Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems

Applied Mathematical Modelling 37 (2013) 7095–7127

Contents lists available at SciVerse ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Multiresolution based adaptive schemes for second orderhyperbolic PDEs in elastodynamic problems

0307-904X/$ - see front matter � 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2012.09.004

⇑ Corresponding author. Tel.: +98 21 66498981; fax: +98 21 66403808.E-mail addresses: [email protected], [email protected] (H. Yousefi), [email protected] (A. Noorzad), [email protected] (J. Farjoodi).

Hassan Yousefi ⇑, Asadollah Noorzad, Jamshid FarjoodiSchool of Civil Engineering, College of Engineering, University of Tehran, P.O. Box 11365-4563, Tehran, Iran

a r t i c l e i n f o

Article history:Received 21 May 2011Received in revised form 14 August 2012Accepted 7 September 2012Available online 18 September 2012

Keywords:Hyperbolic PDEsInterpolating waveletAdaptive solutionTikhonov regularizationSpatially adaptive smoothing

a b s t r a c t

An enhanced interpolation wavelet-based adaptive-grid scheme is implemented for simu-lating high gradient smooth solutions (as well as, discontinuous ones) in elastodynamicproblems in domains with irregular boundary shapes. In the method, spatially adaptivesmoothing is used to improve interpolation property of the solution in high gradient zones.In hyperbolic systems, in fact, there are no certain inherent regularities; hence, the errone-ous adapted grid may be achieved because of small spurious oscillations in the solutiondomain. These oscillations, mainly formed in the vicinity of high gradient and discontinuityzones, make the adaptation procedure strongly unstable. To cover this drawback, enhancedsmoothing splines are used to denoise directly non-physical oscillations in the irregulargrid points, a kind of ill-posed problem. Controllable smoothing is achieved using non-uniform weight coefficients. As the smoothing splines are a kind of the Thikhonovregularization method, they work stably in irregular grid points. Regarding the Thikhonovregularization method, L-curve scheme could be used to investigate trade-off betweenaccuracy and smoothness of the solutions. This relationship, in fact, could not be reliablycaptured by common computational methods. The proposed method, in general, is easyand conceptually straightforward; as all calculations are carried out in the physicaldomain. This concept is verified using a variety of 2D numerical examples.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

Several wavelet-based schemes have been developed for solving Partial Differential Equations (PDEs). These methods cangenerally be divided into: (1) projection schemes, [1]; (2) non-projection ones, [2]. In the first group, the wavelet expansionis used to solve differential equations, as in wavelet (Petrov) Galerkin schemes [3–5]. There, all calculations are performed inthe wavelet spaces. These schemes have difficulties with projecting non-linear operators to wavelet spaces and handlinggeneral boundary conditions [1,2]. In the second group, the wavelet-based adaptive grid approach, the above mentioned dif-ficulties can be handled; since the degrees of freedom are considered as point values in the physical spaces, [2,6,7]. Theseadaptive methods have successfully been developed for PDEs solutions which contain steep moving fronts or sharp transi-tions in small zones [2,6–13]. In this group, the wavelet transforms are mainly used to detect highly non-uniform spatialbehaviors. The wavelets can adequately resolve different scales. They use high resolutions only near sharp transition regions,while moderate resolutions are applied in the regions with smooth and slow varying behaviors. The detail coefficients of(interpolating) wavelet transforms are used to create local refined grids on which the discretizations are performed. Theseschemes have mostly been improved for the elliptic [8,10] and parabolic [2,6–7,9,11] PDEs. It should be mentioned that the

http://dx.doi.org/10.1016/j.apm.2012.09.004

mailto:[email protected]




http://dx.doi.org/10.1016/j.apm.2012.09.004

http://www.sciencedirect.com/science/journal/0307904X

http://www.elsevier.com/locate/apm

http://crossmark.dyndns.org/dialog/?doi=10.1016/j.apm.2012.09.004&domain=pdf

Fig. 1. Procedure of detail coefficient dj;n calculation, where M ¼ 2 (cubic interpolation).

7096 H. Yousefi et al. / Applied Mathematical Modelling 37 (2013) 7095–7127

elliptic systems are stable and therefore they can be solved by fast iterative methods. This is because corresponding condi-tion numbers increase slowly during each iteration.

The hyperbolic PDEs, on the other hand, could not be simulated by common adaptive procedures. These types of PDEsshow no inherent dissipation in comparison with the elliptic and parabolic systems, [14]. Consequently, non-dissipative spu-rious oscillations (very small errors) can be developed. These small non-physical oscillations may be magnified in the solu-tion domain even for simple cases with smooth initial conditions. The small errors do not dissipate in hyperbolic problems

Fig. 2. Effects of spurious oscillations; (a): snapshot of the solution obtained in uniform grid of 257 � 257 grid points; (b): approximation space (the background grid); (c)–(e): adapted grids correspond to levels j 2 f5;6;7g respectively where e ¼ 0:3� 10�6; (f): resultant grid due to superposition of the basegrid and adapted grids.

Fig. 3. Non-uniform estimation, (a) a test function gðxÞ and its corresponding adapted grid points, (b) values of fWjg in case �a ¼ 3.

H. Yousefi et al. / Applied Mathematical Modelling 37 (2013) 7095–7127 7097

and propagate in the solution domain. This could lead to erroneous adapted grid points, i.e., stochastic concentration ofadapted grid points is smooth zones. Such feature can be seen, for example, in Fig. 18 of Ref. [11]. There, an erroneousadapted grid points is illustrated; the grid corresponds to a solution of a first order hyperbolic system. Regarding hyperbolicPDEs, numerical dispersion effects are more bolded in the second order systems, while the numerical dissipation are superiorin the first order ones [15]. Therefore, it seems that in the first order systems, the stability of adaptation procedure is con-served [11]. However this feature should be controlled for enough high gradient solutions; because intensity of the spuriousoscillations is in accordance with gradient of solutions.

One of the main properties of second order hyperbolic equations (elastodynamic problems) is that their solutions prop-agate with finite velocity. Therefore, compact support solutions in space will remain compact in time [16]. Consequentlysimulation of these problems by adaptive methods is appealing.

Several methods are developed for controlling the oscillations in the first order hyperbolic PDEs, some of which are: up-wind methods; essentially non-oscillatory (ENO) schemes [17,18]; weighed ENO methods [18]; filtering schemes [19,20];ones use adaptive order of spatial differentiations in the solution domain [21]. These methods are mainly improved for dis-sipation or reduction of local non-physical oscillations, caused by local spatial sharp discontinuity occurred in first orderhyperbolic PDEs. However, the small spurious oscillations can be produced all over the solutions (even in the smooth ones)regarding the hyperbolic PDEs. These errors are amplified in the vicinity of both smooth high gradient and discontinuouszones. Their sources are: (1) the spatial discretization error, e.g. those resulted by the finite difference approximation (trun-cation error); (2) the temporal discretization error, obtained in time integration methods. These non-dissipative spurious

Fig. 4. Comparison of errors in estimations in energy, Sobolev, and Sup norm senses; the results are obtained by the cubic smoothing spline wherefWjg ¼ 1.

Fig. 5. Comparison of errors in estimations in energy, Sobolev, and Sup norm senses; the results are obtained by the cubic smoothing spline of non-uniformweights, i.e., Wj � 1.


oscillations can act as high frequency vibration modes in dynamic systems (second order hyperbolic problems) with widerange of frequency contents and cause instability in the analysis.

Several wavelet-based adaptive schemes have been developed to solve hyperbolic PDEs. Cai and Wang [22] introduced acollocation method in which the derivatives are calculated in the wavelet space. Holmström [23] proposed an interpolatingwavelet collocation method in which all calculations are carried out in the physical space. However, in this method the finescale features might be missed [23]. A adaptive multiresolution WENO method are employed for simulation first orderhyperbolic PDEs, e.g. [24]; however as mentioned, these methods typically improved for first order hyperbolic PDEs withpiecewise smooth solutions containing some localized spatial discontinuities [18].

Regarding second order hyperbolic systems, some common approaches can directly be used to remedy above mentioneddrawbacks; some of which are: (1) dissipative time integration schemes [25,26]; (2) spatial smoothing (or spectral filtering)[27]. However, using such approaches do not necessarily lead to the smoothest possible solutions, essential for proper adap-tation. Other powerful approaches can be utilized which are basically developed for simulation of the first order hyperbolicsystems. Some of these schemes are: the high resolution methods and the discontinuous Galerkin schemes. The high-reso-lution methods are based on the finite difference and finite volume schemes; they use suitably defined numerical fluxes andslope limiters. Hence, the Gibb’s phenomenon can be controlled and discontinuous solutions can be simulated effectively.These methods are successfully used for simulation of hyperbolic systems on uniform and non-uniform (adaptive) grids/cells

Fig. 6. Influence of threshold values on error-smoothness relationship. Intensity of incorrectness is in accordance with threshold values.

Fig. 7. Effect of locally improved interpolation feature in estimation. (a) uniform smoothing case, (b) adaptive smoothing case.


[28,29]. Recently this approach is employed for simulation of the second order hyperbolic PDEs; for this purpose, the gov-erning equations are rewritten as a system of first order hyperbolic PDEs [30,31]. To incorporate the ideas of the propernumerical fluxes and slope limiters into the finite element context, the discontinuous Galerkin methods are successfullydeveloped; for a good review paper, see Cockburn et al. [32]. These schemes have been employed for both the first order[32] and second order [33] hyperbolic PDEs.

Regarding dissipation and dispersion phenomena in numerical simulations of second order hyperbolic PDEs, effects of theaforementioned two first approaches (the dissipative time integration scheme and spatial smoothing) and the proposedmethod in this work are investigated by a test problem. The proposed method is to smooth adaptively the oscillations bythe smoothing splines directly in non-uniform grids, an ill-posed problem. This simple and cost effective approach is indeedto apply the postprocessors such as low pass filtering schemes. Hence, first the solution is obtained with spurious oscillationsand then non-physical oscillations are reduced or dissipated [14,34]. The test problem is also wave propagation in a unitlength elastic bar; there two different cases are considered: a high gradient smooth solution and a discontinuous one.

