TIME ACCURATE FAST THREE-STEP WAVELET-GALERKIN METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

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1 International Jornal of Wavelets, Mltiresoltion and Information Processing Vol. 4, No. (26) c World Scientific Pblishing Company TIME ACCURATE FAST THREE-STEP WAVELET-GALERKIN METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS MANI MEHRA andb.v.rathishkumar Department of Mathematics, Indian Institte of Technology, Kanpr, U.P., 286, India manimeh2@yahoo.co.in bvrk@iitk.ac.in Received 2 Agst 24 Revised 26 September 25 We introdce the concept of three-step wavelet-galerkin method based on the Taylor series expansion in time. Unlike the Taylor Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it sitable for solving nonlinear problems. Nmerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of fnctions which are smooth, except for localized regions, p to any given accracy are presented. Here nmerical experiments deal with advection eqation with the spiky soltion in one dimension, two dimensions and nonlinear eqation with a shock in soltion in two dimensions. Nmerical reslts indicate the versatility and effectiveness of the proposed scheme. Keywords: Three-step method; wavelets; parabolic eqation; hyperbolic eqation. AMS Sbject Classification: 65M7, 35-xx. Introdction Wavelets method is a new nmerical tool for solving partial differential eqations (PDEs). Wavelet analysis assmed significance de to sccessfl applications in signal and image processing dring the eighties. Compactly spported wavelets which are differentiable were introdced by Dabechies in her celebrated paper, which has had applications in a nmber of areas. Since these fnctions combine orthogonality with localization and scaling properties, it has been a natral idea to attempt to se these fnctions for the nmerical approximation to soltions to partial differential eqations (PDEs). There have been a nmber of papers in this direction inclding stdies of Brgers eqation 2,3 as well as two-dimensional flow. 4 6 Glowinski et al. 7 considered wavelet-based variational methods to solve one-dimension linear and nonlinear ordinary differential eqations. Where mltiresoltion analysis and their associated scale fnction bases are sed as alternative bases in Galerkin methods. 65

2 66 M. Mehra & B. V. Rathish Kmar Ths, sch methods have convergence properties similar to the ones of spectral methods, and simltaneosly, partial derivative operators discretize similarly as the finite-difference methods. In the literatre, many tentatives have been performed, often based on Galerkin or Petrov Galerkin methods which se the compression properties of wavelet bases, and contain specific wavelet methods for PDEs. Some of them take advantage of the wavelet compression of the soltion, 2 others se instead the wavelet compression of the operator. 8 The aim of this paper is to introdce fast three-step wavelet-galerkin method which has the benefit of both these properties. In the conventional nmerical approach to transient problems, the accracy gained in sing the high-order spatial discretization is partially lost de to the se of low-order time discretization schemes. Here, sally spatial approximation precedes the temporal discretization. On the contrary, the reversed order of discretization can lead to better time accrate schemes with improved stability properties. The fndamental idea behind the Taylor Galerkin approach 9 is the sbstittion of space derivatives for the time derivatives in the Taylor series, as sed in derivation of the Lax Wendroff method, the only modification being that the procedre is carried ot to third order. However, its applications are mainly for hyperbolic problems and some convection diffsion eqations, becase too many terms are introdced in the third-order time derivative term, especially for nonlinear mltidimensional eqations, and treatment of the bondary integrations arising from high-order time derivative terms are too complicated. A three-step finite element method based on a Taylor series expansion in time is proposed in Ref.. This scheme involves neither complicated expression nor higher-order derivatives like the Taylor Galerkin method 9 bt the advantage has not been taken of sparsity of matrices which are in evoltionary problems. Time accrate soltion of Korteweg de Vries eqation sing the wavelet-galerkin method is also developed in Ref. 2 and wavelet mltilayer Taylor Galerkin schemes for hyperbolic and parabolic problems are introdced in Ref. 3. Or three-step wavelet-galerkin method is based on fast algorithms like matrix vector prodct in wavelet bases and wavelet compression property of smooth data. Soltions to PDEs often behave differently in different areas. In flid dynamics, we have shocks, bondary layers and trblence. In acostics, an example of PDEs is a low-freqency wave, with a localized high-freqency brst. For these examples, the soltion can be smooth in most of the soltion domain, with small areas where the soltion changes qickly. Where the discretization in space (correlatively in time) oght to handle a hge nmber of degrees-of-freedom. The aim of the present paper is to examine the feasibility of applying fast three-step wavelet- Galerkin method to sch types of PDEs which reqire the advantage of wavelet compression. A combination of sch a time-marching scheme with wavelet approximation in space can lead to simple higher-order space and time accrate nmerical methods.

