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1 Journal of Computatonal Physcs 230 (2011) Contents lsts avalable at ScenceDrect Journal of Computatonal Physcs journal homepage: A grd based partcle method for solvng partal dfferental equatons on evolvng surfaces and modelng hgh order geometrcal moton Shngyu Leung a,, John Lowengrub b, Hongka Zhao b a Department of Mathematcs, Hong Kong Unversty of Scence and Technology, Clear Water Bay, Hong Kong b Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA , USA artcle nfo abstract Artcle hstory: eceved 8 July 2010 eceved n revsed form 16 December 2010 Accepted 16 December 2010 Avalable onlne 28 December 2010 Keywords: Partal dfferental equatons on surfaces Euleran mesh Lagrangan samplng partcles Moton by surface dffuson Wllmore flow CFL condton Sem-mplct scheme CN scheme We develop numercal methods for solvng partal dfferental equatons (PDE) defned on an evolvng nterface represented by the grd based partcle method (GBPM) recently proposed n [S. Leung, H.K. Zhao, A grd based partcle method for movng nterface problems, J. Comput. Phys. 228 (2009) ]. In partcular, we develop mplct tme dscretzaton methods for the advecton dffuson equaton where the tme step s restrcted solely by the advecton part of the equaton. We also generalze the GBPM to solve hgh order geometrcal flows ncludng surface dffuson and Wllmore-type flows. The resultng algorthm can be easly mplemented snce the method s based on meshless partcles quas-unformly sampled on the nterface. Furthermore, wthout any computatonal mesh or trangulaton defned on the nterface, we do not requre remeshng or reparametrzaton n the case of hghly dstorted moton or when there are topologcal changes. As an nterestng applcaton, we study locally nextensble flows governed by energy mnmzaton. We ntroduce tenson force va a Lagrange multpler determned by the soluton to a Helmholtz equaton defned on the evolvng nterface. Extensve numercal examples are also gven to demonstrate the effcency of the proposed approach. Ó 2010 Elsever Inc. All rghts reserved. 1. Introducton The numercal soluton of partal dfferental equatons (PDEs) on evolvng surfaces s a challengng problem as s smulatng the moton of hgh order geometrc evoluton equatons. The nterface may stretch, deform and break apart. In the case of geometrc moton, the velocty of the nterface depends on hgh order dervatves of the nterface poston whch poses challenges for accuracy and stablty of the method. Such problems have applcaton n the bologcal, physcal and engneerng scences. Varous numercal methods have been proposed to solve PDEs on surfaces and geometrc flows. One popular approach s based on explct representaton of surfaces by parametrzng or layng down a mesh on the surface (e.g. [13,25,26, 5,2,10,9,3]). When the surface s evolvng, ths type of approach s also called Lagrangan formulaton snce parametrzaton or meshes on the surface s tracked along the Lagrangan trajectory. One major ssue of ths approach s that t s dffcult (especally n 3 dmensons), to mantan a smooth global parametrzaton or a quas-unform mesh for a movng nterface wth complcated geometry and dynamcs nvolvng large deformatons or topologcal changes. As a result, the resultng algorthm usually requres reparametrzaton/remeshng durng the evoluton, although a recent algorthm developed for geometrc moton of surfaces has the potental to overcome ths problem [3]. In addton, f the topology of the nterface Correspondng author. E-mal addresses: masyleung@ust.hk, syleung@math.uc.edu (S. Leung), lowengrb@math.uc.edu (J. Lowengrub), zhao@math.uc.edu (H. Zhao) /$ - see front matter Ó 2010 Elsever Inc. All rghts reserved. do: /j.jcp