Fig. 8. Functions mðxÞ, u0ðxÞ, and exact solution uxðx; t ¼ 0:2Þjexact and m0 ðx; t ¼ 0:2Þ; (a) artificial viscosity mðxÞ, (b) initial imposed displacement u0ðxÞ, (c)exact solution and corresponding artificial viscosity at t ¼ 0:2. In simulation it is assumed that c ¼ 1 and var ¼ h ¼ 0:04.

Fig. 9. Numerical results obtained by using artificial viscosity (left column) and smoothing spline based denoising method (right column). In eachillustration, numerical and exact solutions are denoted with dashed and solid lines, respectively.


Considering the spatially smoothing approach, the common adaptive procedures could be incorporated by the smoothingsplines with uniform weigh coefficients as denoiser of spurious oscillations from smooth high-gradient solutions [12]. Insuch method, however, errors were denoised uniformly throughout the solution domain. In this work, this constraint is re-

Fig. 10. Representation of smoothness ðkX2k2Þ against solution error ðkq2k2Þ for the proposed method, and implicit generalized-a time integration method.


duced by using spatially adaptive smoothing. This is acquired by using non-uniform weight coefficients which are propor-tional with density of adapted grid points.

Here, the smoothing splines, with uniform and non-uniform weight coefficients, are used to directly remove noises fromirregular data, a kind of ill-posed problem. This method of smoothing is fast [35], and stable – since it is a kind of the Tik-honov regularization method with semi-norm constraint [36]; hence the method is less sensitive to noise ratio [36]. Smooth-ing spline of degree 2m� 1, having continuous derivations up to 2m� 2th, works on the signal f with irregularly spaced data[36–38]. In addition, there are convergence and error bounds in estimating the function f (belonging to Sobolev space) and itscorresponding spatial derivatives for semi-regularly spaced information. These properties are still true even if f is subjectedto some random noises [39]. It is essential to point out that this method gives reliable results in the vicinity of boundaries[40].

Fig. 11. Representation of smoothness ðkX2k2Þ against solution error ðkq2k2Þ for the proposed method, and implicit generalized-a time integration method.

Fig. 12. Adapted grid points and corresponding solutions at different times for p ¼ 0:8, �a ¼ 0 & e ¼ 10�5; (a, b) at time 0.336; (c, d) at time 0.615; (e, f) attime 0.8265.


The smoothing splines with uniform coefficients, in general, are comparable with wavelet based soft threshold denoisingmethod [36]; the former one is a linear scheme while the latter is a nonlinear one. However, fitting properties of smoothingsplines can be improved using variable parameters [41,42]. Here, non-uniform weight coefficients are used to enrich inter-polation properties; thereby controllable dissipations (adaptive estimations) are achieved in the spatial domain. This featureleads to enhanced solutions considering both accuracy and smoothness.

For adaptation the interpolating wavelet transform is used in locally uniform grid points. Wavelet schemes, indeed, aresuccessfully developed and implemented for denoising piecewise continuous data; however most of the corresponding algo-rithms (first generation wavelets, in particular) are established for uniformly sampled data. Second generation wavelets rem-edy this drawback; i.e., capability to process a data directly in irregular grid points [43]. Regarding irregular sampled data,however, reconstruction of thresholded coefficients in common second generation wavelet transforms, may cause consider-able bias, instability in the smoothing procedure (in contrast to the first generation wavelets). This is due the fact that, insuch wavelets small detail coefficients may contain important information [44]. Thereby, in such wavelet transforms stabi-lizing algorithms should be considered [44].

To assess goodness of fitting criterion and smoothing effect, several computational methods – mean square error (MSE),degrees of freedom, and (generalized) cross validation scheme, are developed. However, the results may be unreliable [40].In the smoothing spline case, the methods existed for choosing the smoothing parameters mostly end to over [35] or under[45] smoothing results. Therefore, proper choosing of these parameters should be reviewed. In this regard the trial and errormethod could be used to capture optimum parameter ranges. In this work, smoothness and accuracy in the estimation areinvestigated by the L-curve scheme [46]; the effective parameters in estimations, considered here, are: smoothing parame-ter, smoothing weights, and density of concentrated adapted grid points. It should be noted that estimation on a non-uni-form grid points is essentially a kind of ill-posed problem.

The basis of the proposed scheme is to approximate spatial derivatives directly in the irregular grid points. For this pur-pose the fast and recursive Lagrange polynomials, introduced by Fornberg, is used [47]. Anti-symmetric end padding methodis used to reduce edge effects from the derivatives in 2D simulations. In this method, the second derivatives at end points arezero, like in the relaxed cubic spline one [48]. Therefore, the moving fronts are propagated with constant speed. By end pad-ding a signal, the centered difference formulas, causing less fluctuation than one-sided ones, are used to calculate the deriv-atives at the edge points [12]. Semi-discretized PDEs in spatial domain are solved in time by an explicit integration method(for example 4th Runge–Kutta scheme). The grid points are adapted by the Dubuc–Deslauriers (D–D) interpolating wavelet[49,50], and the noises are reduced by the smoothing splines with constant or non-uniform parameters. As all calculationsare done in the physical domain, the proposed method is simple and conceptually straightforward.

Fig. 13. Numerical and modal solutions of the vibrating membrane. (a) The modal solution and numerical solution for e ¼ 10�5; (a) solutions at spatiallocation ð0:5;0:5Þ; (b) errors in numerical solutions.


This paper is composed of: Section 2 the main concept explanation of multiresolution-based grid adaptation by interpo-lating wavelets; Section 3 a short review of inherent dissipation property of PDEs; Section 4 the wavelet-based adaptive-gridmethod for PDEs solution; Section 5 the issues related to smoothing splines with uniform and non-uniform weight coeffi-cients; Section 6 a brief survey about the ill-posed problems and regularization schemes; Section 7 the concept of the L-curveand smoothness assessing with this method; Section 8, presenting a benchmark; there two cases are studied: a high gradientsmooth solution and one having discontinuities. The results are then compared with those of other common resolving ap-proaches regarding smoothness and accuracy of solutions; Section 9, the examples. The conclusion of this research is pre-sented at the end of the paper.

In wavelet theory context, the complete multiresolution property, fast algorithms and data compression ability [49] makewavelets appealing for the numerical solution of PDEs. In multiresolution analysis, each wavelet coefficient (detail or scale) islinked to a particular point of underlying grid. This distinctive property is incorporated with compression power of the wave-lets and therefore a uniform grid can be adapted by grid reduction technique. In this method a simple criterion is applied in1D grid based on the magnitude of corresponding wavelet coefficients. The existing odd grid points at level j should be re-moved if their corresponding detail coefficients are smaller than predefined threshold (e); wavelet coefficients and gridpoints have one-to-one correspondence [6]. The above mentioned reduction technique can easily be extended to 2D gridpoints. In this research, the scheme explained in [9] is implemented for 2D grids adaptation. Here, Dubuc–Deslauriers (D–D) interpolating wavelets, auto-correlation of Daubechies scaling functions [49], are used for grid adaptation.

Here, multiresolution representation of 1D data by D–D wavelets is briefly reviewed. It should be mentioned that usingsuch wavelet is resulted to very simple and effective algorithms which are physically meaningful [6]. The main idea of pro-cedure is briefly presented in the following.

Fig. 14. Number of grid points used in adaptation procedure for parameters p ¼ 0:8 & e ¼ 10�5.


Several points are assumed in a dyadic grid and mentioned as follows:

Vj ¼ xj;k 2 R : xj;k ¼ k=2jn o

; j 2 Z; k 2 0;1; � � � ;2jn o

ð1Þ

where, j is resolution level (corresponding to scale 1/2j) and k is the spatial position. Such definition of dyadic grid points Vj isended to the xj�1;k ¼ xj;2k condition and the multiresolution representation core: i.e., Vj�1 � Vj.

A function f(x), defined in VJmax , is assumed (i.e., x 2 VJmax ). Regarding the context of multi-resolution representation, it ispossible to show that any continuous function f(x) can be described as follows [6,49]:

f ðxÞ ¼X2Jmin

l¼0

cJmin ;l:uJmin ;lðxÞ þ

XJmax�1

j¼Jmin

X2j

n¼0

dj;n:wj;nðxÞ ¼ PfJminþXJmax�1

Jmin

Qfj ð2Þ

where, uðxÞ and wðxÞ are scaling and wavelet functions, respectively; uj;kðxÞ and wj;kðxÞ are the dilated and shifted versions ofuðxÞ and wðxÞ, respectively, i.e., uj;kðxÞ ¼ uð2jx� kÞ, wj;kðxÞ ¼ wð2jx� kÞ; cj;k and dj;k are approximation and detail coefficients,respectively which are defined as:

Fig. 15. Schematic plot of domain of simulation. Initial imposed deformation is subjected at point r1.

Fig. 16. The number of grid points (ng) used in adaptation algorithm during simulations for different values of thresholds and smoothing parameters, p incase fWjg ¼ 1; number of grid points in the uniform case is 49,407.


cj;l ¼Z 1

�1f ðxÞ � ~uj;lðxÞdx; dj;n ¼

Z 1

�1f ðxÞ � ~wj;nðxÞdx ð3Þ

where, ~uj;lðxÞ and ~wj;nðxÞ are dual function of uj;lðxÞ and wj;nðxÞ, respectively, in bi-orthogonal wavelet systems. In case oforthogonal wavelets, ~uj;lðxÞ ¼ uj;lðxÞ and ~wj;nðxÞ ¼ wj;nðxÞ.

Pfj ¼P2j

l¼0cj;l:uj;lðxÞ shows the approximation of f ðxÞ, defined in grid points Vj, and Qfj ¼P2j

n¼0dj;n:wj;nðxÞ shows the detailsthat should be added to Pfj for obtaining Pfjþ1 (approximation of f ðxÞ in Vjþ1).