3 Time Accrate Fast Three-Step Wavelet-Galerkin Method for PDEs Wavelet Preliminaries Compactly spported wavelets have several properties that are qite sefl for representing soltions of PDEs. The orthogonality, compact spport and exact representation of polynomials of a fixed degree allow the efficient and stable calclation of regions with strong gradients or oscillations. Dabechies defines the class of compactly spported wavelets. Briefly, let φ be a soltion of the scaling relation φ(x) = k a k φ(2x k). The a k are a collection of coefficients that categorize the specific wavelet basis. The expression φ is called the scaling fnction. The associated wavelet fnction ψ is defined by the eqation ψ(x) = k ( ) k a k φ(2x k). The normalization φdx = of the scaling fnction leads to the condition k a k = 2. The translates of φ are reqired to be orthonormal, i.e. φ(x k)φ (x m)dx = δ k,m. The scaling relation implies the condition N k= a ka k 2m = δ,m where N is the order of wavelet. For coefficients verifying the above two conditions, the fnctions consisting of translates and dilations of the wavelet fnction, ψ(2 j x k), form a complete, orthogonal basis for sqare integrable fnctions on the real line, L 2 (R). If only a finite nmber of the a k are nonzero, then φ will have compact spport. Since φ(x)ψ(x m)dx = ( ) k a k a k 2m =, k the translates of the scaling fnction and wavelet define orthogonal sbspaces V j = {2 j/2 φ(2 j x k); m =...,,,,...}, W j = {2 j/2 ψ(2 j x k); m =...,,,,...}. The relation V j+ = V j + W j implies the Mallat transform V V V j+, V j+ = V W W W j.

4 68 M. Mehra & B. V. Rathish Kmar Smooth scaling fnctions arise as a conseqence of the degree of approximation of the translates. The reslt that the polynomials,x,...,x p be expressed as linear combinations of the translates of φ(x k) is implied by the conditions ( ) k k m a k = for m =,,...,p. The following are eqivalent reslts. k (i) {,x,...,x p } are linear combinations of φ(x k); (ii) f c j k φ(2j x k) C2 jp f p,wherec j k = f(x)φ(2 j x k)dx; (iii) x m ψ(x)dx =form =,,...,p ; (iv) f(x)ψ(2 j x)dx c2 jp. For the Dabechies scaling/wavelet fnction, DN have p = N/2. In Fig., we see an example of a compactly spported scaling fnction and its associated fndamental wavelet fnction. By rescaling and translation, we obtain a complete orthonormal system for L 2 (R) which has a sfficient smoothness to also be a basis for H (R). This wavelet system then yields a basis for soltion methods for second-order elliptic bondary problems on intervals on the real line. The illstrated example has fndamental spport [, 5]. For arbitrarily large even N, there is the Dabechies example of a fndamental scaling fnction defining a wavelet family with spport in the interval [,N ]. As pointed ot by Meyer (99), the complete toll box bilt in L 2 (R) can be sed in the periodic case L 2 ([, ]) φ.5 ψ Fig.. Dabechies scaling and wavelet fnctions for N = 6 with spport on [, 5].

