Journal of Wavelet Theory and Applications. ISSN 0973-6336 Volume 1 Number 1 (2007), pp. 83–96 c Research India Publications http://www.ripublication.com/jwta.htm Wavelet optimized finite difference method using interpolating wavelets for solving singularly perturbed problems Vivek Kumar Tata Institute of Fundamental Research, IISc Campus, Bangalore 560012, India, [email protected]Mani Mehra Department of Mathematics and Statistics, McMaster University, Hamilton, Canada, [email protected]Abstract A wavelet optimized finite difference (WOFD) method is presented for adaptively solving a class of singularly perturbed elliptic and parabolic problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples for elliptic and parabolic problems are provided. The proposed method proves to be a better alternative for dealing singu- lar perturbation problems in terms of automatic grid generation and CPU time. Keywords: Wavelets, Singularly perturbed problems, Interpolating wavelet trans- form, Lagrange finite difference. 1. Introduction In this paper we consider semilinear singularly perturbed reaction diffusion problem of elliptic and parabolic types. Elliptic problem is defined as −u (x)= f (x, u ) where 0 ≤ x ≤ 1, (1.1a) with boundary conditions u (0) = u a , u (1) = u b , ∂f ∂u ≥ c> 0, for all x ∈ (0, 1), (1.1b)
A wavelet optimized finite difference (WOFD) method is presented for adaptivelysolving a class of singularly perturbed elliptic and parabolic problems. The methodis based on an interpolating wavelet transform using polynomial interpolation ondyadic grids. Adaptive feature is performed automatically by thresholding thewavelet coefficients. Numerical examples for elliptic and parabolic problems areprovided. The proposed method proves to be a better alternative for dealing singu-lar perturbation problems in terms of automatic grid generation and CPU time.
In this paper we consider semilinear singularly perturbed reaction diffusion problem ofelliptic and parabolic types. Elliptic problem is defined as
−εu′′ε (x) = f(x, uε) where 0 ≤ x ≤ 1, (1.1a)
with boundary conditions
uε(0) = ua, uε(1) = ub,∂f
∂uε
≥ c > 0, for all x ∈ (0, 1), (1.1b)
84 Vivek Kumar and Mani Mehra
where ε is a small parameter and f(x, uε) is sufficiently smooth. For ε � 1, the solutionhas boundary layers at x = 0 and x = 1. If all the above conditions are satisfied, then
|uε(x)| ≤ max{1
cmaxx∈[0,1]
|f(x, 0)|, |ua|, |ub|} for all x ∈ [0, 1],
and the solution will be unique [1]. The parabolic problem is defined as
Lεuε = −ε u′′ε (x, t) + a(x, t)uε(x, t) + b(x, t)
∂uε(x, t)
∂t= f(x, t, uε), (1.2a)
where (x, t) ∈ Q = (0, 1) × (0, T ] and
uε(x, 0) = s(x) on Sx = (x, 0) : 0 ≤ x ≤ 1 (1.2b)
uε(0, t) = q0(t) on S0 = (0, t) : 0 < t ≤ T (1.2c)
uε(1, t) = q1(t) on S1 = (1, t) : 0 < t ≤ T . (1.2d)
Here ε is a parameter satisfying 0 < ε � 1. We assume that a(x, t) ≥ a0 > 0 andb(x, t) ≥ b0 > 0 on Q, where a0, b0 are some constants and Q = [0, 1] × [0, T ] denotesthe closure of Q. The solution u has in general boundary layers of parabolic type alongthe sides x = 0 and x = 1 of Q [1].