In order to calculate the approximation and detail coefficients, alternative other procedures can be provided in D–D wave-let system other than using Eq. (3). The approximation coefficients (cJmin ;k) are equal to sampled values of f ðxÞ at points xJmin ;k

(the points included in VJmin ), i.e., cJmin ;l ¼ f ðxJmin ;lÞ [6,49], by considering the interpolating properties of D–D wavelets (i.e.,uðkÞ ¼ dk0 8k 2 Z).

The detail coefficients are evaluated using multi-resolution analysis and interpolation property of wavelet function;namely, due to the properties VJmin

� � � � � Vj � Vjþ1 � � � � � VJmaxand uðn� kÞ ¼ dn;k. Due to the facts that Pfjðxj;kÞ ¼ f ðxj;kÞ,

and xj;k ¼ xjþ1;2k, then it is clear that:

Pfjðxj;kÞ ¼ Pfjþ1ðxjþ1;2kÞ ¼ f ðxjþ1;2kÞ

For odd-numbered grid points xjþ1;2kþ1 including in Vjþ1, in general, pfjþ1ðxjþ1;2kþ1Þ – f ðxjþ1;2kþ1Þ. The differences betweenf ðxjþ1;2kþ1Þ and Pfjþ1ðxjþ1;2kþ1Þ are measured as the magnitudes of detail coefficients. Thereby, for calculating detail coefficientsin j resolution, at first the values of odd-numbered grid points, included in Vjþ1 (i.e., points xjþ1;2kþ1), are estimated by locallyLagrange interpolation by the known even-numbered grid points in Vjþ1 (i.e., the points belong to Vj and Vjþ1). Regardinglocally Lagrange interpolation, in D–D wavelet case of order 2M � 1, 2M most neighbor points, including in Vj, are selectedin the vicinity of xjþ1;2kþ1 point, and are presented as:

xjþ1;2k�2n

� �n 2 f�M þ 1;�M þ 2; . . . ;Mg

Using this set, the estimation at the point xjþ1;2kþ1 is denoted by Pfjþ1ðxjþ1;2nþ1Þ. The detail coefficients are simply the dif-ferences between known values of f ðxÞ at odd-numbered grid points included in Vjþ1 (i.e., f ðð2nþ 1Þ=2jþ1Þ) and Pfjþ1ðxjþ1;2nþ1Þ[6], viz.:

dj;n ¼ f ðxjþ1;2nþ1Þ � Pfjþ1ðxjþ1;2nþ1Þ ð4Þ

Schematic shape of detail coefficient calculation is illustrated in Fig. 1 for case M ¼ 2.For adaptation a grid, in each level of resolution j 2 Jmin; Jmin þ 1; � � � ; Jmax � 1f g where f ðxÞ 2 VJmax

, odd-numbered pointsxjþ1;2nþ1 are omitted from the calculating grid points, if corresponding dj;n satisfy the condition dj;n � e.

The boundary wavelets, introduced by Donoho [51], are used in the vicinity of edge points in case of finite grid points.The already mentioned strategy was shown to be efficient in the resolution of single equation system. For resolution of

PDE in vector system, the previous procedure is modified to reflect the behavior of the solutions of all equations. In otherwords, the resultant adapted grid is simply the superposition of all adapted grids.

Fig. 17. Solutions observed at receivers r1 (top figure) and r2 (bottom figure), with different values of thresholds and smoothing parameters in casefWjg ¼ 1.


2. Surveying the inherent dissipation of PDEs

The diffusion state is expressed by the below parabolic system:

@u=@t ¼ @2u=@x2; ðx; tÞ 2 ½0;1 � ½0;1Þ ð5Þ

where, u and v are both the solutions of Eq. (5) having the identical boundary conditions and different initial values. Theparameter WðtÞ, defined bellow, is to assess the differences between the above mentioned solutions:

WðtÞ ¼ 12

Z 1

0uðx; tÞ � vðx; tÞð Þ2dx ð6Þ

The timely non-increasing property of WðtÞ can easily be shown using the integration by parts, [52]:

dWðtÞ=dt ¼Z t

0uðx; tÞ � vðx; tÞð Þ @uðx; tÞ=@t � @vðx; tÞ=@tð Þdx ¼ �

Z t

0@uðx; tÞ=@x� @vðx; tÞ=@xð Þ2dx � 0 ð7Þ

Fig. 18. Snapshots of solutions at 0.352, 0.535, 0.742, 0.97, 1.084, and 1.3 s, illustrated in figures (a)–(f), respectively.


According to this formula, small errors, eðx; tÞ ¼ vðx; tÞ � uðx; tÞ, could not develop and they dissipate in time [52]; there-fore, the stable numerical computations are obtainable.

These errors may cause erroneous adapted grid points in the hyperbolic cases in comparison with the parabolic equations.As an example, a square membrane subjected to an initial imposed deformation is considered; the governing equations are:

PDE@2uðx; z; tÞ

@x2 þ @2uðx; z; tÞ@z2

!¼ @

2uðx; z; tÞ@t2 ; X 2 ½0;1 � ½0;1 0 � t <1

ICs uðx; z;0Þ ¼ Uðx; zÞ; _uðx; z; 0Þ ¼ 0BCs uð0; z; tÞ ¼ uð1; z; tÞ ¼ uðx;0; tÞ ¼ uðx;1; tÞ ¼ 0

ð8Þ

where, Uðx; zÞ ¼ expð�1000:ððx� 0:5Þ2 þ ðz� 0:5Þ2ÞÞ. This example is analyzed by the method of lines scheme with uniformgrid of 257� 257 points and time step of size dt ¼ 0:0043. Finally at t ¼ 0:35 s the uniform grid is adapted by the D–D wave-let of order 3 with threshold value e ¼ 0:3� 10�6; the results are shown in Fig. 2. There the solution is illustrated in Fig. 2(a),and back ground grid and corresponding adapted grid points are shown in Figs. 2(b)–(e) for different levels of resolution. Theresultant adapted grid, Fig. 2(f), is obtained by superposition of Fig. 2(b)–(e). The results show that noises are mainly con-centrated in the finest resolution with the highest frequency content. Applying ordinary adaptive grid-based methods, thesespurious oscillations may lead to improper adapted grid points.

3. Wavelet-based adaptive-grid method for solving PDEs

At the time step (t ¼ tn), if the solution of PDE is f ðx; tÞ, then the procedure for PDE solution in adaptive wavelet-basedframework is followed as:

1. Determining the grids, adapted by adaptive wavelet transform, using f ðx; tn�1Þ (step n � 1). The values of points withoutf ðx; tn�1Þ, are obtained by smoothing spline fitting method (Section 5).

2. Computing the spatial derivatives in the adapted grid using Fornberg fast and recursive scheme [47] improved by anti-symmetric end padding method [12]. In this regard, extra non-physical fluctuations deduced by one sided derivatives arereduced. Here, five points are locally chosen to calculate derivatives and therefore a high-order numerical scheme isachieved [2,53].

3. Discretizing PDEs in spatial domain and then solving semi-discrete systems which are discrete in space and continuous intime. To solve ODEs at the time t ¼ tn, the standard time-stepping methods such as Runge–Kutta schemes can be used.

Fig. 19. Snapshots of adapted grid points at 0.352, 0.535, 0.742, 0.97, 1.084, and 1.3 s, illustrated in figures (a)–(f), respectively; where e ¼ 10�5.


4. Denoising the spurious oscillations directly performed in non-uniform grid by smoothing splines (Section 5).5. Repeating the steps from the beginning.

In practice, the grid is not adapted at each time step regarding computation time. For 1D data of length n, smoothingspline of 2m � 1 degree, needs m2:n operations [35], and a wavelet transform (employing pyramidal algorithm) uses n oper-ations. Therefore both procedures are fast and effective. In fact, it is adapted after several time steps (e.g. 10–20 steps) basedon the velocity of moving fronts. These moving fronts are captured by adding some extra points to the fronts of adapted gridat each resolution level (e.g. 2 points to each end at each level).

4. Smoothing splines

4.1. General definitions

The noisy data are recommended not to be fitted exactly, causing significant distortion particularly in the estimation ofderivatives. Adaptive solutions, in particular, are sensitive even to small noises (noises may lead to erroneous adapted grid).The smoothing fit is used to remove noisy components in a signal; therefore, interpolation constraint is relaxed. In this re-search, the spurious oscillations are denoised by smoothing spline directly from the irregular sampled points throughout thesolution.

The discrete values of n observations yj ¼ yðxjÞ where j ¼ 1;2; . . . ;n and x1 < x2 < � � � < xn are assumed in order to deter-mine a function f ðxÞ, that yj ¼ f ðxjÞ þ ej. ej are random, uncorrelated errors with zero mean and variance r2

j . Here, f ðxÞ is thesmoothest possible function in fitting the observations to a specific tolerance. It is well known that the solution to this prob-lem is minimizer, f ðxÞ, of the functional [37,38]:

Q ¼Xn

j¼1

Wjjyj � f ðxjÞj2 þ kZjðdmf ðxÞ=dxmÞj2dx; 0 � k <1 ð9-aÞ

where k is a Lagrangian parameter (known as the smoothing parameter), Wj is a weight factor at point xj and m is the deriv-ative order. To remap range ½0;1Þ of k to [0,1], it is assumed k ¼ ð1� pÞ=p; by substituting this in Eq. (9-a), this equation canbe rewritten as:

Fig. 20. Snapshots of adapted grid points at 0.352, 0.535, 0.742, 0.97, 1.084, and 1.3 s, illustrated in figures (a)–(f), respectively; where e ¼ 10�6.

Fig. 21.case W


Q ¼ pXn

j¼1

Wjjyj � f ðxjÞj2 þ ð1� pÞZjðdmf ðxÞ=dxmÞj2dx; 0 � p � 1 ð9-bÞ

It can be shown that spline of degree k ¼ 2m� 1, having 2m� 2 continuous derivatives, is an optimal solution for Eq. (9);where, n � 2m [37,38]. In this paper the cubic smoothing spline is chosen to have a minimum curvature property; hence,m ¼ 2 (2m� 1 ¼ 3) and f 2 C2½x1; xn [36].