5 Time Accrate Fast Three-Step Wavelet-Galerkin Method for PDEs 69 by introdcing a standard periodization techniqe. This techniqe consists at each scale in folding, arond the integer vales, the wavelet ψ j,k and the scaling fnctions φ j,k centeredin[, ]. It writes φ j,l (x) = n= φ j,l(x + n) and ψ j,l (x) = n= ψ j,l(x + n) and generates V Pj and W Pj. A fnction f V Pj in pre periodic scaling fnction expansion f(x) = 2 j k= c j φ k j,k (x) and the periodic wavelet expansion f(x) = 2 J k= c J φ J 2 k J,k(x)+ j j=j k= dj ψ k j,k (x); where J satisfies J J and the decay of the wavelet coefficient is given by the following theorem 4 : Theorem 2.. Let P = D/2 be the nmber of vanishing moments for a wavelet ψ j,k and let f C P (R). Then the wavelet coefficients decay as d j,k C P 2 j(p + 2 ) max ξ Ij,k f (P ) (ξ). 3. Three-Step Wavelet-Galerkin Method Before introdcing three-step method, it is necessary to have a brief statement of the two-step Lax Wendroff method. Performing a Taylor series expansion in time, we have (t + δt) =(t)+δt (t) + δt2 2 (t) t 2 t 2 + δt3 3 (t) 6 t 3 + O(δt 4 ). (3.) By approximating Eq. (3.) to second-order accracy, the formlation of two-step method can be derived as ( t + δt ) = (t)+ δt (t), 2 2 t (3.2) (t + δt/2) (t + δt) =(t)+δt. t Now, we introdce the three-step method. 3.. Principle of T-WGM for advection-diffsion eqation Consider the linear advection-diffsion eqation t = a x + ν x, 2 (3.3) where a and ν>are positive constant coefficients. Let s leave the spatial variable x continos and discretize (3.3) in time. To obtain an improved order of accracy in δt, we shall apply a three-step method based on the Taylor series expansion (3.). By approximating Eq. (3.) p to third-order accracy, the formlations of the three-step method can be written as ( t + δt ) = (t)+ δt (t), 3 3 t ( t + δt ) = (t)+ δt (t + δt/3), (3.4) 2 2 t (t + δt/2) (t + δt) =(t)+δt. t

6 7 M. Mehra & B. V. Rathish Kmar In Eq. (3.4), by replacing the time derivative with spatial derivatives, the associated three-step wavelet-galerkin eqations give the T-WGM scheme of n+ 3 = n + D n, n+ 2 = n + D 2 n+ 3, (3.5) n+ = n + D 3 n+ 2, ( ) where the operator D will look like D = δt 3 a x + ν x 2.Now,thewavelet- Galerkin discretization trns the problem into a finite-dimensional space. d n+ 3 = d n + D d n, d n+ 2 = d n + D 2d n+ 3, (3.6) d n+ = d n + D 3 d n+ 2. In this finite-dimensional space, n is to be replaced by the vector d n along a wavelet finite basis, and D are replaced by, respectively, D (finite) matrices. Now, to solve Eq. (3.6) in the wavelet basis, we will compte D once and store in compressed form. We can now give a comptational procedre for compting (3.6) sing wavelet compression. Algorithm (i) trnc(d,ɛ M ) (D) ɛm (ii) compte initial gess in wavelet basis d (iii) trnc ( ) ( ) d,ɛ V d ɛv (iv) for n ( =, ),...,n (v) D ɛm d n ɛv d n+ (vi) trnc ( ) ( ) d n+,ɛ V d n+ ɛv where trnc(d,ɛ V )= { d j k, dj k >ɛ V} and trnc(d,ɛm )={[D m,n ], [D m,n ] >ɛ M }. A frther property of the wavelet representation of operators is that the sccessive powers D n of the time iteration matrix become sparser and sparser with increasing n. This property is very specific to wavelets, as the opposite occrs with finite difference where D n becomes a more and more dense matrix as shown in Fig. 2. It is seen from Fig. 2 that in the wavelet-galerkin approach, compression in the matrix D n is larger than finite difference approach. From this property, we can obtain iterative speed of the three-step wavelet-galerkin scheme. Algorithm 2 (i) Initialize (D ) ɛm and ( ) d ɛv (ii) for n =, (,...,n ) (iii) (D n ) ɛm d n ɛv ( ) d n+ ɛv (iv) Dn 2 D n+. (v) endfor Then the approximate soltion of PDE is at t =2 n δt is d (2n ).