In singular perturbation problems we have shocks as boundary layers for ε → 0. Forsuch kinds of problems the solution can be smooth in most of the solution domain withsmall area where the solution changes fast. Solution to such problems suffers a localbreakdown at the boundary layer as ε → 0. These kind of behavior is also called dis-sipative because of the rapidly varying component of the solution decays exponentiallyaway from the point of local breakdown. When solving such problems numerically,we would like to adjust the discretization to the solution. In term of mesh generation,we want to have many points in area where the solution have strong variations and fewpoints in area where the solution is smooth. With a very small perturbation parameterε, large Nj (total no. of mesh points at jth level) is required to obtain accurate solu-tion. For good resolution of the numerical solution, at least one of the collocation pointsshould lie in the boundary layer region. For example, if the problem possesses a bound-ary layers of width κε, κ = ln(Nj) > 0, then on a uniform grid with h < κε spacingbetween the points we need Nj > 1/(κε), which is not practically applicable for ε � 1.Mainly, there are two kinds of approaches to deal with such problems the first one isfitted mesh, it consists of choosing fine grids in the layer region’s, and another is fittedoperator method in which grid remains uniform and difference operators reflects thesingularly perturbed nature of the differential operators. These kinds of problems usingone or both of the strategies have been discussed in [2] and [3]. Farrell et al. [3] devel-oped a successful upwind central difference scheme on a piecewise uniform mesh givenby Shishkin. One of the drawbacks with Shishkin mesh generation is that it requires apriori knowledge of the location and width of the boundary layers, therefore, motivatesus to look for adaptive methods. More literature can be found in [3] and [4]. Wavelet
Wavelet optimized finite difference method 85
optimized finite difference [5] works by using adaptive wavelet to generate an irregulargrid which is then exploited for the finite difference method.
Wavelets have been making their appearance felt in many pure and applied areaof science and engineering [5, 6]. Wavelets detect information at different scales and atdifferent locations throughout a computational domain. Wavelets can provide a basis setin which the basis functions are constructed by dilating and translating a fixed functionknown as the mother wavelet (first generation wavelets). The mother wavelet can beseen as a high pass filter in the frequency domain. One of the key strength of thewavelet methods is data compression. An efficient basis is one in which a given setof data can be represented with as few basis elements as possible. Suppose we havewavelet representation of a function
∑
k
cj,kϕj,k(x) +∑
j,k
dj,kψj,k(x),
where ϕj,k(x) are scaling functions and ψj,k(x) are wavelets, the scaling function co-efficients cj,k, deals with smooth part of the function, while the wavelet coefficient dj,k
contains information of the function behavior on successive finer scales. The most com-mon way of compressing such a representation is thresholding. We generally delete allwavelet coefficients of magnitude less than some threshold, say τ . If the total no. ofcoefficients in the original representation was NJ , we have Ns significant coefficientsleft after the thresholding. Note that by thresholding a wavelet representation we havea way to find an adaptive feature and we can also use this representation to computefunction values at any point.
We discuss interpolating wavelet transform in section 2. Section 3 gives a brief intro-duction about adaptivity and the wavelet optimized finite difference method. Numericalresults have been discussed in section 4.
2. The interpolating wavelet transform
Here we briefly describe the interpolating wavelets of Donoho and Harten [7] [8]. In-terpolating wavelets are constructed on a set of dyadic grids on the line,
Aj = {xj
k ∈ R : xjk = 2−jk, k ∈ Z, j ∈ Z}, (2.1)
where xjk are the grid (collocation) points and j is the level of resolution. Since xj−1
k =
xj2k, it follows that A
j−1 ⊂ Aj . Essentially, Interpolating wavelets can be formally in-
troduced through the interpolating subdivision scheme of Deslauriers and Dubuc [9],which consider the problem of building an interpolant f j(x) on a grid A
j+1 for a givendata sequence f(xj
k). The algorithm proceeds by interpolating the data f(xjk) to the
points on the grid Aj+1 which don’t belong to A
j . The even numbered grid points xj2k
already exist in Vj , and the corresponding function values are kept unchanged. Values atodd numbered grid points xj
2k+1 are computed from the polynomial interpolation from
86 Vivek Kumar and Mani Mehra
0 2 4 6 8 10−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
φ(x
)
Figure 1: Scaling function φ(x).
the values at the even numbered grid points. The interpolation is achieved by construct-ing local polynomials, P2N−1(x) of order 2N−1, which uses 2N closest points. For ex-ample, to find the value of the interpolant at location xj+1
2k+1 we construct the polynomialof order 2N−1 based on the values of the function at locations xj
k+1(l = −N +1, ....N)
and evaluate it at location xj+12k+1. Evaluate this polynomial at point xj+1
2k+1 and substitutethe values of polynomial coefficient expressed in terms of values f(xj
k), we can easilyget that
f j(xj+12k+1) =
N∑
l=−N+1
wjk,lf(xj
k+1). (2.2)
The main attraction of interpolating subdivision scheme is that the values of theseweights are the same for evenly space grids and procedure can be easily extended tothe nonuniform grids, which will result in location dependent weights. The adaption tothe boundaries is simple. We use the closest point inside the boundary on the coarsergrid to define the interpolating polynomial.