According to Eq. (9), the natural cubic spline interpolation is obtained by p ¼ 1 and the least-squares straight line fit byp ¼ 0. As p decreases the interpolating property is vanished while the smoothing property is increased. In Eq. (9), the errorsare measured by the summation and the roughness by the integral. Therefore, the smoothness and accuracy are obtained

The number of grid points (ng) used in adaptation algorithm during simulations for different values of thresholds and smoothing parameters, p inj � 1; number of grid points in the uniform case is 49,407.

Fig. 22. Solutions observed at receivers r1 and r2 for different values of (�a&p), in case e ¼ 10�5.


simultaneously. In the mentioned equations, the trade-off between smoothness and goodness of fit to the data is controlledby the smoothing parameter.

In this research, effects of variable weight coefficients in interpolating feature are studied for different p values. Thesecoefficients are concentrated with large values around high gradient zones. Here, weight coefficients are chosen inverselyproportional to density of adapted grid points. The N + 1 points uniformly sampled by the function ff ðxÞjx 2 ½x0; x0 þ Dxgat uniform grid fxig, i ¼ 1; . . . ;N þ 1, is considered. It is assumed that after adaption, mþ 1 (m � N) adapted points are ob-tained; i.e., points fxjg : j ¼ 1; . . . ;mþ 1. In this case, Wj can be defined as follows:

Wj ¼ 1þ�a

xjþ1�xj�12

� �NDx

; �a � 0; 2 � j � m� 1

W1 ¼ 1þ�a

ðx2 � x1Þ NDx

; Wm ¼ 1þ�a

ðxm � xm�1Þ NDx

; �a � 0ð10Þ

where, �a is an amplification factor. For the case �a ¼ 0 (fWjg ¼ 1), the uniform weight condition is obtained. It is recom-mended to replace the weight Wj by its average value obtained by averaging each Wj with its neighborhood coefficients;e.g. Wmodified

j ¼ ðP2

i¼�2WjþiÞ=5. This leads to more uniform variation of weigh coefficients. It should be noted that finite datacan firstly be extended by the mirror end padding scheme for edge coefficients and then corresponding edge weights can beevaluated.

Fig. 23. Solutions observed at receivers r1 and r2 for values {p ¼ 0:5&�a ¼ 3} and {p ¼ 0:7&�a ¼ 0} in the case e ¼ 10�5.


4.2. Test problems

In this subsection, at first, a test function (gðxÞ) and corresponding weigh coefficients obtained on an adapted grid is pre-sented; this is to confirm that the weights are properly concentrated around high gradient zones. After that, rates of conver-gence in estimations are investigated for different p and �a values.


To control proper concentration of weigh coefficients on an irregular grid, a test function gðxÞ is considered. The test func-tion, gðxÞ, its corresponding adapted grid points and weight values (fWjg), are shown in Fig. 3; there, it is assumed �a ¼ 3 ande ¼ 10�5. The function gðxÞ is defined as:

gðxÞ ¼ f ðx; x2; x3Þ þ f ðx; x1; x2Þx� x1

x2 � x1þ f ðx; x3; x4Þ 1� x� x3

x4 � x3

� �ð11Þ

where, f ðx; a; bÞ ¼ 1� H ðx� aÞðx� bÞð Þ, HðxÞ is the Heaviside step function, x1 ¼ 0:44, x2 ¼ 0:46, x3 ¼ 0:54 and x4 ¼ 0:56. Theresults confirm that the adapted grid points, the dots in Fig. 3(a), and their associated weight coefficients, Fig. 3(b), are appro-priately concentrated in the vicinity of high gradient zones to improve interpolating property.

In the following, the convergence in estimation of both a smooth but high gradient function (f ðxÞ) and its correspondingfirst derivative are studied; considered norm senses are: L2, Sobolev and L1. All calculations are carried out by the cubicsmoothing spline (i.e., m ¼ 2). The test function is:

f ðxÞ ¼ sinð2pxÞ þ expð��bðx� 0:5Þ2Þ ð12Þ

where, �b ¼ 104 and 0 � x � 1. Errors in estimations of the test function are shown in Fig. 4(a)–(c). In calculations, it is as-sumed that fWjg ¼ 1, i.e., �a ¼ 0. In all figures, the results are plotted against the number of significant adapted points(Ns), obtained by adaptive interpolating wavelet transform (D–D wavelet of order 3).

It is obvious that the error in estimations are decreased in all senses by increasing the Ns & p values. However, the con-vergence rate is approximately independent for Ns of large values and only depends on the p values.

The results imply that if p! 1, then the convergence rate increases, Ragozin [39]; if p! 1, then the over smoothing effectis bolded especially in the derivative with Ns of large values (Sobolev norm kek1;2). Furthermore, for small ps, the errors inestimations are less sensitive to Ns of large values.

The effects of adaptive estimation, obtained by non-uniform weight coefficients, are studied and presented in Fig. 5. Theresults indicate that the accuracy increases in accordance with �a for certain p values. For example, two different parametersof adaptive and non-adaptive estimations {p ¼ 0:7 & �a ¼ 6} and {p ¼ 0:9 & �a ¼ 0} are considered, respectively.

The estimation accuracy is measured by kek2 and kek1 senses and the smoothness by kek1;2 for both adaptive and non-adaptive estimations. The adaptive estimation (Fig. 5) shows higher accuracy and smoothness in comparison with non-adap-tive one (Fig. 4).

The results obtained by the parameters, {p ¼ 0:8&�a ¼ 0} and {p ¼ 0:7&�a ¼ 1} can also be compared as mentioned above.In approximation, proper values of p are 0.75–0.95 in case of fWjg ¼ 1, according to the results. The lower values of p are

applicable for non-uniformly weighed data, i.e., Wj � 1.

4.3. Implementation algorithm

Considering the finite difference approach, the order-m derivative of discrete values f ðxiÞ; i ¼ 1;2; . . . ;N can be approxi-mated as:

Fig. 24. Schematic plot of a half space domain. Initial imposed deformation is subjected at point r1.

Fig. 25. Snapshots of solution ux at 0.16, 0.4, 0.646, 0.802, 0.928, and 1.174 s, illustrated in figures (a)–(f), respectively.


fðmÞ DðmÞ � f ð13Þ

From statistical and signal processing perspectives, the DðmÞ coefficients are usually evaluated by considering uniform dis-tribution of f ðxiÞ (regardless of the spacing of xi). In this regard, the first and second derivative matrixes can be approximatedas [46]:

Dð1Þ ¼ 1Dx1

1 �1. .

. . ..

1 �1

0B@

1CA 2 RðN�1Þ�N ð14Þ

and

Dð2Þ ¼ 1Dx2

1 �2 1. .

. . .. . .

.

1 �2 1

0B@

1CA 2 RðN�2Þ�N ð15Þ

where Dx ¼ xiþ1 � xi is the spatial uniform sampling step.The integral of a squared integrand can also be approximated as [54,55]:
Z xN
x1

f ðxÞj j2 dx fT:B:f ð16Þ

where B is the midpoint rule integration matrix, which can be evaluated as:
Z xN
x1

jf ðxÞj2 dx 12ð�x1 þ x2Þf 2

1 þXN�1

i¼2

12ð�xi�1 þ xiþ1Þf 2

i þ12ð�xN�1 þ xNÞf 2

N ð17Þ

or equivalently:

Z xN

x1

jf ðxÞj2 dx 12

f1

..

.

fN

2664

3775

T

�

�x1 þ x2 00 �x1 þ x3 0

. ..

0 �xN�2 þ xN 00 �xN�1 þ xN

0BBBBBBB@

1CCCCCCCA�

f1

..

.

fN

2664

3775 ð18Þ

Fig. 26. Snapshots of solution ux at 0.16, 0.4, 0.646, 0.802, 0.928, and 1.174 s, illustrated in figures (a)–(f), respectively.

Fig. 27. Snapshots of adapted grids correspond to the solutions ux and uz at 0.16, 0.4, 0.646, 0.802, 0.928 and 1.174 s, illustrated in figures (a)–(f),respectively.



or
Z xN
x1

f ðxÞj j2 dx fT:B:f ð19Þ

If the distribution is assumed to be uniform, then B Dx I. In general, Dx in Eqs. (13) and (19) can be absorbed by theregularization parameter k, and it is ignored in calculations [54,55].

Considering above mentioned algebra based formulations, the functional Q (Eq. (9-a) where Wj ¼ 1) can be rewritten indiscrete form as [54,55]:

Q ¼ ðy � fÞT :B:ðy � fÞ þ k ðDðmÞfÞT :~B:ðDðmÞfÞ ð20Þ

In the above equation, size of B is N � N, and if the order of derivative is m, then size of ~B must be ðN �mÞ � ðN �mÞ. ~B canbe considered as a subset of B, where both rows and columns from the beginning and end are properly dropped. Consideringuniform distribution assumption (regardless of irregularity of xi), the functional Q can also be represented as:

Q ¼ ðy � fÞT :I:ðy � fÞ þ k ðDðmÞfÞT �~I � ðDðmÞfÞ ð21Þ

If each goodness-of-fit value jyi � fij2 is weighted by a coefficient Wi, then the functional Q can be rewritten as:

Q ¼ ðy � fÞT :W:ðy � fÞ þ k ðDðmÞfÞT �~I � ðDðmÞfÞ ð22Þ

In the above equation, W is a diagonal weighting matrix where Wði; iÞ ¼Wi (see Eq. (9)); size of this matrix is: N � N.To find the solution of Eq. (22), the Q is minimized with respect to f:

@Q@f¼ 2W:ðy � fÞ þ 2k ðDðmÞÞT �~I � ðDðmÞfÞ ¼ 0 ð23Þ

Hence the minimizing (regularized) solution is [54,55]:

f ¼ Wþ k ðDðmÞÞT :~I:DðmÞn o�1

W � yð Þ ð24Þ

From the above equation, it is clear that since the weighing matrix W is a diagonal matrix, then the solution algorithmremains fast and effective even for non-uniform weighting cases (i.e., W–I). It should be mentioned that in the above equa-tion, the inverse of matrix exists always [46,56], or equivalently the regularized solution can be obtained in all situations.

