*Corresponding author.E-mail address: fdmagalh@fe.up.pt (F. D. Magalha� es).

Chemical Engineering Science 56 (2001) 3305}3309

Shorter communication

Using wavelets for solving PDEs: an adaptivecollocation method

Paulo Cruz, AdeH lio Mendes, Ferna� o D. Magalha� es*LEPAE, Chemical Engineering Department, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal

Received 7 August 2000; received in revised form 27 November 2000; accepted 18 December 2000

1. Introduction

Most of the partial derivative equations (PDEs) thatdescribe the physical and chemical phenomena commonin chemical engineering are not susceptible of being sol-ved analytically. When the PDE's solution has a regularbehavior, essentially any implementation of the `tradi-tionala numerical methods (e.g. "nite di!erences, collo-cation) can be applied successfully in its resolution.However, singularities and steep changes often emerge inmany phenomena, like concentration and/or temper-ature fronts in "xed-bed sorption/reaction processes,shock wave formation in compressive gas #ux, etc. Suchsharp transitions in an otherwise `smootha solution aretypically moving with time along a spatial coordinate.This demands for the use of non-uniform grids or movingelements, that dynamically adapt to the changes in thesolution (Sereno, 1989). Such strategies are often basedon the knowledge of the solution itself, on empiricalcriteria or on front-tracking schemes. As a consequence,one or more adjustable parameters are introduced in theproblem and it is di$cult to de"ne how the solutionshould be accurately computed at each new added point.There is still a need for an e$cient and fully adaptive

method for solving this kind of problems. That is wherewavelets play a role. This concept was introduced inapplied mathematics and physics by the end of the 1980s(Daubechies, 1988, 1992; Mallat, 1989). For an introduc-tion to wavelets, the reader is referred to Strang (1989,1994); DeVore and Lucier (1992), Jawerth and Sweldens(1994) and Graps (1995).The term `waveleta is generally used to describe

a function that features compact support, i.e. it is non-zero only on a "nite interval. The representation of a setof time (or space)-dependent data on a wavelet basis

leads to an unique structure of information that is local-ized simultaneously on the frequency and time domains.This does not occur in a Fourier representation, wherespeci"c frequencies cannot be associated to a particulartime interval, since the basis functions have constantresolution on the entire domain * absence of compactsupport. A wavelet basis representation originates a set ofwavelet coe$cients, structured over di!erent levels ofresolution. Each coe$cient is associated to a resolutionlevel (frequency) and a point in the time (or space) do-main. The coe$cients involved in the lowest-resolutionlevel describe the low-frequency features (`smootha fea-tures) of the data, spanning over broad time (or space)intervals. At the highest level, the coe$cients are asso-ciated to highly localized high-frequency features (`de-taila features). It is then evident how this concept can beapplied to signal (sound, image, etc.) compression. Afterthe wavelet transform is applied to the data, speci"cwavelet coe$cients (associated to a given frequencyrange and a given domain region) can be appropriatelyidenti"ed and rejected, so that super#uous detail is re-moved from `uninterestinga regions. After inversion, thedata set obtained contains only essential information.But how can wavelets be applied in the resolution of

PDEs? In the case of a moving steep front, using thewavelet transform one can track its position and increasethe local resolution of the grid by adding higher resolu-tion wavelets in that region. On the other hand, theresolution level in the smoother regions can be appro-priately decreased, avoiding an unnecessarily dense grid.A number of papers have been published in the last yearsdescribing this strategy for solving PDEs (Bertoluzza,1996a, b, 1997; Vasilyev & Paolucci, 1996,1997; Jameson,1998; HolmstroK n & WaldeH n, 1998; WaldeH n, 1999;HolmstroK n, 1999; Kaibara & Gomes, 2000). Even thoughall these works are based on the same theoreticalprinciples, they di!er on practical aspects like the type ofwavelets used, the grid adaptation criteria or the strategyto compute the space derivatives. This shows that the

0009-2509/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 5 5 1 - 0

�See Bertoluzza (1996) for the detailed treatment at the interval'sboundaries.

method is not yet completely established/optimized. Inaddition, the type of language and reasoning used bymany of these authors is closely tied to the "eld of appliedmathematics, lacking a more practical and explicit (en-gineering-like) perspective. To our knowledge, the "rstattempt to bring wavelet-based methods into a chemicalengineering context was done very recently by Liu,Cameron, and Wang (2000). However, it must be notedthat these authors used a constant resolution method,which does not take full advantage of the wavelets' po-tential. Such approach is actually equivalent to centered"nite di!erences, as shown by Jameson (1993), and doesnot use an adaptive grid.The purpose of the authors in the present communica-

tion is to introduce the chemical engineering communityto the basic features of wavelet-based adaptive collo-cation methods for solving PDEs and to give an illustra-tion of its powerful capabilities.