The interpolating scaling function φjk(x) can be formally defined by setting f(xj
l ) =δl,k, where δl,k is the Kronecker delta, then performing the interpolating subdivisionscheme up to an arbitrary high level of resolution j. Now using the linear superposition,it is easy to show that
f j(x) =∑
k
cjkφ
jk(x), (2.3)
where cjk = f(xj
k). An example of an interpolating scaling function φ(x) is shown inFig 1. It is easy to show that the interpolating function has the following properties:
• compact support of [−2N + 1, 2N − 1];
• φ(x) is cardinal interpolating; i.e φ(x) = δk,0;
• linear combination φjk(x) reproduce the polynomials up to degree 2N − 1;
Wavelet optimized finite difference method 87
• φ(x) satisfies a refinement relation φjk =
∑l∈Aj+1 hj
k,lφj+1l ;
• φ(x) is the autocorrelation of the Daubechies scaling function of order 2N [10].
Used recursively, the interpolating subdivision scheme generates function values on afine grid, given values on coarse grid. On the contrary, when we go from a fine to acoarse grid, could just throw away half of the grid points at each level, but we wouldthen lose information. Instead at each level, for odd numbered grid points, compute thedifference between the known function value and the function value predicted by theinterpolation from the coarser grid. We call this differences in function values waveletcoefficients dj
k. Repeating these recursively we have an algorithm for wavelet transformas follows. The forward interpolating wavelet transform can be written as
djk =
1
2(cj+1
2k+1 −∑
l
wjk,lc
j+12k+2l), (2.4)
cjk = cj+1
2k , (2.5)
while the inverse wavelet interpolation transform is given by
cj+12k = cj
k, (2.6)
cj+12k+1 = 2dj
k +∑
l
wjk,lc
jk+l, (2.7)
where wjk,l are the interpolating coefficients as introduced in (2.2). The algorithm for
constructing interpolating wavelets on an interval and on a uniform grid are same, exceptthe modification of weights at the boundaries.
3. Wavelet optimized finite difference method
In this section we outline a completely different approach to solving singularly per-turbed problems with the aid of wavelets. The method is due to Leland Jameson [5] andis called wavelet optimized finite difference method. It works by using the wavelets onirregular grids which is then exploited for the finite difference method. Hence it fallsunder the category of fitted mesh methods.
3.1. Adaptive Mesh Generation
Adaptive mesh is generated in the following ways:
• First we solve the given problem on a uniform mesh for the initial solution profile.
• We apply the discrete wavelet transform to the solution profile and calculate thewavelets coefficients. Wavelets coefficients will be smaller where the solution issmooth and large at the place of singularity like boundary layers in our case.
88 Vivek Kumar and Mani Mehra
• We remove the mesh points where |djk| < τ (prescribed threshold) and keep the
remaining.
• We apply the finite difference on the irregular grid as given in the next section onthe remaining mesh points and get the final solution.
3.2. Finite difference on irregular grid
Using adaptive mesh points from the interpolating wavelet, derivatives on non-uniformgrid are approximated using Lagrangian interpolating polynomial through p points [5].We consider only odd p ≥ 3 because it makes the algorithm simpler. Let w = p−1
2and
define
uI(x) =i+w∑
k=i−w
u(xk)Pw,i,k(x)
Pw,i,k(xk), (3.1)
where
Pw,i,k(x) =i+w∏
l=i−w,l �=k
(x − xl).
It follows that uI(xi) = u(xi) for i = 0, 1, 2....NJ − 1, NJ i.e uI interpolates u at thegrid points. Differentiation of uI(x) d times yields
udI(x) =
i+w∑
k=i−w
u(xk)P d
w,i,k(x)
Pw,i,k(xk). (3.2)
The derivatives udI(x) can then be approximated at all the grid points by
udI = Dd
pu,
where the differentiation matrix Ddp is defined by
[Ddp]i,k =
P dw,i,k(xi)
Pw,i,k(xk); d = 1, 2.