5. Surveying regularization methods

According to the Hadamard classic definition, a problem is ill-posed (or incorrect) if at least one of the following three con-ditions is not satisfied:

1. The solution exists.2. The solution is unique.3. The solution is stable, in other words, arbitrarily perturbation of coefficients, parameters, initial or boundary conditions

lead to arbitrarily small solution changes [56].

Ill-posed problems could be solved by special (stable, or regular) schemes, called regularization methods. They incorpo-rate further information about desired solutions to stabilize them. Hence, an ill-posed problem is replaced by a close well-posed problem [46,56].

One of the most famous approaches to regular the ill-posed problem is to minimize the following functional:

minfq2ðf Þ þ k�X2ðf Þg ð25Þ

where f is the unknown function that would be estimated, q(f) denotes the residual norm and X(f) is the additional infor-mation, known as the smoothing norm. Furthermore, k is a regularization parameter that controls trade off between theresidual and smoothing norms. The already mentioned scheme is well known as the Tikhonov regularization method [46,56].

The smoothing splines, indeed, are a type of the Tikhonov regularization method [36], wherePn

j¼1Wjjyj � f ðxjÞj2 andRþ1�1 jf ðmÞðxÞj

2dx are the weighted residual norm and additional information (a semi norm), respectively. Consequentlysmoothing splines lead to stable estimations for irregular grid points, a kind of ill-posed problem. In the successive section,by taking regularization concept into consideration, the L-curve scheme is reviewed to assess the smoothing effects.

6. Assessment the smoothing effects by L-curve

As mentioned before, choosing smoothing parameters by the current computational methods are not reliable. Conse-quently, to investigate regularization effects, here, L-curve is used – a graphical representation [46]. In this context, residualnorm in estimation is presented against regularization (semi) norm, in log–log scale for all valid regularization parameters.


Hence trade-off between these two quantities is apparently exhibited. Indeed, this relationship is the heart of any regular-ization method.

The L-curve for Tikhonov regularization plays a central role in regard to other regularization methods. Any regularizedsolution, indeed, lies on or above this curve; namely, the Tikhonov regularization method leads to the smoothest resultfor a certain error in estimation [46].

Working within the smoothing splines framework, influence of effective parameters in estimations (i.e., e, p and �a) areinvestigated with respect to accuracy and smoothness. For this purpose, a smooth and high gradient test function is chosenas:

f ðxÞ ¼ expð�500ðx� 0:5Þ2Þ ð26Þ

At first, a non-adaptive estimation is considered (i.e., fWjg ¼ 1). In Fig. 6, the corresponding L-curves are shown, wheree ¼ f10�4;10�5;10�7g and 0:025 � p � 0:975. There, the additional information (XðxÞ) is the semi-norm of the second deriv-ative of the regularized solution (xreg), i.e., XðxregÞ ¼ kLxregk. In Fig. 6, the operator L (in the vertical axis) denotes the secondderivative operator of order 3-being obtained by differentiating the local Lagrange interpolation using 5 neighbor points. Forthe uniform grid case, it can be evaluated as:

f 00ðxiÞ �f ðxi�2Þ þ 16f ðxi�1Þ � 30f ðxiÞ þ 16f ðxiþ1Þ � f ðxiþ2Þ

12h2

where h is the spatial step.The results indicate that for thresholds of large enough values (e = 10�4), the associated L-curves approach from upward

to a unique L-curve (correspond to small thresholds). In other words, for large values of threshold, stronger ill-posed problemis formed. However, by decreasing threshold values, corresponding curves approach to a unique L-curve (see curves corre-spond to e ¼ f10�5;10�7g in Fig. 6). Besides, by increasing threshold values over-smoothing phenomenon is more bolded (seevertical branch of the L-curve for large values of e). This feature is clearly revealed in Fig. 7. In this figure, L-curves correspondto different thresholds with various local interpolating properties (controlled by �a values) are presented as 3D curves.Fig. 7(a) and (b) correspond to locally un-enhanced (�a ¼ 0) and enhanced (�a ¼ 4) interpolating properties in estimations.In this figure, the second derivative of order 3 (same as the Fig. 6) is used as the operator (L) of the additional condition.

From the results it is apparent that: (1) for e of large value, the corresponding L-curve approaches to over smoothing por-tion; (2) for a p value, there exists a critical e value (ecr) such that for smaller thresholds (e < ecr), the over smoothing phe-nomenon increases in accordance with e values; (3) for e < ecr , smoothing and residual norms are independent of e values;(4) for e � ecr , the residual norms decrease as p values increase. (5) By locally enhancing the interpolating properties, the oversmoothing phenomenon is reduced.

In fact, for e < ecr , the ill-posed problem turns to be a semi well-posed one. This is due to concentration of locally uniformgrid points in the vicinity of high-gradient zones; hence, the fitting results are not so sensitive as the cases where e � ecr .

7. Benchmark problems

In the following, performance of the proposed method and some other common approaches are studied with respect totheir dissipation and dispersion properties. The considered examples are: (I) a smooth but high-gradient solution, (II) a dis-continuous one.

In this regard, a wave propagation problem is considered in an elastic rod of unit length. It is assumed that, the rod is onlysubjected to an imposed initial displacement u0ðxÞ and an initial velocity _u0ðxÞ ¼ 0. For the cases I & II (i.e., smooth and dis-continuous solutions), a smooth and a discontinuous initial conditions are assumed, respectively. For the case I, the smoothinitial condition u0ðxÞ is:

u0ðxÞ ¼ Exp �1000ðx� 0:5Þ2�

; x 2 0;1½ ð27Þ

and for the case II is:

u0ðxÞ ¼1=f30ðx� 0:5Þ þ 1g2 for x � 0:5

�1=f30ðx� 0:5Þ � 1g2 for x < 0:5

(ð28Þ

where u0ðxÞ has a discontinuity at location x ¼ 0:5 and x 2 ½0;1. Besides it is assumed that in the all cases the wave velocityin rod is unit, i.e., c2 ¼ E=q ¼ 1; where E & q are the elastic modulus and density, respectively.

For comparison purpose, two other methods used commonly for resolution of high gradient/discontinuous solutions areconsidered. They are: (1) adding artificial viscosities around discontinuities in the numerical simulations; (2) employingalgorithmic dissipative time integration methods to filter out high frequency components of solutions. For the case 1, theKelvin–Voigt visco-elastic solid is considered for simulation of discontinuous solutions. In the case 2, the implicit general-ized-a temporal integration method is used. In this case, both smooth high-gradient and discontinuous solutions are inves-tigated. In each case, the results are compared with the regularized based (the Tikhonov method) outputs.

Fig. 28. Number of grids points, ng, used by adaptation algorithm during analyses for different values of p in case: fWjg ¼ 1 and e ¼ 10�5; here number ofgrid points in the uniform case is 65,536.


7.1. Performance of the artificial viscosity in discontinuous solutions

To control spurious oscillations in second order hyperbolic PDEs, one of the common approaches is to use the artificialviscosities around discontinuities [57–59]. For viscoelastic solids the ordinary choice is the Kelvin–Voigt type viscosity[58,59]. The linearized continuum equations of motion for an isotropic Kelvin–Voigt viscoelastic solid are [58]:

rij ¼ kþ lð Þ uk;k þ m _uk;k

� �dij þ l ui;j þ m _ui;j þ uj;i þ m _uj;i

� �;

q€ui ¼ rij;jð29Þ

where: r is stress in solid; u is displacement vector; q is density; k and l are the Lamé constants; m denotes the viscosity; dij

is the Kronecker delta.Artificial viscosity v is a function of spatial domain (m ¼ mðxÞ) and is defined locally around discontinuity; its midpoint is

almost centered at discontinuous point. The function m could be either constant or variable (such as the Gaussian function).

Fig. 29. Comparison of solutions at receiver r1, with different values of smoothing parameter for case fWjg ¼ 1 and e ¼ 10�5.


The latter case, in general, leads to better results than the constant viscosity [60]. In Eq. (29), midpoint of the artificial vis-cosity is continuously concentrated around discontinuities during numerical simulation. Namely, the local viscosities prop-agate with speed c in each direction; this propagating viscosity is denoted by function m0ðx; tÞ.

In the following, the wave equation is numerically simulated by the Kelvin–Voigt viscosity approach; the results are thencompared with the regularized based method (the Thikonov based regularization obtained by the cubic smoothing spline).

For the Kelvin–Voigt visco-elastic solid, the equation of motion is:

Fig. 3

q €ux ¼ ðkþ lÞ ux;xx þ m0ðx; tÞ _ux;xx þ m00ðx; tÞ _ux;x� �

; x 2 ½0;1ICs : uxðx; t ¼ 0Þ ¼ u0ðxÞ; _uxðx; t ¼ 0Þ ¼ 0BCs : uxð0; tÞ ¼ uxð1; tÞ ¼ 0

ð30Þ

The function m0ðx; tÞ is assumed to be: m0ðx; tÞ ¼ 0:5� ðmðx� ctÞ þ mðxþ ctÞÞ. The function mðxÞ is assumed to be the Gauss-ian function [60], viz.:

mðxÞ ¼ mmax Exp �0:5x� lvar

� 2� �

ð31Þ

where, l and var is center and variance of function mðxÞ, respectively. l represents the discontinuity location; considering thediscontinuous initial condition u0ðxÞ, it is clear that l ¼ 0:5. Functions mðxÞ, u0ðxÞ, uxðx; t ¼ 0:2Þjexact (the exact solution) andm0ðx; t ¼ 0:2Þ are illustrated in Fig. 8. There, u0ðxÞ corresponds to a discontinuous solution and it is assumed that c ¼ 1 andvar ¼ 0:04.