7 Time Accrate Fast Three-Step Wavelet-Galerkin Method for PDEs 7 9 x 4 Finite difference Wavelets x 4 Fig. 2. Nmber of coefficients in the sccessive powers of D n based on CN times stepping in wavelets and in finite differences, verss x =2 n, n = 5, N = 24, δt = 3, ɛ M = 8. There is no need to change from classical to wavelet coordinates ntil some time steps. In classical coordinates, the evoltion operator changes from very sparse to dense. In the wavelet representation, we may start the sqaring in the classical coordinates and change to the wavelet basis at the point where the wavelet representation is sparser. Ths, we have the following algorithm: Algorithm 3 (i) for n =, 2,...,p (ii) D n n+ (iii) endfor (iv) Initialize D ɛm and ( ) d p ɛv (v) for n ( = ) p +,p+2,...,p+ n (vi) D ɛm d n ɛv ( ) d n+ (i) trnc ( ) ( ) d n+,ɛ V d n+ ɛv (ii) endfor It is essential for the sccess of these algorithms that the comptation of the matrix vector prodct flly exploits the compressed form of both matrix and vector. This can be done sing the algorithm in Ref. 4 or fast mltiplication based on a general sparse format for both matrix and vector. Or techniqe is more favorable for parabolic problems in terms of taking the advantage of sparsity of operator. For hyperbolic (st-order systems of) PDEs, the sitation is different. We no longer have a sparse representation of the operator in wavelet space for the hyperbolic

8 72 M. Mehra & B. V. Rathish Kmar problem. However, a characteristic for hyperbolic problems is the presence of shocks in the soltion. A soltion might be smooth and nearly constant over the whole domain except at one point where it is discontinos. Therefore, for hyperbolic problems, wavelets cold be an efficient way to represent the soltion instead of, the soltion operator for parabolic and elliptic problems. In contrast with the Taylor Galerkin method, the three-step method does not contain any new higher-order spatial derivatives and can ths be applied to solve nonlinear mltidimensional flows with easy Theoretical stability of the linearized schemes We se the notion of asymptotic stability of a nmerical method as it is defined in Ref. 5 for a discrete problem of the form d/dt = L where L is assmed to be a diagonal matrix. This is becase for many evoltion eqations, it is necessary to adapt the time steps to the spatial resoltion in order to maintain the stability and precision of the nmerical scheme. The region of absolte stability of a nmerical method is defined for the scalar model problem d/dt = λ to be set of all λδt sch that n is bonded as t. Finally, we say that a nmerical method is asymptotically stable for a particlar problem if, for small δt >, the prodct of δt times every eigenvales of L lies within the region of absolte stability Nmerical simlation of advection eqation Here, we examine the performance of the T-WGM on a linear advection eqation on the nit interval 7 with periodic bondary conditions. The initial condition (x) =sin(2πx)+exp α(x /2)2, which is smooth in most of the domain except near x =.5, where we have a spike and the thresholded wavelet expansion of the soltion (x, t) is shown in Fig. 3 for α = 4. This wavelet expansion will have few coefficients, except for in the neighborhood of the spike at x =.5 t. We stepped forward to t =.3, where the soltion for j =8andj =9isshown in Fig. 4 sing wavelet-galerkin method (WGM). Here, oscillation are prodced throghot the domain de to the qick change of soltion near spike for j =8 and oscillation are confined near the neighborhood of spike for j =9.Intermsof finite difference methods, we want to have many points in areas where the soltion has strong variation and few points in area where the soltion is smooth. If we se a Galerkin method, this corresponds to the representation of the soltion having fewer basis fnctions in the smooth areas. Note that by thresholding a wavelet representation, we have a way to atomatically find a sparse representation of smooth part. Holmstrom and Walden have applied adaptive wavelet methods on sch type of PDEs. 8,7 However, by or approach of T-WGM scheme, we are taking the advantage of the time accrate scheme as well as wavelet capabilities of compression to prodce fast algorithm based on fast matrix vector prodct in terms of sparsity. We are getting oscillation-free soltion as shown in Fig. 5 withot and