First and second derivatives are
P(1)w,i,k(x) =
i+w∑
l=i−w,l �=k
i+w∏
m=i−w,m �=k,l
(x − xm), (3.3)
and
P(2)w,i,k(x) =
i+w∑
l=i−w,l �=k
i+w∑
m=i−w,m �=k,l
i+w∏
n=i−w,n �=k,l,m
(x − xn). (3.4)
Wavelet optimized finite difference method 89
4. Numerical Results and Discussion
We solve an elliptic and two parabolic problems to demonstrate the efficiency of theproposed method. Please note that we take J instead of j.
4.1. Elliptic problem
We consider linear elliptic problem as
−εuxx + u = − cos2(πx) − 2επ2 cos(2πx), (4.1)
with boundary conditionsu(0) = 0 and u(1) = 0 (4.2)
This problem has earlier been discussed in [11]. The exact solution is
u(x) =exp(−1(1 − x)/
√ε) + exp(−x/
√ε)
1 + exp(−1/√
ε)− cos2(πx),
which has boundary layers at x = 0 and x = 1.We denote uJ
τ (x) be the approximation of u(x) at J level for prescribed threshold τusing interpolating wavelet. Let us define L∞ error as
‖uJτ − u‖∞. (4.3)
Table 1 shows the L∞ error obtained by the proposed method and even better resultsare obtained for very small ε. This problem has also been discussed in [4] using B-splinewith Shishkin meshes. Fig. 2(a) shows the solution for elliptic problem for different ε.It is clear from the figure that mesh points are concentrated more at the boundaries(where the boundary layers occur). Fig. 2(b) shows the wavelet coefficients at differentresolutions. As the resolution is increased, non zero wavelet coefficients can be seenonly at the place of solution with high gradient.
Eigenvalues for the diffusion operator has been plotted in Fig. 3(a) which shows thatall the eigenvalues for the diffusion operator are non positive and real. Fig. 3(b) verifiesthe theoretical results between τ and Ns (number of significant points after adaptivity)as given by Donoho [7]
τ ≤ c1N−2Ns . (4.4)
Fig. 4 (a) gives the L∞ error for elliptic problem for various values of ε (perturbationparameter) and Fig. 4 (a) gives the L∞ error for elliptic problem for different wavelets(N ). It verifies the theoretical results given in [7]
Equation (1.2) with a(x, t) = q0(t) = q1(t) = 0, b(x, t) = 1, f(x, t, u) = e−u − 1and initial condition s(x) = 1, 0 ≤ x ≤ 1 makes the first parabolic problem withnonlinear term. Nonlinearity is dealt with quasilinearization technique of Bellman andKalaba [12]. In the quasilinearization technique, the non-linear differential equation issolved recursively by a sequence of linear differential equations. The main advantage ofthis method is that if the procedure converges, it converges quadratically to the solutionof the original problem. The linear equation is obtained by using the first and secondterm of the Taylor’s series expansion of the original non-linear differential equation.The non-linear term f(x, un+1(x)) can be expanded as
f(x, un+1(x)) = f(x, un(x)) + (un+1 − un)f ′(x, un(x)) + .... where n = 0, 1, 2......(4.6)
We consider an implicit two-level time difference scheme on the nonuniform adap-tive mesh. In both parabolic problems we have calculated all the results with δt = .05time step size. Fig. 5 (a) and Fig. 5 (b) show the numerical solution using WOFD atvarious times for different ε = 10−3 and ε = 10−8 respectively. Adaptive mesh hasbeen generated after two initial iterations. Moreover, we see that more mesh points areconcentrated in the area with large gradient. Fig. 6 (a) shows the solution as time (t)increases from 0 to 10 and corresponding number of significant mesh points. As timeincreases, solution becomes almost zero which leads to fall in number of significant gridpoints starting from 512 to minimum 25. The number of significant coefficients (Ns)with respect to number of iterations (t : 0 → 10 corresponds to iterations : 1 → 200)
91Wavelet optimized finite difference method
101
102
103
104
10−12
10−10
10−8
10−6
10−4
10−2
100
Ns
||yJ−y||
= 10−3
= 10−5
= 10−7
(a) The L∞ of the solution as a func-tion of Ns for various ε .
101
102
103
104
10−12
10−10
10−8
10−6
10−4
10−2
100
Ns
||yJ−y||
N = 2N = 3N = 4
(b) The L∞ error of the solution as afunction of Ns for various N and ε =10−5 .
Figure 4: Elliptic Problem.
is given in Fig. 6 (b). It further strengthen the fact that proposed scheme is stable andreduce the memory requirement.