The comparison of numerical results is shown in Fig. 9 for different p and mmax values. In the numerical simulations, it isassumed that: dx ¼ 1=256 (a uniform grid with 256 grid points), c ¼ 1 (speed of propagating wave), var ¼ 0:04 (variance ofartificial viscosity). Temporal integration is done by 4th order Runge–Kutta time-integration method, where dt ¼ 0:001. InFig. 9, in each illustration, numerical and exact solutions are presented for x 2 ½0:25;0:5 at time t ¼ 0:2. There, in eachrow, p and mmax values are selected in such a way that their corresponding solutions have the same maximum values.

The results indicated that:

(1) Without reducing spurious oscillations, instability in the solutions are formed, Fig. 9(a) and (b).(2) Existence of a very small amount of the spline based smoothing, leads to a very smoother solution than those of the

artificial viscosity based one (Fig. 9(c) and (d)).(3) In the all cases, the smoothing spline based method leads to more smoothed results than those of the Kelvin–Voigt

viscoelastic approach.(4) For large values of smoothing, the smoothing splines based method leads to more numerical dissipation than those of

the artificial viscosity based scheme (Fig. 9(g) and (h)).

7.2. Performance of the implicit generalized-a time integration method for smooth high-gradient/discontinuous solutions

To use this scheme, by using the finite element method, the wave equation is converted to form:

0. Schematic shape of an infinite media containing a square tunnel subjected to SH waves. Initial imposed deformation is subjected at point S.

Fig. 31. Snapshots of solution uy at 0.07185, 0.17985, 0.26985, 0.35986, 0.44985, 0.61185 s, illustrated in figures (a)–(f), respectively.


M€uþ Ku ¼ F; uð0Þ ¼ d & _uð0Þ ¼ v ð32Þ

then at time step n + 1 the family of generalized-a time integration algorithms are [25]:

Manþ1 þ ð1þ aÞ Kdnþ1 � aKdn ¼ Fnþ1

dnþ1 ¼ dn þ Dt vn þ Dt2 0:5� bð Þan þ banþ1½ vnþ1 ¼ vn þ Dt 1� cð Þan þ canþ1½ d0 ¼ d & v0 ¼ v & a0 ¼M�1 F0 � Kd0ð Þ

ð33Þ

where a, b and c are free parameters which control the stability and numerical dissipation of the algorithm. For case c > 0:5numerical dissipation is present; and for b � 0:25ðcþ 0:5Þ2 the mentioned algorithm is unconditionally stable. Here it is as-sumed that: c ¼ 0:6 and b ¼ 0:25 cþ 0:5ð Þ2 [25].

In the finite element formulation, to minimize Gibbs phenomenon, elements of the linear shape functions are used. Num-ber of equal length elements are 255 (to be comparable with the previous solution); hence, more than ten elements are usedper wave-length. To minimize oscillations in numerical solutions, it is assumed Dt Le=C, where Le is the element length andc is the wave velocity (here Dt ¼ 0:80� fðLe ¼ 1=255Þ=ðc ¼ 1Þg ¼ 0:003) [57].

The performance of the two method, regarding smoothness and error in estimation, is presented in Fig. 10 at t ¼ 0:2 s.There, smoothness (the regularization additional condition) and error in estimation (residual error) are measured bykX2k2 and kq2k2, respectively. The condition kX2k2 measures the semi-norm of the second derivative; i.e., at time t ¼ tn itis: kX2k2 ¼

RjðdnÞð2Þj2dx. In this formula, it is assumed that the derivative has third order accuracy (obtained by the Lagrange

interpolation of five neighbor points, as mentioned before). The residual error kq2k2 at t ¼ tn can also be computed by:Pmj¼1jðdnÞj � ðuexact

n Þjj2, where uexact

n represents the exact solution (being obtained by the D’Alembert solution) at t ¼ tn andthe symbol j denotes the spatial positions.

Illustrations (a) & (b) are for implicit generalized-a method, and figures (c) & (d) are for the Tikhonov based method. In allillustrations the solid and dashed lines correspond to the exact and numerical solutions, respectively. The results indicatethat:

(1) Spurious oscillations of numerical solutions are approximately comparable with each other.(2) While in the proposed method (the Tikhonov based method) a strong form formulation is used, it leads to smoother

results (in the whole of the domain) than those of the finite element method with linear shape functions, a weak formformulation.

(3) In the proposed method, the solution u has the feature u 2 C2, while in the finite element formulation u 2 C0.

Fig. 32. Snapshots of adapted grids correspond to the solutions uy at 0.07185, 0.17985, 0.26985, 0.35986, 0.44985, 0.61185 s, illustrated in figures (a)–(f),respectively.


Corresponding L-curve is presented in Fig. 11 at t ¼ 0:2; definitions of the norms kX2k2 and kq2k2, are same as the Fig. 10.The results indicate that the cubic smoothing spline based method leads to more accurate and smoother results than

those of the dissipative time integration scheme.

8. Numerical examples

The following examples are to study the effectiveness of the proposed method concerning some phenomena in elastody-namic problems. Here, two kinds of waves, scalar and vector, are modeled. The scalar wave propagation is modeled in asquare and a L-shaped membrane and a square tunnel in the infinite domain subjected to SH waves. Propagation of theP–SV waves (systems of PDEs) are simulated for two cases: half space media, and a semi-infinite domain with a rectangularvalley.

The main assumptions in numerical simulations are: (1) applying D–D interpolating wavelet of order 3; (2) decomposingthe grid (sampled at 1=28 step in the finest resolution) in three levels; (3) repeating re-adaptation and smoothing processesevery ten time steps.

Example 1. In this example, vibration of a rectangular membrane with four fixed sides subjected to an initial imposeddisplacement is presented, where the solution satisfies

PDE c20 u;xx þ u;yy� �

¼ u;tt ; X 2 ½0;1 � ½0;1 0 � t <1

ICs uðx; y;0Þ ¼ Uðx; yÞ; _uðx; y; 0Þ ¼ Vðx; yÞ ð34Þ

BCs uð0; y; tÞ ¼ uð1; y; tÞ ¼ 0; uðx;0; tÞ ¼ uðx;1; tÞ ¼ 0

Numerical solutions are compared with the modal analysis, with mode shapes mnm ¼ sinðmpxÞ: sinðnpyÞ.The parameters used for the simulation are: c0 ¼ 1, Vðx; yÞ ¼ 0, e ¼ 10�5, dt ¼ 0:00015, p ¼ 0:8, fWjg � 1, m ¼ 3 (level of

decomposition) and initial uniform grid of size ð28 þ 1Þ � ð28 þ 1Þ points. Initial displacement is defined as:

Uðx; yÞ ¼ exp �500 ðx� 0:5Þ2 þ ðy� 0:5Þ2n o�

. Re-adaptation and smoothing are frequently done after each ten time steps.

Fig. 33. Solutions observed at receivers r1 and r8, where fWjg ¼ 1 and p ¼ 0:8.


In Fig. 12, snapshots of solutions (bottom row) and their corresponding adapted grids (top row) are illustrated forfWig ¼ 1 or equivalently �a ¼ 0. There, illustrations {(a) & (b)}, {(c) & (d)} and {(e) & (f)} correspond to times 0.336, 0.615and 0.8265, respectively.

The adapted grids confirm that proper inner and outer moving fronts and nearly symmetric adapted grid points are devel-oped before and after reflection from the fixed boundaries.

To study effects of non-uniform weighting in accuracy of solutions, Fig. 13 shows comparisons between the modal anal-ysis with n;m 2 f1;2; . . . ;40g and numerical results for two values of �a (i.e., 0 and 2) at receiver with location (0.5, 0.5). Theresults of this figure show that for smaller values of �a, less accuracy in solutions are occurred. This figure demonstrates thatsmoothing has accumulative effect in numerical solutions, and rate of it increases as �a decreased.

The numbers of grid points during simulation used by adaptation algorithm for different values of �a are presented inFig. 14. This plot implies that for smaller values of parameter �a, smaller numbers of grid points are needed.

Example 2. Here, wave propagation is studied in an L-shaped membrane with fixed boundaries. The governing equationsare the same as the previous example, where, c0 ¼ 1 and Uðx; yÞ ¼ expð�500ððx� 0:25Þ2 þ ðy� 0:25Þ2ÞÞ.

In the simulations, dt is assumed to be 0.0002. The effects of different values of smoothing spline parameters (p & fWjg)and thresholds (e) are studied. The schematic shape of computational domain is illustrated in Fig. 15.

Different p and e values are considered to identify effects of smoothing with uniform weight coefficient, i.e., fwig ¼ 1(�a ¼ 0). The considered parameters are: {p 2 f0:9;0:7;0:5g&e ¼ 10�5} and {p 2 f0:9;0:7g&e ¼ 10�6}. The results are com-pared with those of a conventional method (the finite difference scheme) where an uniform grid is used with sampling step1=28 in each dimension. The numbers of grid points (ng) used in the adaptation algorithm are shown in Fig. 16. The resultsindicate that for case e ¼ 10�5 both ng and p values increase accordingly. However, the sensitivity of ng to p decreases as edecreases. As shown in Fig. 16, their curves approach to each other when e = 10�6.

The solutions obtained at the receivers 1 and 2 (denoted by r1 and r2, respectively), are shown in Fig. 17. As the figureshows, the smoothness increases as p decreases and smoothing effects are mainly appeared in the high gradient portions.

Snapshots of solutions at various times are shown in Fig. 18, where assumed parameters are: {p ¼ 0:9&e ¼ 10�5}. Theseresults are obtained at 0.352, 0.535, 0.742, 0.97, 1.084, and 1.3 s, illustrated in figures (a)–(f), respectively.

Their corresponding adapted grid points obtained by two set of parameters: {p ¼ 0:9&e ¼ 10�5} and {p ¼ 0:9&e ¼ 10�6}are presented in Figs. 19 and 20, respectively. According to the results, the grid points are more concentrated around the highgradient zones in the smaller threshold value.