9 Time Accrate Fast Three-Step Wavelet-Galerkin Method for PDEs (a) (b) Fig. 3. (a) Initial fnction (x); (b) Trncated approximate soltion by D 6 with the threshold ɛ =., which leads to 3 retained wavelet coefficients, ot of the original (a) (b) Fig. 4. Soltion of advection eqation at t =.3 by WGM (a) (b) Fig. 5. Soltion of advection eqation at t =.3, j =9byT-WGM.

10 74 M. Mehra & B. V. Rathish Kmar with trncation, where WGM is nable to prodce oscillation-free soltion near the spike with the same degrees-of-freedom. Error in approximated soltion for WGM, two-step Lax Wendroff wavelet Galerkin method (L-WGM) and T-WGM withot trncation is shown in Fig. 6; where near the spike error in the T-WGM scheme is small compared to both L- WGM and WGM scheme. From this, we can conclde that near the sharp gradients, we can take the advantage of time accracy and compression properties of wavelet in the T-WGM scheme. These nice properties are also observed for all the experiments in two dimensions. For some ɛ M >, ɛ v >, the density of D n is small bt the trncation error D n Dn ɛm <C ɛm and n ɛv n <C ɛv is bonded for large n. Figre7givesan indication of the error committed by trncation of the soltion. In all the schemes,.25 D=6 sing WGM.25 D=6 sing LWGM (a) (b). D=6 sing T WGM (c) Fig. 6. Comparison of errors in approximated soltion.

11 Time Accrate Fast Three-Step Wavelet-Galerkin Method for PDEs e-6 e n=sqaring(log(time)) Fig. 7. Trncation errors n ɛv n for different threshold ɛ v. 4 x x 4 Fig. 8. For T-WGM, δt times the eigenvales of D for j =8. δt is chosen sch that it shold satisfy stability condition. The λδt for the T-WGM scheme is plotted in Fig T-WGM for 2D advection eqation As in the one-dimensional case, we first examine a linear advection eqation. Again it will test the method s ability to follow featres of the soltion: t = x + y. (3.7)

12 76 M. Mehra & B. V. Rathish Kmar Here, we are sing the third-order accrate T-WGM scheme based on (3.6). Let t = and ptting this vales in Eq. (3.6): n+/3 = n + D n, n+/2 = n + D 2 n+/3, (3.8) n+ = n + D 3 n+/2, where D = δt 3 a, D 2 = δt 2 a and D 3 = δta. All the algorithm in two dimensions will also take the advantage of wavelet compression like the one-dimensional case Nmerical simlation of 2D advection eqation The initial fnction is the smooth fnction with a localized spike (x, y) = exp α((x /2)2 +(y /2) 2).2sin(2πx)sin(2πy) and periodic bondary conditions. In Fig. 9, the inital fnction and soltion at t =.3 withot and with trncation is shown for α =2 4. (a) (b) (c) Fig. 9. (a) The initial fnction; (b) Soltion at t =.3 whenj =8,ɛ M = 5,ɛ v =;(c) Soltion at t =.25 for ɛ M = 5, ɛ v = 4.