2. Overview of interpolating wavelet theory

The following is based on Bertoluzza's formal treat-ment (1996a). A more qualitative description is given byHolmstroK n (1999).Start by considering a set of dyadic grids of the type

x��3R: x�

�"2��k, where k, j3Z. We shall use in the col-

location method the Deslaurier}Dubuc interpolatingfunction (Deslaurier & Dubuc, 1989), which is the auto-correlation of the Daubechies scaling functions (Saito& Beylkin, 1993). This function, �, veri"es the interpola-ting property �(0)"1, �(k)"0 ∀kO0, k being an inte-ger, and features compact support: it is non-zero onlywithin the interval [!M, M], M being the order of theinterpolating function.The interpolating basis functions are de"ned by trans-

lations and dilations of the mother function:�

���(x)"�(2�x!k). (1)

The ���verify the interpolation property at the dyadic

points: ���(x�

�)"�

��, where � designates the KroK necker

delta. An interpolation operator can then be de"ned as

I�f (x)"��

����

f (x��)��

�(x). (2)

Notice that the interpolation involves only M#1 near-est points, since � features compact support.Consider now the dyadic points belonging to level

j, x��. These will be present in level j#1 as x���

��, as

follows from the earlier de"nition of the dyadic grid. But

level j#1 is denser than level j and, therefore, containsadditional points: the odd points x���

����. f can be esti-

mated at these locations by interpolation from thecoarser level j:

I�f (x�������

)"��

����

f (x��)��

�(x���

����). (3)

Obviously, there is a di!erence between this interpolatedvalue and the real one, f (x���

����). This di!erence is desig-

nated as the wavelet coezcient:

d��"f (x���

����)!I�f (x���

����). (4)

The wavelet coe$cient is, therefore, a measurement of theaccuracy of the interpolator I�f in estimating f at the oddpoints of level j#1 based on the points at level j. Eachcoe$cient is associated with an wavelet function, ��

�, de-

"ned as

���"����

����. (5)

It is possible to show that any function continuous inthe interval can be described in terms of the correspond-ing wavelet coe$cients at levels above j

���:

f (x)"�����

����

f (x����

�)�����

�(x)#

����

�������

��

����

d�����(x). (6)

An essential idea that should be withdrawn from thisdiscussion is the fact that each coe$cient d�

�describes the

`irregulara behavior of the function, at a certain domainregion and resolution level, relative to interpolating fromthe lower level. A `smootha function might be solelydescribed by low-level coe$cients, i.e. a coarse repres-entation is su$cient. If, however, local irregularities exist(e.g. steep gradients), corresponding higher-level coe$-cients in the corresponding domain region cannot beneglected, so that detailed local information is includedin the wavelet representation. This is the basis for the gridadaption strategy, described next.

3. Grid adaption

The distinctive feature of the method lies on thestraightforward way how the non-uniform collocationgrid is dynamically adjusted, taking into account thebehavior of the solution in the interval. The strategy usedcomes as a natural consequence of the use of a waveletrepresentation.The grid is divided into distinct wavelet resolution

levels. The minimum and maximum levels, j���

and j���

,are prede"ned by the user. All the grid points corre-sponding to level j

���(the sparser level) are maintained

throughout the integration. Points at higher levels are

3306 P. Cruz et al. / Chemical Engineering Science 56 (2001) 3305}3309

removed or added according to a criteria based onthe magnitude of the corresponding wavelet coe$cient. Itis assumed that if the coe$cient is lower than aprede"ned threshold, then the point can be removedfrom the grid (grid reduction). Otherwise, the point mustbe maintained. In order to guarantee that the grid canaccommodate transient changes in the solution's features(say, a moving concentration front), extra adjacent pointsare added at the same and at lower levels (grid extension).The adaptation procedure is summarized in the followingalgorithm:

1. For each existing odd grid point, x�������

, compute theinterpolation from the lower level (Eq. (3)).

2. Compute the corresponding wavelet coe$cients d��

(Eq. (4)). And apply the grid reduction/extension cri-teria:� If d�

�(�, then point x���

����must be removed from

the grid.� If d�

�*�, then point x���

����is maintained in the grid.

Adjacent points at the same and lower (coarser) levelsare marked to be added to the new grid.