The non-uniform dissipating effects are studied using different values of p and �a. The assumed parameters are:{p ¼ 0:9; �a ¼ 0 & e ¼ 10�5}, {p ¼ 0:7; �a 2 f0;1g & e ¼ 10�5} and {p ¼ 0:5&�a 2 f0;1;3;6g & e ¼ 10�5}. The relevant ng values

Fig. 34. Number of grids points, ng, used by adaptation algorithm during analyses in case: fWjg ¼ 1.


and the solutions obtained at the receivers 1 and 2, are illustrated in Figs. 21 and 22, respectively. Regarding Fig. 21, it shouldbe mentioned that number of grid points in the uniform case is 49,407.

The results show that for a certain p value, ng values and the interpolating properties increase accordingly with �a. Thesolutions obtained by the parameters {p ¼ 0:5&�a ¼ 3} and {p ¼ 0:7&�a ¼ 0} show almost equal accuracies, Fig. 23. However,the ng values obtained by the former parameter (the smoother one) are lower than those gained by the latter one, Fig. 23. Ingeneral, the accuracy of the results can be improved in the high gradient zones by enhancing the local interpolating prop-erties (increasing �a values) for a certain p value.

Example 3. Here, propagation of P–SV waves is simulated in a half space media with three absorbing boundary conditionsand one free surface.

The absorbing boundaries are usually used for presenting infinite boundaries. The defect of numerical simulations isoccurrence of artificial boundaries which reflect incoming energies to the computation domain. In this study, the absorbingboundary introduced in [61] is used to simulate infinite boundaries, where the absorbing boundary condition is consideredexplicitly. Therefore, the wave equation is modified by a damping term Qðx; yÞ � _uðx; y; tÞ where, Qðx; yÞ is an attenuation

Fig. 35. Schematic shape of an semi infinite media with a rectangular valley subjected to P–SV waves. Initial imposed deformation is subjected at point S.

Fig. 36. Snapshots of solution ux at 0.0829, 0.3599, 0.4799, 0.7199, 0.9599, 2.1958 s, illustrated in figures (a)–(f), respectively.


factor. This factor is zero in computation domain and increases gradually approaching to the artificial boundaries.Consequently, the waves incoming towards these boundaries are gradually diminished. In general, no absorbing boundarycan dissipate all incoming energies, i.e., some small reflections will always remain.

The mentioned above modification, performed for P–SV wave equations is as follows:

PDEs :1qðkþ 2lÞ @

2ux

@x2 þ l @2ux

@z2

!þ 1

qðkþ lÞ @

2uz

@x@z

!¼ @

2ux

@t2 þ Q@ux

@t1qðkþ 2lÞ @

2uz

@z2 þ l @2uz

@x2

!

þ 1qðkþ lÞ @

2ux

@x@z

!¼ @

2uz

@t2 þ Q@uz

@tX 2 ½0;1 � ½0;1 0 � t <1 ð35Þ

ICs uxðx; z; t ¼ 0Þ ¼ Uðx; zÞ; _uxðx; z; t ¼ 0Þ ¼ 0

uzðx; z; t ¼ 0Þ ¼ 0; _uzðx; z; t ¼ 0Þ ¼ 0

where, q ¼ 2, VP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkþ 2lÞ=q

p¼ 1, VS ¼

ffiffiffiffiffiffiffiffiffil=q

p¼ 0:67, and Uðx; zÞ ¼ expð�500ððx� 0:5Þ2 þ ðz� 0:7Þ2ÞÞ. In numerical sim-

ulations, dt is assumed to be 0.0002. Here, the simulations are carried out by the parameters {p 2 0:6;0:8;0:9f g, �a ¼ 0 &e ¼ 10�5}. The schematic shape of the computational domain, the corresponding absorbing boundaries and the free surfaceare illustrated in Fig. 24.

The factor Q is designed to guarantee the gradual reduction of incoming wave energy without reflection. This factor isbounded and twice differentiable with smooth derivatives [61]. Here, Q is considered as:

Qðx; zÞ ¼ Qdðx; zÞ þ Q bðx; zÞ

Q dðx; zÞ ¼ axðebx :x2 þ ebx :ð1�xÞ2 Þ þ azðebz :z2 Þ

Q bðx; zÞ ¼ abxðebb

x :x2 þ ebb

x :ð1�xÞ2 Þ:ebbz :ð1�zÞ2 :ð1� Hððz� 0:6Þ:ðz� 1ÞÞ

ð36Þ

where: ax ¼ az ¼ abx ¼ 30; bx ¼ �110; bz ¼ �70; bb

x ¼ bbz ¼ �50; H(x) is the Heaviside function, Q bðx; zÞ denotes the corner

boundary absorber, and the Q dðx; zÞ is the boundary absorber (used for three infinite boundaries, in this example). The ab-sorber Qb reduces unphysical surface waves reflected from artificial (absorbing) boundaries, which simulated by Qd. Finally,these two absorbers are added together to reduce effects of un-physical reflected long period waves.

Fig. 37. Snapshots of solution ux at 0.0829, 0.3599, 0.4799, 0.7199, 0.9599, 2.1958 s, illustrated in figures (a)–(f), respectively.


The snapshots of solutions ux and uz are shown in Figs. 25 and 26, respectively, where, p ¼ 0:8&e ¼ 10�5. The snapshotsare obtained at 0.16, 0.4, 0.646, 0.802, 0.928, and 1.174 s, illustrated in figures (a)–(f), respectively.

Fig. 38. Snapshots of adapted grids correspond to the solutions at 0.0829, 0.3599, 0.4799, 0.7199, 0.9599, 2.1958 s, illustrated in figures (a)–(f), respectively.

Fig. 39. Solutions observed at receivers r1 and r6, where fWjg ¼ 1 and p ¼ 0:8. Right and left figures correspond to ux and ux, respectively.


The adapted grids corresponded to Figs. 25 and 26 are shown in Fig. 27. According to this figure, there are still some smallsurface waves after the incoming waves being reflected from the free surface. These reflected surface waves can lead to anerroneous adapted grid points in the vicinity of the free surface, Fig. 27. The existence of such waves is reported by otherresearchers, as well [61].

The numbers of grid points, used by the adaptation algorithm, are plotted in Fig. 28; there, number of grid points in theuniform case is 65536.The ux component, measured at receiver 1 (r1), is compared with those obtained by a conventionalmethod (the finite difference scheme) where an uniform grid of sampling step 1/28 is applied, Fig. 29. According to the re-sults, both ng and accuracy (see Figs. 28 and 29) have lower values in the smoother solutions; the accuracy is mostly lower inhigh gradient zones. As the waves propagate toward the absorbing boundaries, the incoming waves diminish gradually andtherefore the ng values reduce gradually during simulation, Fig. 28.

Example 4. The capability of modeling a more complex computational domain is provided by simulating a tunnel within aninfinite media. The tunnel is subjected to propagating SH waves. The schematic shape of the domain is illustrated in Fig. 30.

In this example, it is assumed that p ¼ 0:8; �a ¼ 0, e ¼ 10�5, and dt ¼ 0:00015. The initial imposed deformation is:uyðx; z; t ¼ 0Þ ¼ expð�500ððx� 0:5Þ2 þ ðz� 0:78Þ2ÞÞ.

The snapshots of the solution and corresponding adapted grid points are illustrated in Figs. 31 and 32, respectively. Theyare taken at times 0.07185, 0.17985, 0.26985, 0.35986, 0.44985, and 0.61185 s, and are sorted from figures (a)–(i), respec-tively. The results offer: (1) adapted grid points are correctly concentrated around propagating waves; (2) small magnitude


long-period waves are concentrated in the vicinity of the tunnel, even after the incident waves propagate outward. This alsocan be observed by studying responses recorded at receivers r1–r6 (Fig. 33). Besides, number of effective adapted grid pointsused by adaptation procedure (ng) is shown in Fig. 34.

It is apparent that after incident waves propagate outward, small fluctuations remain still due to presence of small butlong period waves.

Example 5. Here a rectangular valley subjected to P–SV waves is studied (illustrated in Fig. 35). The parameters used incomputation are the same as the previous example, except for dt = 0.0001. The initial imposed displacement is:

uzðx; z; t ¼ 0Þ ¼ expð�500ððx� 0:375Þ2 þ ðz� 0:625Þ2ÞÞ:

The ux, uz and corresponding adapted grid points are shown in Figs. 36–38, respectively. The snapshots are for times0.0829, 0.3599, 0.4799, 0.7199, 0.9599, and 2.1958 s, illustrated in figures (a)–(i) respectively. By investigating records r1–r6 (Fig. 39), it is clear that small long period waves exist around the valley, same as the previous example.

9. Conclusions

A multiresolution based adaptive scheme has been developed to solve hyperbolic PDEs. Here, the novelty is applyingsmoothing splines (with constant or variable coefficients) as a postprocessor for smoothing the spurious oscillations fromsolutions, obtained in a simple or complex computational domain. Furthermore, anti-symmetric end padding method isimplemented for accurate calculation of derivatives at boundary points. The adaptive solutions of the presented examplesshow good agreement with the results of conventional methods (i.e., the finite difference schemes) using uniform grids.The advantages of this improved wavelet based adaptive method are:

� It is simple, straightforward and applicable as all calculations are done in the physical space.� As this method eliminates non-physical oscillations, the proper and stable adaptation is achieved during simulations.� The smoothing splines are directly implemented in irregular grids and no extra effort is needed for remapping non-uni-

form data to uniform one.� The suggested method can handle any kind of boundary conditions and systems of PDEs due to its simplicity.� In contrast to wavelet-based projection methods, more complex boundary shapes could be simulated by non-projected

ones.� It is to be mentioned that despite all advantages of this type of smoothing, the proper selection of parameters should be

reconsidered. Here, variable weight coefficients with constant smoothing parameters as well as wide varieties of smooth-ing parameters are studied. The performance of such smoothing, however, could be investigated by the L-curve method.

References

[1] G. Beylkin, J.M. Keiser, An adaptive pseudo-wavelet approach for solving nonlinear partial differential equations, in: W. Dahmen, A. Kurdila, P. Oswald(Eds.), Multiscale Wavelet Methods for Partial Differential Equations, Wavelet Analysis and its Applications, vol. 6, Academic Press, San Diego, 1997, pp.137–197.