2 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) changes, then local nterface surgery s requred whch can be hghly problematc to perform n three dmensons. However, these Lagrangan type methods are relatvely easy to mplement and are effcent and accurate numercally. An alternatve approach s based on an mplct representaton of surfaces usng an auxlary functon on a fxed Euleran mesh. For example, the level set method can be used for mplct representaton whle the PDEs on the surface, whch s the zero level set of the level set functon, are extended off the nterface onto neghborng level sets (e.g. [4,1,44,15,43,11]). There are three man advantages usng an mplct formulaton. Frst, there s no need to parametrze or trangulate the surface. Second, t s easy to handle topologcal changes. Thrd, the surface PDEs s extended off the surface to a correspondng PDE n a neghborhood of the zero level set on a fxed Euleran mesh for whch many standard numercal methods for PDEs can be appled. Examples of other recent Euleran approaches nclude [31,23], whch are based on the closest pont method, [17], whch s based on volume of flud method, [32] whch s based on a grd-free partcle level set method, and [28,12,37], whch are based on dffuse nterface method. Typcally the Euleran formulaton has an ncreased computaton cost compared to Lagrangan formulaton due to embeddng of the equaton nto one hgher dmenson. Also t s dffcult to deal wth open surfaces. For applcatons of Euleran methods to geometrc evoluton equatons, we refer the reader to the recent revew [22] for references. In [20], we have recently proposed a novel approach, the grd based partcle method (GBPM), to represent and model an nterface and ts moton. The dea s to sample the nterface by meshless and non-parametrzed Lagrangan partcles accordng to an underlyng unform or adaptve Euleran mesh. Ths results n a quas-unform samplng of the nterface. As the nterface moves, we contnuously update the locaton of these partcles by solvng ordnary dfferental equatons (ODEs), rather than partal dfferental equatons (PDEs). Usng extra Lagrangan nformaton defned on these samplng ponts, we can naturally capture topologcal changes such as mergng or breakup of surfaces. Unlke usual Lagrangan methods dependng on surface parametrzatons or meshes, the GBPM does not requre any connectvty nformaton among the partcles. Ths feature allows one to easly add or delete partcles, whch s mportant for mantanng a consstent resoluton of the surface as well as dealng wth topologcal changes. Moreover, the samplng of the partcles has a one-to-one reference to the underlyng grd ponts whch are n the neghborhood of the nterface. Ths Euleran reference provdes both a quas-unform samplng of the nterface and neghborhood nformaton among meshless partcles. Ths s useful for local constructon of the surface and detecton of nterface collson or self-ntersecton. The Euleran reference s contnuously updated wthout usng a PDEs approach. Even though t s not demonstrated n the current paper, an adaptve samplng of the nterface can be acheved easly through local grd refnement of the underlyng grd takng advantage of the fact that no PDE s solved on the grd. For a more detaled descrpton, we agan refer nterested readers to some recent publcatons [20,19]. We have successfully appled ths technque to varous velocty models n [20], ncludng moton by mean curvature. We have also demonstrated that the method can be used to capture the vscosty soluton or the multvalued soluton when two nterfaces come across each other. Numerous test examples have been shown n [20] that demonstrate the flexblty of the approach. We can also deal wth curves/surfaces wth hgh codmenson as well as open curves/surfaces easly wth ths method [19,21]. In ths paper, we extend our prevous work and use the GBPM to solve advecton dffuson equatons on surfaces. In partcular, we develop mplct methods that remove tme step restrctons due to dffuson. We then further extend these technques and consder hgh-order geometrc flows ncludng surface dffuson and Wllmore flow [42]. As an nterestng applcaton of these technques, we further mpose a local nextensblty condton on an evolvng nterface. Ths constrant s mportant n many felds ncludng lpd vescle modelng n mathematcal bology (e.g. [30,8,36,40,39,35]), flexble fber nteractons (e.g. [38]) and robotcs (e.g. [6]). To enforce local nextensblty, we ncorporate a Lagrange multpler whch acts as the local tenson of the evolvng nterface. Ths Lagrange multpler satsfes a surface Helmholtz equaton relatng the geometry of the evolvng nterface wth the tangental and the normal veloctes of the unconstraned flow. The paper s organzed as follows. In Secton 2, we frst summarze the grd based partcle method we have ntroduced n [20]. We explan how we generalze the method for solvng PDEs on evolvng surface n Sectons 3.1,3.2,3.3. Wth these technques, we then model hgher order motons of the nterface ncludng the moton by surface dffuson and the Wllmore flow n Sectons 3.4 and 3.5. In Secton 4, we apply the technques we have developed to smulate local nextensble flows. Secton 5 shows examples to demonstrate the performance of our method. 2. Grd based partcle method (GBPM) In ths secton, we gve a bref revew of the grd based partcle method. For a complete descrpton of the algorthm, we refer nterested readers to [20]. The nterface s represented by meshless partcles whch are assocated to an underlyng Euleran mesh. In our current algorthm, each samplng partcle on the nterface s chosen to be the closest pont from each underlyng grd pont n a small neghborhood of the nterface. Ths one to one correspondence gves each partcle an Euleran reference durng the evoluton. The closest pont to a grd pont, x, and the correspondng shortest dstance can be found n dfferent ways dependng on the form n whch the nterface s gven. At the frst step, we defne an ntal computatonal tube for actve grd ponts and use ther correspondng closest ponts as the samplng partcles for the nterface. A grd pont p s called actve f ts dstance to the nterface s smaller than a gven tube radus, c, and we label the set contanng all actve grds C. To each of these actve grd ponts, we assocate the

3 2542 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) correspondng closest pont on the nterface, and denote ths pont by y. Ths partcle s called the foot-pont assocated to ths actve grd pont. Ths lnk between the actve grd ponts and ts foot-ponts s kept durng the evoluton. Furthermore, we can also compute and store certan Lagrangan nformaton of the nterface at the foot-ponts, ncludng normal, curvature and etc., whch may be useful n varous applcatons. As a result of the nterface samplng, the densty of partcles on the nterface wll be roughly nversely proportonal to the local grd sze. Ths relaton makes t easy to ncorporate adaptvty n the current grd based partcle method. In some regons where one wants to resolve the nterface better by puttng more marker partcles, one mght smply locally refne the underlyng Euleran grd and add the new foot-ponts accordngly. Ths ntal set-up s llustrated n Fg. 1(a). We plot the underlyng mesh usng sold lnes, all actve grds are plotted usng small crcles and ther assocated foot-ponts are plotted usng squares. To each grd pont near the nterface (blue 1 crcles), we assocate a foot-pont on the nterface (red squares). The relatonshp of each of these pars s shown by a sold lne lnk. In practce, we do not necessarly use such a thck computatonal tube. Snce the thcker the tube, the more the nterface s over-sampled. Usually, we use c = 1.1h whch already gves a relatvely good samplng. To track the moton of the nterface, we move all the samplng partcles accordng to a gven moton law. Ths moton law can be very general. Suppose the nterface s moved under an external velocty feld gven by u = u(y). Snce we have a collecton of partcles on the nterface, we smply move these ponts just lke all other partcle-based methods, whch s smple and computatonally effcent. We can smply solve a set of ordnary dfferental equatons usng hgh order scheme whch gves a very accurate locaton of the nterface. For more complcated motons, the velocty may depend on the geometry of the nterface. Ths can be done through a local nterface reconstructon as descrbed n [20]. Here, we extend ths approach to solve PDEs on surfaces and hgher order geometrc equatons. It should be noted that a foot-pont y after moton may not be the closest pont on the nterface to ts assocated actve grd pont p anymore. For example, Fg. 1(b) shows the locaton of all partcles on the nterface after the constant moton u = (1,1) T wth a small tme step. As we can see, these partcles on the nterface are not the closest pont from these actve grd ponts to the nterface anymore. More mportantly, the moton may cause those orgnal foot-ponts to become unevenly dstrbuted along the nterface. Ths may ntroduce both stffness, when partcles become clustered, and large error, when partcles become far apart. To mantan a quas-unform dstrbuton of partcles, we need to resample the nterface by recomputng the foot-ponts and updatng the set of actve grd ponts (C) durng the evoluton (see Fg. 1(c) (e)). Durng the resamplng process, we locally reconstruct the nterface, whch nvolves communcaton among dfferent partcles on the nterface. Ths local reconstructon also provdes geometrc and Lagrangan nformaton at the recomputed foot-ponts on the nterface. The key step n the method s a least squares approxmaton of the nterface usng polynomals at each partcle n a local coordnate system, {(n 0 ) \,n 0 }, wth y as the orgn, see Fg. 2(a). Usng ths local reconstructon, we fnd the closest pont from ths actve grd pont to the local approxmaton of the nterface, Fg. 2(b). Ths gves the new foot-pont locaton. Further, we also compute and update any necessary geometrc and Lagrangan nformaton, such as normal, curvature, and also possbly an updated parametrzaton of the nterface at ths new foot-pont. For a detaled descrpton, we refer nterested readers to [20]. To end ths secton, we summarze the algorthm here. Algorthm 1. Intalzaton [Fg. 1(a)]. Collect all grd ponts n a small neghborhood (computatonal tube) of the nterface. From each of these grd ponts, compute the closest pont on the nterface. We call these grd ponts actve and ther correspondng partcles on the nterface foot-pont. 2. Moton [Fg. 1(b)]. Move all foot-ponts accordng to a gven moton law. 3. e-samplng [Fg. 1(c)]. For each actve grd pont, re-compute the closest pont to the nterface reconstructed locally by those partcles after the moton n step Updatng the computatonal tube [Fg. 1(d) and (e)]. Actvate any grd pont wth an actve neghborng grd pont and fnd ther correspondng foot-ponts. Then, nactvate grd ponts whch are far away from the nterface. 5. Adaptaton (Optonal). Locally refne the underlyng grd cell f necessary. 6. Iteraton. epeat steps 2 5 untl the fnal computatonal tme. 3. Tme and spatal dervatves In the frst part of ths secton, we dscuss how we apply the GBPM to solve a PDE on an evolvng surface. Then n the second part we apply the GBPM to compute hgh order flow ncludng moton by surface dffuson and Wllmore flow Advecton equaton In ths secton, we consder f : ðtþ! defned on the surface (t) represented by quas-unformly dstrbuted but meshless partcles. Assumng that the functon f s advected by the flow, we have 1 For nterpretaton of color n Fgs. 1, 3, 19 and 20, the reader s referred to the web verson of ths artcle.

4 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 1. Grd based partcle method. From left to rght: (a) Intalzaton, (b) after moton, (c) after re-samplng, (d) after actvatng new grd ponts wth ther foot-ponts and (e) after nactvatng grd ponts wth ther foot ponts that are too far from the nterface. Df Dt ¼ 0; where D/Dt + u r s the materal dervatve along any trajectory of a partcle on the nterface. Solvng ths equaton s relatvely straghtforward n our formulaton. Snce we are representng the nterface (t) usng partcles, the correspondng f at each of these partcles s constant durng the moton. Denote y (t n ) the locaton of the footpont assocated to an actve grd pont x at the tme t = t n. Accordng to the moton u = u(y ), we solve the ODE ð1þ

5 2544 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 2. (a) Defnton of a local coordnates and (b) the way how we determne the new foot-pont usng a local least squares reconstructon of the nterface. dyðtþ ¼ uðyðtþþ; dt for y(t n+1 ) wth the ntal condton y(t n )=y (t n ). Denotng ths locaton by y ðtn Þ, we obtan the functon value f after ths moton phase f ðy ðtn ÞÞ ¼ f ðy ðt n ÞÞ: The more mportant step s the resamplng phase. In ths step, we locally reconstruct the nterface and then recompute the foot-pont assocated to a partcular actvated grd pont. Thus, we have to update the functon value at the new footpont locaton y (t n+1 ) usng the functon values at y j ðtn Þ. Ths s an approxmaton problem. Gven f ðy j ðtn ÞÞ for j =1,...,m, one approxmates f(y (t n+1 )). In ths paper, we consder the local coordnates at y ðtn Þ and then the approxmaton can be easly done on the tangent plane at the pont y ðtn Þ. Here we dscuss the