This is performed periodically along time integration.It must be noted that there is a correlation between thefrequency of the grid adaptation and the number ofadjacent points added when the adaptation is performed.Considering the problem of a moving front, if the gridadaptation is not done often enough, a large number ofadjacent points must be added so that the front does not`move outa of the grid in-between adaptations. Con-versely, more frequent adaptations demand less extrapoints. The algorithm can incorporate a criteria to adjustthe time interval between adaptations and the number ofadjacent points added, considering the amount of grid-point changes performed along the previous steps.Any traditional method can be used for time advance-

ment (e.g. Runge}Kutta). Calculation of the space deriva-tives can be done using centered "nite di!erences at thegrid points. Every time a point necessary for the compu-tation is absent from the grid, it is interpolated from thecoarser level. Alternatively, "nite di!erences on a non-uniform grid can be used, making it unnecessary tointerpolate extra points. An additional cost is then asso-ciated to the use of a local di!erential operator that isdi!erent for each grid point, but its form can still bestored during the time interval in-between grid adapta-tion steps, saving some computation time.

4. Numerical example

We shall exemplify the method's application using thesame problem as Liu et al. (2000): propagation of a con-centration peak along a chromatographic column with

axial dispersion and linear adsorption isotherm. Themodel's equation is

1

Pe

��y�x�

"

�y

�t#

�y

�x(7)

with initial condition

y"0, t"0 (8)

and boundary conditions

x"0,1

Pe

�y

�x"y!y

��(t), (9)

x"1,�y

�x"0, (10)

where x and t are dimensionless space and time variables,respectively.The analytical solution of the problem is known (Liu et

al., 2000). We consider a value of Pe equal to 10�, anda Dirac pulse input:

y��(t)"�(t)"�

1/¹, 0)t)¹,

0, t'¹,¹"10��. (11)

Fourth-order interpolating wavelets were used witha minimum resolution level j

���"4. In this case, grid

adaptation was performed at prede"ned time intervals of0.5% of the problem's time scale. Ten extra adjacentpoints were added to each side of the grid points selectedby the algorithm. The threshold value used was �"10��.Time integration was performed with routine LSODA(Petzold & Hindmarsh, 1997). The spatial derivativeswere computed using cubic splines on a non-uniformgrid.Fig. 1 shows the results obtained at two dimensionless

times (0.2 and 0.8). The concentration peak is showntogether with the corresponding adapted grids (showndiscretized in terms of resolution levels). It is interestingto see how the grid changes along the integration. In theinitial stages, when the peak is narrower, higher detail(denser grid) is necessary to describe it. The adaptiondoes this by allocating grid points up to the highestresolution level ( j"11) in the region where the peak islocated. As the peak approaches the end of the column, itbecomes slightly broader due to axial dispersion. Sinceless resolution is necessary, the algorithm places pointsonly up to level j"10. This dynamic adaption of the gridto the evolution of the solution's gradients is obviouslynot possible with a constant resolution grid, as used byLiu and co-workers (2000). Actually, these authors dis-cuss the accuracy of the computed solution only in termsof the maximum value of the peak at the column's outlet.They fail to recognize that a de"cient representationof the solution in the initial integration stages, due to

P. Cruz et al. / Chemical Engineering Science 56 (2001) 3305}3309 3307

Fig. 1. Results obtained for integration of Eq. (7) using an adaptive waveletcollocation method: (a) concentration peak for t"0.2; (b) collocation gridfor t"0.2; (c) concentration peak for t"0.8; (d) collocation grid for t"0.8.The parameters used in the adaption algorithm are mentioned in the text.

Fig. 2. Number of grid points used in the collocation method for twodi!erent values of the threshold parameter �.

Table 1Comparison of CPU times for the constant grid and adaptive gridalgorithms, with �"10��, for di!erent values of the grid resolutionj ( j

���in the adaptive algorithm). The computer used was an Intel

Pentium III at 700 MHz

j CPU time (s)

Const. grid Adapt. grid

9 12.1 5.110 27.7 7.111 619.7 13.0

using a grid that is too sparse for such a narrow peak,originates an error that is propagated up to the exit.Trying to achieve a compromise between grid spacingand accuracy, when using a constant uniform grid, al-ways implies having to use an excessive number of points.Fig. 2 shows the number of grid points selected by the

adaption procedure along integration, for two di!erentvalues of the threshold parameter �: 10�� and 10��. For�"10��, the maximum number of points, i.e. of equa-tions being integrated, is only slightly above 200. Obvi-ously, when � is increased, a larger number of points isneglected: for �"10�� the maximum number of gridpoints barely exceeds 125. The threshold parameter � isa direct measurement of the error involved approximat-ing the solution by a reduced set of wavelet coe$cients(HolmstroK n, 1999). Higher � implies less equations tointegrate, but lower accuracy.Table 1 shows the CPU times obtained for the con-

stant grid and adaptive grid algorithms with di!erentgrid resolutions. It is evident that the grid adaptionstrategy leads to signi"cantly lower CPU times. It isinteresting to note that, for the constant grid algorithm,there is an abrupt increase in CPU time for j"11. This isdue to Jacobian evaluations. When j"11, each Jacobiancomputation implies 4098 function evaluations. With theadaptive grid strategy, the Jacobian does not have such

3308 P. Cruz et al. / Chemical Engineering Science 56 (2001) 3305}3309

a signi"cant toll on CPU time, since the grid is muchsparser throughout the integration.