[2] L.M. Jameson, T. Miyam, Wavelet analysis and ocean modeling: a dynamically adaptive numerical method ‘‘WOFD-AHO’’, Mon. Weather Rev. 128 (5)(2000) 1536–1548.

[3] K. Amaratunga, J.R. Williams, Wavelet–Galerkin solution of boundary value problems, Arch. Comput. Method Eng. 4 (3) (1997) 243–285.[4] M. Griebel, F. Koster, Adaptive wavelet solvers for the unsteady incompressible Navier–Stokes equations, in: J. Malek, J. Necas, M. Rokyta (Eds.),

Advances in Mathematical Fluid Mechanics, Springer, Berlin, 2000, pp. 67–118.[5] T.K. Hong, B.L.N. Kennett, On a wavelet-based method for the numerical simulation of wave propagation, J. Comput. Phys. 183 (2002) 577–622.[6] P. Cruz, A. Mendes, F.D. Magalhães, Wavelet-based adaptive grid method for the resolution of nonlinear PDEs, AICHE J. 48 (4) (2002) 774–785.[7] P. Cruz, A. Mendes, F.D. Magalhães, Using wavelets for solving PDEs: an adaptive collocation method, Chem. Eng. Sci. 56 (10) (2001) 3305–3309.[8] O.V. Vasilyev, N.K.R. Kevlahan, An adaptive multilevel wavelet collocation method for elliptic problems, J. Comput. Phys. 206 (2) (2005) 412–431.[9] J.C. Santos, P. Cruz, M.A. Alves, P.J. Oliveira, F.D. Magalhães, A. Mendes, Adaptive multiresolution approach for two-dimensional PDEs, Comput. Methods

Appl. Mech. Eng. 193 (3) (2004) 405–425.[10] M. Xiang, Z. Nicholas, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys.

228 (2009) 3084–3113.[11] M. Mehra, N.K.R. Kevlahan, An adaptive wavelet collocation method for the solution of partial differential equations on the sphere, J. Comput. Phys. 227

(2008) 5610–5632.[12] H. Yousefi, A. Noorzad, J. Farjoodi, Simulating 2D waves propagation in elastic solid media using wavelet based adaptive method, J. Sci. Comput. 42

(2010) 404–425.[13] R. Burger, R. Ruiz-Baier, K. Schneider, Adaptive multiresolution methods for the simulation of waves in excitable media, J. Sci. Comput. 43 (2010) 261–

290.[14] D. Gottlieb, J.S. Hesthaven, Spectral methods for hyperbolic problems, J. Comput. Appl. Math. 128 (1–2) (2001) 83–131.[15] T. Cebeci, J.P. Shao, F. Kafyeke, E. Laurendeau, Computational Fluid Dynamics for Engineers, Springer, California, 2005.[16] P. Joly, The mathematical model for elastic wave propagation, in: N.A. Kampanis, V.A. Dougalis, J.A. Ekaterinaris (Eds.), Effective Computational

Methods for Wave Propagation, Chapman & Hall/CRC, New York, 2008, pp. 247–279.[17] A. Harten, B. Engquist, S. Osher, S. Chakravarthy, Uniform high-order accurate essentially non-oscillatory schemes III, J. Comput. Phys. 71 (1987) 231–

303.[18] J.A. Ekaterinaris, ENO and WENO schemes, in: N.A. Kampanis, V.A. Dougalis, J.A. Ekaterinaris (Eds.), Effective Computational Methods for Wave

Propagation, Chapman & Hall/CRC, New York, 2008, pp. 247–279.


[19] Y. Gu, G.W. Wei, Conjugate filter approach for shock capturing, Commun. Numer. Methods Eng. 19 (2003) 99–110.[20] G.W. Wei, Y. Gu, Conjugate filter approach for solving Burgers’ equation, J. Comput. Appl. Math. 149 (2002) 439–456.[21] I. Fatkullin, J.S. Hesthaven, Adaptive high-order finite-difference method for nonlinear wave problems, J. Sci. Comput. 16 (1) (2001) 47–67.[22] W. Cai, J. Wang, Adaptive multiresolution collocation methods for initial boundary value problems of nonlinear PDEs, SIAM J. Numer. Anal. 33 (3)

(1996) 937–970.[23] M. Holmström, Solving hyperbolic PDEs using interpolating wavelets, SIAM J. Sci. Comput. 21 (2) (1999) 405–420.[24] R. Bürger, A. Kozakevicius, Adaptive multiresolution WENO schemes for multi-species kinematic flow models, J. Comput. Phys. 224 (2007) 1190–1222.[25] H.M. Hilber, T.J.R. Hughes, R.L. Taylor, Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Eng. Struct.

Dynam. 5 (1977) 283–292.[26] G.H. Hulbert, J. Chung, Explicit time integration algorithms for structural dynamics with optimal numerical dissipation, Comput. Methods Appl. Mech.

Eng. 137 (1996) 175–188.[27] M. Galis, P. Mocoz, J. Kristek, M. Kristekova, An adaptive smoothing algorithm in the TSN modeling of rupture propagation with the linear slip-

weakening friction law, Geophys. J. Int. 180 (2010) 418–432.[28] J.A. Trangenstein, Numerical Solution of Hyperbolic Partial Differential Equations, Cambridge University Press, 2007.[29] R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.[30] A. Kurganov, M. Pollack, Semi-Discrete Central-Upwind Schemes for Elasticity in Heterogeneous Media, SIAM J. Sci. Comput. (Submitted for

publication).[31] T. Fogarty, Finite volume methods for elastic-plastic wave propagation in heterogeneous media, PhD Thesis, University of Washington, 2001.[32] B. Cockburn, G.E. Karniadakis, C.W. Shu, The development of discontinuous Galerkin methods, in: B. Cockburn, G.E. Karniadakis, C.W. Shu (Eds.),

Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer-Verlag, Berlin, Heidelberg, 2000.[33] S. Srivastava, Mixed discontinuous Galerkin methods: application to nonlinear elastodynamics, PhD Thesis, The State University of New York at

Buffalo, 2008.[34] J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University Press, 2007.[35] M.F. Hutchinson, F.R. de Hoog, Smoothing noisy data with spline functions, Numer. Math. 47 (1) (1985) 99–106.[36] M. Unser, Splines: a perfect fit for signal/image processing, IEEE Signal Process. Mag. 16 (6) (1999) 22–38.[37] C.H. Reinsch, Smoothing by spline functions. II, Numer. Math. 16 (1971) 451–454.[38] C.H. Reinsch, Smoothing by spline functions, Numer. Math. 10 (1967) 177–183.[39] D.L. Ragozin, Error bounds for derivative estimates based on spline smoothing of exact or noisy data, J. Approx. Theory 37 (1983) 335–355.[40] C. Loader, Smoothing: local regression techniques, in: J.E. Gentle, W. Härdle, Y. Mori (Eds.), Handbook of Computational Statistics Concepts and

Methods, Springer, Berlin, 2004, pp. 539–564.[41] T.C.M. Lee, Improved smoothing spline regression by combining estimates of different smoothness, Stat. Probab. Lett. 67 (2) (2004) 33–140.[42] T. Prvan, Smoothing splines with variable continuity properties and degree, Appl. Math. Lett. 10 (2) (1997) 75–80.[43] W. Sweldens, P. Schröder, Building your own wavelets at home, in: Wavelets in Computer Graphics, ACM SIGGRAPH Course Notes, ACM, 1996, pp. 15–

87.[44] E. Vanraes, M. Jansen, A. Bultheel, Stabilized wavelet transforms for non-equispaced data smoothing, Signal Process. 82 (12) (2002) 1979–1990.[45] T.C.M. Lee, Smoothing parameter selection for smoothing splines: a simulation study, Comput. Stat. Data Anal. 42 (1–2) (2003) 139–148.[46] P.Ch. Hansen, Rank-deficient and discrete ill-posed problems, SIAM, Philadelphia, 1998.[47] B. Fornberg, Calculation of weights in finite difference formulas, SIAM Rev. 40 (3) (1998) 685–691.[48] D. Salomon, Curves and Surfaces for Computer Graphics, Springer, New York, 2006.[49] S. Mallet, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.[50] Z. Shi, D.J. Kouri, G.W. Wei, D.K. Hoffman, Generalized symmetric interpolation wavelets, Comput. Phys. Commun. 119 (2–3) (1999) 194–218.[51] D.L. Donoho, Interpolating wavelet transforms, Technical Report 408, Department of Statistics, Stanford University, 1992.[52] J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, West Sussex, 2003.[53] L.M. Jameson, A wavelet-optimized, very high order adaptive grid and order numerical method, SIAM J. Sci. Comput. 19 (6) (1998) 1980–2013.[54] J.J. Stickel, Data Smoothing and numerical differentiation by a regularization method, Comput. Chem. Eng. 34 (2010) 467–475.[55] P.H.C. Eilers, A perfect smoother, Anal. Chem. 75 (2003) 3631–3636.[56] Yu.P. Petrov, V.S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications, VSP, Netherlands, 2005.[57] L.L. Wang, Foundations of Stress Waves, Elsevier, New York, 2007.[58] S.M. Day, L.A. Dalguer, N. Lapusta, Y. Liu, Comparison of finite difference and boundary integral solutions to three-dimensional spontaneous rupture, J.

Geohys. Res. 110 (2005).[59] S.M. Day, G.P. Ely, Effect of a shallow weak zone on fault rupture: numerical simulation of scale-model experiments, B. Seismol. Soc. Am. 92 (2002)

3022–3041.[60] G.E. Barter, Shock capturing with PDE-based artificial viscosity for an adaptive, higher-order discontinuous Galerkin finite element method, PhD

dissertation, MIT, 2008.[61] J. Sochacki, R. Kubichek, J. George, W.R. Fletcher, S. Smiths, On Absorbing boundary conditions and surface waves, Geophysics 52 (1) (1987) 60–71.

Documents

Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems