with x 2[�1; 4] ; t 2 [0; 2] ; a = 1; h = 1=64 and b = ha��1=2: For other values of hthe same value of b� will perform well. Figure 3 illustrates this example for

15

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.2

0.4

0.6

0.8

1

x

u

Initial ProfileExactComputed

Fig. 3. Convection-Dominated Example at t = 2

� = 0:5; � = 6 and b� = 1:5.

� (21); � = 4 (21); � = 6 (21); � = 8

0:25" = 9:4212e� 003

b� = 2:25

" = 1:1833e� 002

b� = 3:0

" = 1:2065e� 002

b� = 4:0

0:5" = 1:0714e� 002

b� = 1:125

" = 1:2710e� 002

b� = 1:5

" = 7:1431e� 003

b� = 2:0

0:75" = 1:0637e� 002

b� = 0:75

" = 1:3850e� 002

b� = 1:0

" = 1:7370e� 002

b� = 1:375

1:0" = 1:2062e� 002

b� = 0:5625

" = 2:0333e� 002

b� = 0:75

" = 1:9226e� 002

b� = 1:0

Example 3 (Viscous Burger�s equation) Molli�ed Forward-time Central-space scheme for the nonlinear equation

ut +1

2

�u2�x= buxx

with

u (x; t) = a� c tanh�c

2b(x� at)

�: (25)

16

-1 -0.5 0 0.5 1 1.5 2

1

1.2

1.4

1.6

1.8

2

x

u

ExactComputed

Fig. 4. Molli�ed FTCS (20)

-1 -0.5 0 0.5 1 1.5 2

1

1.2

1.4

1.6

1.8

2

x

u

ExactComputed

Fig. 5. Molli�ed FTCS (21)

In this case, the two di¤erent implementations (20) and (21) stabilize but the�rst one does not prevent oscillations. Figures 4 and 5 illustrate this behaviorat t = 1:0 for a = 1:5; c = 0:5; h = 1=64; b = h(a + c); � = 4 and b� = 1:9.This selection implies (� = 0:5) :It is a matter of further study that scheme (21) satis�es the conditions of anessentially non oscillatory (ENO) scheme. Our results on this matter will bepresented elsewhere.

Finally, some remarks should be pointed out. The �rst example illustrates howstabilization can reduce the computing time by relaxing the stability bound.

17

The second example shows that scheme (21) is more desirable than scheme(20) for advection dominated problems. The third example shows that thestabilization procedure presented here can also be successfully implementedfor non-linear problems.

References

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