13 Time Accrate Fast Three-Step Wavelet-Galerkin Method for PDEs 77 (a) (b) (c) Fig.. (a) The initial fnction; (b) Soltion at t =.25 when j =7,ɛ M = 5, ɛ v =; (c) Soltion at t =.25 for ɛ M = 5, ɛ v = T-WGM for 2D nonlinear eqation Now, we wold like to examine the ability of or method in two-dimensions when the gradient has developed in the soltion. We examine the following two-dimensional conterpart to the one-dimensional Brgers eqation: t + ( x + y )=ν( xx + yy ). (3.9) Here, we are sing the third-order accrate T-WGM scheme based on (3.6). Let f(x, y, ) =( x + y ), then the original eqation is t = ν f(x, y, ) and ptting this vale in Eq. (3.6), where D = νδt 3 n+/3 = n + D n (δt/3)f, n+/2 = n + D 2 n+/3 (δt/2)f, (3.) n+ = n + D 3 n+/2 δtf, νδt, D = 2 and D = νδt.

14 78 M. Mehra & B. V. Rathish Kmar Nmerical simlation of 2D nonlinear eqation If we choose the initial fnction as a two-dimensional sine wave (x, y) = sin(2π(x + y)) and periodic bondary conditions, the gradient in the shock will reach its maximm at time t =.25whentheextremaofthesinewavehave advected into the shock. In Fig., the inital fnction and soltion at t =.25 withot and with trncation is shown for ν = Conclsion In the three-step wavelet-galerkin method, the precedence of time discretization to space discretization in conjnction with wavelet bases for expressing spatial terms renders robstness to the proposed schemes and makes them space and time accrate. The three-step wavelet-galerkin method retains the third-order accracy and stability property of the Taylor Galerkin method. The concepts are introdced throgh linear advection eqation and Brgers eqation where we can show the power of wavelet compression as well as time accrate schemes. Since no new higherorder derivative term occrs in the nmerical formlations, the present method is sitable for nonlinear problems. Frther, the method can be directly extended to three-dimensional problems. The nmerical reslts show that the present method is comptationally efficient. References. I. Dabechies, Orthonormal basis of compactly spported wavelets, Comm. Pre Appl. Math. 4 (988) J. Liandrat, V. Perrier and Ph. Tchmitchian, Nmerical resoltion of nonlinear partial differential eqations sing wavelet approach. Wavelets Appl. (992) A. Latto and E. Tenenbam, Les ondelettes à spport compact et la soltion nmériqe de l èqation de Brgers, C. R. Acad. Sci. Paris Sér. I 3 (99) J. Weiss, Wavelets and the stdy of two dimensional trblence, in Proc. USA-French Workshop Wavelets and Trblence (Princeton University, Jne, 99). 5. M. Farge, N. Kevlahan, V. Perrier and E. Goirand, Wavelets and trblence, Proc. IEEE 84 (996). 6. J. Frohlich and K. Schneider, An adaptive-vagelette algorithm for the soltion of PDEs, J. Comp. Phys. 3 (997) R. Glowinski, W. Lawton, M. Ravachol and E. Tenenbam, Wavelet soltions of linear and nonlinear elliptic, parabolic and hyperbolic problems in D, Compting Methods in Applied Sciences and Engineering (SIAM, PA, 99), pp B. Engqist, S. Osher and S. Zhong, Fast wavelet based algorithms for linear evoltion eqations, SIAM J. Sci. Compt. 5(4) (994) J. Donea, A Taylor Galerkin method for convective transport problems, Internat. J. Nmer. Methods Engrg. 2 (984) 9.. P. D. Lax and B. Wendroff, Systems of conservation laws, Comm. Pre Appl. Math. 3 (96) C. B. Jiang and M. Kawahara, The analysis of nsteady incompressible flows by a three-step finite element method, Internat. J. Nmer. Methods Flids 6 (993)

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