Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August Time-dmain lifted wavelet cllcatin methd fr mdeling nnlinear wave prpagatin Kelvin Chee-Mun Lee and Wn-Seng Gan Digital Signal Prcessing Labratry, Schl f Electrical and Electrnic Engineering, Nanyang Technlgical University, S-Ba-, Nanyang Ave Singapre 998 ecmlee@ntu.edu.sg and ewsgan@ntu.edu.sg Abstract: A time-dmain adaptive numerical methd fr mdeling nnlinear wave prpagatin is develped. This methd is based n a secnd-generatin wavelet cllcatin using a lifting scheme and makes use f the multilevel decmpsitin nature f the scheme t allw fr autmatic grid refinement accrding t the magnitude f wavefrm steepening. The multiplicatin in the nnlinear term is als easy due t the cllcatin nature. With threshlding, the slutin is cmpact at every level f reslutin and cmputed nly at cllcatin pints assciated with the remaining significant wavelet cefficients. The errr tlerance and cmpressin rati f the new methd are ttally cntrlled by the threshld value used. This brings substantial savings in cmputatin time when cmpared t the cnventinal finite difference scheme n a unifrmly fine grid. Acustical Sciety f America PACS numbers:..c,..l Date Received: March Date Accepted: April. Intrductin The time-dmain numerical methd f simulating nnlinear wave prpagatin in a sund beam by using an implicit backward finite difference (IBFD) scheme t slve the Khkhlv- Zabaltskaya-Kuznetsv (KZK) equatin has been described in previus literature. The accuracy f the IBFD scheme depends n the implicit integratin slver and the fineness f the cmputatinal grid used and is therefre prprtinal t the cmputatin time required t slve the prblem. The purpse f this paper is t intrduce a wavelet cllcatin apprach that slves in a cmpact manner withut the redundant cmputatins in the wellapprximated slutin regins. We use an adaptive cmputatinal grid that is able t accmmdate lcal structures thrugh dynamic grid refinement. We first study the new methd in a -D Burgers prblem and then apply it in an axisymmetric -D frm f the KZK equatin. A cmparisn between the new results and that frm a cnventinal finite difference scheme is als made t access the perfrmances f bth.. Lifted Interplating Wavelet Transfrm Wavelets frm a versatile tl fr representing general functins r data sets because they are highly capable f capturing the essence f a functin with nly a small set f cefficients. Generally, wavelets are defined as the dyadic translates and dilates f a particular mther wavelet functin. Hwever, in this sectin, a new class f wavelets termed secndgeneratin wavelets is intrduced. These birthgnal wavelets are nt necessarily translates and dilates f each ther but rather cnstructed frm a spatial dmain apprach knwn as the lifting scheme,. They allw interplatin prcedures t be defined n mre cmplicated nnunifrm cmputatinal grid structures since the filter cefficients used in these wavelets ARLO (), Oct 9-8//()///$9. (c) Acustical Sciety f America
Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August can be psitin- and level-dependent. The simplest frm f such wavelets is the lifted interplating wavelet transfrm and is the basis set fr the wavelet cllcatin methd because f its simplicity and its in-place cmputatin withut the need f auxiliary memry. Essentially, the transfrm is cnstructed ut f the spatial dmain using basic steps: split/merge, predict (dual lifting) and update (regular lifting). The first step splits the data int subsets (dd and even); the secnd step predicts the even set based n the dd ne and finds the difference between them; and the third step updates the dd set t preserve the mean f the functin. The resulting apprximatin and detail values (r λ andγ cefficients respectively) are then btained, and the prcess is iterated using the apprximatin cefficient λ as the new input until a specified level f carsest reslutin is reached. Fig.. The frward lifted interplating wavelet transfrm (left) and its inverse (right). The multireslutin analysis prperty f wavelets is still valid fr secnd-generatin J- j j wavelets and can be expressed as V W W... W L, wherev andw are the rthgnal subspaces that crrespnd t the scaling and wavelet functins respectively, and L is the space spanned by the riginal functin. Thus, when wavelet decmpsitin is perfrmed n the riginal functin P, its representatin at the finest reslutin level J can be btained as J J j j ( ) = λφ k k ( ) + γlφl ( ). () P x x x k K j = j l L The strength f this multilevel decmpsitin is then bvius: the functin is decmpsed int a set f rthgnal cmpnents at different reslutin levels. It is crucial fr a fast way t cmpute the slutin via the cnstructin f an adaptive cmputatinal grid, which will be elabrated in the next sectin. Lcal structures f the riginal functin due t wavefrm steepening at different scales can be islated and recrded in the magnitude f theγ cefficients. We use the lifted interplating wavelet transfrm with the number f dual vanishing mments N = and the number f real vanishing mments N = thrughut ur discussin in this paper.. Numerical Methd A crdinate transfrmatin is applied t the KZK equatin fr an axisymmetric and unfcused sund beam that prpagates in the psitive z directin (see Fig. ) t arrive at the fllwing dimensinless frm in terms f the nrmalized acustic pressure P : P = P + P d ' + A P + NP P ( + σ) ρ ( ). () σ ρ ρ + σ Fr -D nnlinear wave prpagatin, a dimensinless Burgers equatin identical t Eq. () withut the diffractin term is used. The prblem is cnveniently transfrmed int a regular and dimensinless cmputatinal grid represented by the axial distanceσ and the radial distance ρ and is discretized fr finite difference calculatins as Pijk,, = P( σk, min+ i, j ρ). As we have pint values, the derivative peratrs can be implemented using cnventinal finite difference stencils whereas the integral peratr in the diffractin term ARLO (), Oct 9-8//()///$9. (c) Acustical Sciety f America
Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August can be implemented using an adaptive Simpsn quadrature w.r.t. '. Multiplicatin in the nnlinear term is als straightfrward and a lcal peratin since we are wrking with cllcatin pints. A th rder explicit Runge-Kutta slver is als used t march frward in space, integrating Eq. () numerically with respect tσ t get the slutin at the next prpagatin step. Fig.. Radiatin gemetry frm an axisymmetric sund surce lcated in the plane z =.. Adaptive Cmputatinal Grid The multilevel decmpsitin f Eq. () allws us t cnstruct a set f nested grids with different reslutins. Assuming the slutin is nrmalized, a relative errr threshld ε can be set t split the slutin P J int regins: ne with cefficients abve the threshld and ne with cefficients belw. This can be expressed as ( σ,, ρ ) ( σ, j, ρ ) j ( σ,, ρ ) P = P + P j =,... J, J J J l l γ ε γ ε l L j. () Threshlding the latter remves a substantial amunt f cllcatin pints frm the grid because they cntain trivial details and results in lesser subsequent cmputatins. This prvides a simple way f cntrlling the cmpressin factr f the cmputatinal grid and the apprximatin errr f the slutin. The selectin f the threshld value is easy by specifying the percentage amunt f signal energy t be retained after threshlding. The finite difference and quadrature schemes are then applied lcally n each level where there are nγ cefficients n finer scales. Cnvlutin with the finite difference stencil als leads t an increase in supprt, s we need t add extra neighbring cllcatin pints next t the edge pints in the cmputatinal grid at every level s as t crrect fr this. Furthermre, the differentiated functin might als have new significant cefficients appearing in the neighbring regins. This manner f dynamically allcating cllcatin pints at every level allws the cmputatinal grid t adapt dynamically t the lcal changes as harmnics are intrduced int the slutin due t the nnlinear term. Upn inverse wavelet transfrm, the full slutin at all cllcatin pints can be btained fr the current integratin step.. Results and Discussin The results fr the prpsed numerical methd are presented. As mentined earlier, a th rder Runge-Kutta integratin slver (MATLAB s de) is used tgether with a relative errr threshld f ε = fr the first example using Burgers equatin and a relative errr threshld f ε = fr the next example using KZK equatin. Fr stability in the integratin, we use = ρ =. and an initial prpagatin step-size f σ = e, which will then be adjusted by de s variable step-size cntrl algrithm during the simulatin. Fr higher efficiency, the prgram is cmpiled as a MEX (MATLAB-EXecutable) file befre executin in MATLAB.. -D Dimensinless Burgers Equatin ARLO (), Oct 9-8//()///$9. (c) Acustical Sciety f America
Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August The dimensinless Burgers equatin fr simulating -D nnlinear wave prpagatin is generally expressed as P P P = A NP () σ where, fr this example, A =., N =, and the initial functin atσ = is represented by P( σ =, ) = P sin( π) given.. is the range f the time windw used and the bundary cnditins are P( σ, =±.) =. The number f decmpsitin levels is fixed at and, because the functin is largely smth, we chse a large errr threshld f ε =. The slutin starts develping a sharp gradient at = and evlves int a sawtth-like wavefrm as it prpagates away frm the surce. This is a gd test f the new methd s adaptivity t refine the slutin at lcal regins where shcks are frming, i.e., t refine the cmputatinal grid arund = as the shck develps. Nte that increasingly higher reslutin is allcated t such lcal structures in the slutin. The new methd is able t refine the grid in the vicinity f the shck accrding t the magnitude f the gradient, even after much attenuatin. The number f significant cefficients in the higher levels als increases as the gradient increases until the shck is finally reslved at the finest reslutin level. The results btained are cmpared t results frm the cnventinal finite difference scheme (i.e., ε = ). The methd demnstrates sufficient rbustness in handling nnlinearities thrughut the whle cmputatin prcess. Snapsht samples f these results, taken at selected intervals σ = (,.,.,, ) are shwn belw..8.. x -. -. -. -. -.8 -...8... -. -. -. -.8 -...8... -. -. -. -.8 -...... -. -. -. -. -.... -.. -.. -.. -.. - - - - -.. x - - - - - -.. x - - - - -.. x - 8 - - - -.. x - -. - -. -.. -.. - - -.. Fig.. Adaptivity f the lifted wavelet cllcatin methd in handling evlving shcks: refinement n a finer grid is dne as the shck gradient increases. Cmparisn with the cnventinal finite difference slutin f the Burgers equatin shws similar results within the tlerance f the given relative errr threshld ε =. ARLO (), Oct 9-8//()///$9. (c) Acustical Sciety f America
Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August. Dimensinless Axisymmetric KZK Equatin The initial surce cnditin fr the KZK equatin in Eq. () is given as + ρ P( σ =,, ρ) = exp sin( π ( + ρ )) () π / with as the range f the time windw used and ρ as the radial distance range (Fig. ). This can als be seen as the size f the initial cmputatinal grid t be used. Initial Surce Cnditin (nrmalized).8.. Retarded Time -. -. - -. - -. - -.8 -..... Radial Distance ρ Fig.. Initial surce cnditin f cmputatinal grid. With the number f decmpsitin levels fixed at, the results btained frm the lifted wavelet cllcatin methd are shwn and cmpared with thse btained frm the cnventinal finite difference scheme in Figs. and. The first set f results in Fig. shws the n-axis slutins f the prpagating wave at varius prpagatin distancesσ whereas the secnd in Fig. shws the slutins alng the ρ axis at = at the same distances. The algrithm is stable with ε = thrughut the integratin range and is seen by the cnvergence f the abslute apprximatin errrs in the far-right clumns f Figs. and t within the specified ε. Hwever, it may be pssible t cntrl the errr threshld adaptively accrding t the percentage amunt f signal energy retained after threshlding s as t further reduce the cmplexity f the cmputatins. σ=, ρ=. σ=., ρ=.8... -. -. -. -.8 -....... -..8... -. -. -. -.8 -. - x -. σ=, ρ=... -. -. -. -. -.... -. -. - -. - x -... -. -.... - -.. - - - - - Fig.. On-axis slutins at varius axial distances using the wavelet cllcatin methd ( N =, A =.). 8 ARLO (), Oct 9-8//()///$9. (c) Acustical Sciety f America 8
Lee et al.: Acustics Research Letters Online [DOI./.] Published Online 8 August σ=, = σ=, = σ=., =.8... -. -. -. x - - - - -8 - - - - ρ........... ρ..8... -. -. -. -.8 -. x - - - - ρ x - 8... -. -... - - - -.. -8 -. -. Cnclusin Fig.. Slutins alng ρ axis at = at varius axial distances using the wavelet cllcatin methd ( N =, A =.) An adaptive numerical methd based n secnd-generatin wavelets is develped. The adaptivity f the methd is achieved by using a lifted interplating wavelet basis t apprximate the slutin and refining the cmputatinal grid in lcal regins t accmmdate develping shcks. By specifying the errr threshld value, it is pssible t actively cntrl the accuracy f the slutin and the size f the adaptive cmputatinal grid. The sparse representatin leads t significant savings in the memry strage and the number f cmputatins required t slve the Burgers and KZK equatins numerically. This methd uses an adaptive cmputatinal grid and utperfrms the cnventinal finite difference methd, which uses a unifrm fine grid thrughut. Due t the cllcatin nature, the methd can als handle general bundary cnditins and nnlinearities effrtlessly. References and links Daubechies, I., Sweldens, W., Factring Wavelet Transfrms int Lifting Steps, J. Furier Anal. Appl., (), - (998). Hlmstrm, M., Slving hyperblic PDEs using interplating wavelets, SIAM J. Sci. Cmput., (), - (999). Lee, Y. S., Hamiltn, M. F., Time Dmain Mdeling f pulsed finite-amplitude sund beams, J. Acus. Sc. Am, 9(), 9-9 (99). Sweldens, W., The lifting scheme: A cnstructin f secnd-generatin wavelets, SIAM J. Math. Anal., 9(), - (998). Vasilyev, O. V., Bwman, C., Secnd generatin wavelet cllcatin methd fr the slutin f partial differential equatins, J. Cmp. Phys,, -9 (). 9 ARLO (), Oct 9-8//()///$9. (c) Acustical Sciety f America 9