5. Conclusions

The adaptive wavelet collocation method is able todynamically track the evolution of the solution's `irregu-lara features and to allocate higher grid density to thenecessary regions. Therefore, the number of collocationpoints needed is optimized, without damaging the accu-racy of the solution. Further optimization of the algo-rithm is being worked on. This is a recent technique, stillunder study and development within di!erent areas ofmathematics, physics and engineering, that deserves at-tention from the chemical engineering community.

Acknowledgements

The work of Paulo Cruz was supported by FCT(Grant BD/21483/99).

References

Bertoluzza, S. (1996a). Adaptive wavelet collocation method for thesolution of Burgers equation. Transport Theory and Statistical Phys-ics, 25, 339}352.

Bertoluzza, S. (1996b). Adaptive wavelet collocation for nonlinearBVPs. In M.-O. Berger (Ed.), Proceedings of the ICAOS 96, LectureNotes in Control and Information Sciences. London: Springer.

Bertoluzza, S. (1997). Multiscale wavelet methods for partial di!erentialequations. In: W. Dahmen, A. Kurdila & P. Oswald (Eds.),Waveletsanalysis and its applications, vol. 6. New York: Academic Press.

Daubechies, I. (1988). Orthogonal bases of compactly supportedwavelets. Communications on Pure and Applied Mathematics, 41,225}236.

Daubechies, I. (1992). Ten Lectures on Wavelets. Philadelphia, PA:SIAM.

Deslaurier, G., & Dubuc, S. (1989). Symmetric iterative interpolationprocesses. Constructive Approximation, 23, 1015}1030.

DeVore, R., & Lucier, B. (1992). Wavelets. In A. Iserles (Ed.), ActaNumerica, vol. 92 (pp. 1}56). New York: Cambridge UniversityPress.

Graps, A. (1995). An introduction to wavelets. IEEE ComputationalSciences and Engineering, 2, 50}61.

HolmstroK n, M. (1999). Solving hyperbolic PDEs using interpolationwavelets. Journal of Scientixc Computing, 21, 405}420.

HolmstroK n, M., & WaldeH n, J. (1998). Adaptative wavelet methods forhyperbolic PDEs. Journal of Scientixc Computing, 13, 19}49.

Jameson, L. (1993). On the Daubechies-based wavelet diwerentiationmatrix. ICASE Report No. 93}95.

Jameson, L. (1998). A wavelet-optimized, very high order adaptiveorder adaptive grid and order numerical method. Journal of Scient-ixc Computing, 19, 1980}2013.

Jawerth, B., & Sweldens, W. (1994). An overview of wavelet basedmultiresolution analyses. SIAM Review, 36, 377}412.

Kaibara, M. K., & Gomes, S. M. (2000). Fully adaptive multiresolutionscheme for shock computations. In E. F. Toro (Ed.), Godunovmethods: Theory and applications. Dordrecht, New York: KluwerAcademic/Plenum Publishers.

Liu, Y., Cameron, T. F., & Wang, Y. (2000). The wavelet-collocationmethod for transient problems with steep gradients. Chemical En-gineering Science, 55, 1729}1734.

Mallat, S. (1989). Multiresolution approximation and wavelet ortho-gonal bases of ¸�(R). Transactions of the American MathematicalSociety, 315, 69}87.

Petzold, L. R., & Hindmarsh, A. C. (1997). LSODA. Computing andMathematics Research Division, Lawrence Livermore National La-boratory.

Saito, N., & Beylkin, G. (1993). Multiresolution representations usingthe auto-correlation functions of compactly supported wavelets.IEEE Transactions on Signal Processing, 41, 3584}3590.

Sereno, C. (1989). Me& todo dos elementos xnitos mo& veis* Aplicac7 oJ es emengenharia qun&mica. Ph.D. thesis, FEUP.

Strang, G. (1989). Wavelets and dilation equations: A brief introduc-tion. SIAM Review, 31, 613}627.

Strang, G. (1994). Wavelets. American Scientist, 82, 250}255.Vasilyev, O. V., & Paolucci, S. (1996). A dynamically adaptative multi-

level wavelet collocation method for solving partial di!erentialequations in a "nite domain. Journal of Computational Physics, 125,498}512.

Vasilyev, O. V., & Paolucci, S. (1997). A fast adaptive wavelet collo-cation algorithm for multidimensional PDEs. Journal of Computa-tional Physics, 125, 16}56.

WaldeH n, J. (1999). Filter bank methods for hyperbolic PDEs. Journal ofNumerical Analysis, 36, 1183}1233.

P. Cruz et al. / Chemical Engineering Science 56 (2001) 3305}3309 3309