4.2.1 Parabolic problem 2
The second parabolic problem is given as f(x, t, u(x, t)) = 0, a(x, t) = 0, b(x, t) = 1with initial and boundary conditions s(x) = 0, q0(t) = t2, q1(t) = t. Boundary layerswidth moves with respect to time and becomes very thin as time increases.
Fig. 7(a) shows the solution of parabolic problem 2 and Fig. 7(b) shows the corre-sponding adaptive mesh at various times for ε = 10−3. It is observed from Fig. 7 (a)that the boundary layer becomes thin with increasing time, therefore adaptive grid alsobecomes fine in the layer region with increasing time in Fig. 7 (b). This confirm theautomatic evolution of adaptive grid with respect to time. Furthermore, as ε decreases,we need more points NJ = O(ε−1) to resolve the boundary layer region of width O(ε)on non adaptive grid. Fig. 8 shows the solution of parabolic problem 2 and corre-sponding adaptive mesh at various times for ε = 10−8. It is clear from Fig. 8 that gridhas automatically adapted to resolve the thin region of boundary layer and confirms themoving feature of mesh with respect to to the singular perturbation parameter ε. It alsoshows the advantage of proposed method using wavelet adaptive grid over the Shishkinmeshes for singular perturbation problems which require a prior knowledge of locationand width of the boundary layers.
92 Vivek Kumar and Mani Mehra
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
u(x,)
t=.5
t=1
t=1.5
(a) ε = 10−3, J = 8
0 0.2 0.4 0.6 0.8 1−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
xu(x,)
t=.5
t=1
t=2
(b) ε = 10−8, J = 10
Figure 5: Numerical solution for parabolic problem 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
u(x,)
(a) Numerical solution at various timesas t ∈ (0, 10)
0 50 100 150 2000
10
20
30
40
50
60
70
80
90
100
iterations
Ns
(b) Number of significant grid points(Ns) as t ∈ (0, 10)
Figure 6: Parabolic problem 1 for ε = 10−5, J = 9
93Wavelet optimized finite difference method
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
20
25
x
u(x,) t=5
t=2.5t=.5
(a) Numerical solution at varioustimes.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
xt
(b) Adaptive mesh as t ∈ (0, 5).
Figure 7: Parabolic problem 2 for ε = 10−3, J = 8.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
20
25
x
u(x,)
(a) Numerical solution at varioustimes.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x
t
(b) Adaptive mesh as t ∈ (0, 5).
Figure 8: Parabolic problem 2 for ε = 10−8, J = 9.
94 Vivek Kumar and Mani Mehra
5. Conclusion
We have presented wavelet optimized finite difference for adaptively solving singularlyperturbed reaction diffusion elliptic and parabolic problems. The adaptivity is achievedby using an interpolating wavelet basis. The novelty of this work lies in the threshold-ing scheme used to reduce the CPU time with automatic adaptive feature of waveletand method does not require the pre-knowledge of boundary layers present in the sin-gular perturbation problems. The adaptive grid works fine when solution of singularperturbation problems evolves with time as well as for the features of shrinking bound-ary layers when ε decreases. Finally, the wavelet optimized finite difference method ismore efficient and easy to implement for singular perturbation problems.
References
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[2] Miller, J.J.H., O’Riordan, E. and Shishkin, I.G. “Fitted numerical methods forsingular perturbation problems”, world Scientific, 1996.
[3] Farrell, P.A., Hegarty, A.F., Miller, J.J.H., O’Riordan, E. and Shishkin, I.G. “Ro-bust computational techniques for boundary layers”, Chapman and Hall, CRC,2000.
[4] Kadalbajoo M.K, Aggarwal Vivek K. “Fitted mesh B-Spline collocation methodfor solving singularly perturbed reaction-diffusion problems ”, J. of Concrete andApplicable Mathematics, 4, 3, 2006, 349-365.
[5] Jameson, L. “A wavelet-optimized, very high order adaptive grid and numericalmethod”, Siam J. Sci. Comp. , 19, 1998, 1980-2013.
[6] Kumar, V., Mehra, M. “Cubic spline adaptive wavelet scheme to solve singularlyperturbed reaction diffusion problems”, Int. J. of Wavelet, Multi. and InformationProcessing, Vol. 5, 1, 2007, 1-15.
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