two dmensonal formulaton. Extenson to hgher dmensons s straghtforward. In the local coordnate {(n 0 ) \,n 0 } wth n 0 the normal vector assocated to the foot-pont y (t n ), we express the functon f n terms of ths local coordnates, f ðx; yþ ¼f ð~x; ~yð~xþþ ¼ f ð~xþ; where ~yð~xþ s the least squares polynomal we obtaned to locally approxmate the nterface. Let ~x be the mnmzer whch gves the locaton for the new foot-pont, we have the followng approxmaton problem. Gven m data ponts ð~x j ; f ð~x j ÞÞ for j =1,...,m, we want to approxmate f ð~x Þ. Varous methods could be used. In the current formulaton, we agan use the least squares technque and determne a polynomal ~ f approxmatng the functon f accordng to the data ponts. Then the functon value at the new foot-pont s gven by ~ f ð~x Þ,.e. f ðy ðt nþ1 ÞÞ ¼ ~ f ð~x Þ: Although ths s not conservatve, t s found to gve very good results. To smplfy the notaton n the followng sectons, we defne the followng operators for solvng the advecton equaton. Let f( n,y n,t n ) be the functon evaluated at the Lagrangan partcles y n whch samples the surface n at t n. We defne the advecton operator A Dt by A Dt f ð n ; y n ; t n Þ¼f ð nþ1 ; y ; t Þ: The superscrpt ( ) accounts for the fact that the samplng partcles y are locatons of the foot-ponts y n rght after moton. These ponts are n general not the correspondng closest ponts from the underlyng grd anymore. But snce these ponts y stll sample the nterface at t = t n+1, we denote the surface by n+1. To restore the quas-unform samplng of the nterface, we apply the resamplng step as descrbed n Secton 2. The functon value at these new samplng ponts y n+1 are obtaned by nterpolaton, denoted by the operator I, ð2þ ð3þ ð4þ ð5þ ð6þ f ð nþ1 ; y nþ1 ; t nþ1 Þ¼If ð nþ1 ; y ; t Þ: To sum up, the soluton of the advecton equaton on the evolvng surface s f ð nþ1 ; y nþ1 ; t nþ1 Þ¼IA Dt f ð n ; y n ; t n Þ: ð7þ ð8þ 3.2. Surface Laplacan Consder a surface parametrzed by (s 1,s 2 ). The surface Laplacan of a functon f :! s gven by D f ¼ X2 1 pffffff ffffff g g g j ;j¼1 ð9þ where the coeffcents g j are the components of the nverse of the metrc tensor [g,j ], whose components are gven by

6 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) g j ; ð10þ wth the surface s gven by X(s 1,s 2 ). To locally parametrze the surface n our formulaton, we use agan the local coordnate system of the tangent plane. At each partcle, we translate and rotate the coordnate system accordng to ts normal vector n so that the foot-pont s now at the orgn of the new coordnate. The local parametrzaton of the surface s then gven by the tangent plane {t 1,t 2 } where n, t 1 and t 2 are orthogonal. If we use a second order polynomal to locally approxmate the nterface,.e. Xðs 1 ; s 2 Þ¼ X2 X ¼0 06þj62 a ;j s 1 sj 2 ; the metrc tensor [g j ] n ths coordnate system can be found va @X 2 C ½g j Š¼@ 2 @s ¼ a 1;0 þ 2a 2;0 s 1 þ a 1;1 s 2 ¼ a 0;1 þ 2a 0;2 s 2 þ a 1;1 s 1 : Next, snce the functon f s defned only on the foot-ponts y, we also ft f usng a local least squares quadratc polynomal f ðs 1 ; s 2 Þ¼ X2 X ¼0 06þj62 b ;j s 1 sj 2 : The correspondng dervatves n Eq. (9) can then be approxmated by the dervatves of ths local least squares approxmaton. A smlar approxmaton can be found n [41] where the authors proposed an approach to fnd the surface Laplacan usng local polynomal estmaton. However, ther approach s appled on a fxed surface whch s not evolvng. Moreover, all calculatons are done on the tangent plane at each of the samplng ponts,.e. the local metrc [g j ] s flat, whch makes the overall approxmaton of low order Advecton dffuson equaton In ths secton, we apply the technques for computng the surface Laplacan to solve the advecton dffuson equaton. We develop explct and mplct schemes. Numercal detals are also dscussed An explct scheme The easest way to solve the advecton dffuson equaton s to apply operator splttng technques and to compute each operator explctly. We frst consder the advecton part gven by Df Dt ¼ 0; wth the ntal condton f(y (t n )) defned on the nterface (t n ). Ths can be solved by the technques descrbed n Secton 3.1. Once we have obtaned the soluton f(y (t n+1 )) defned on the surface (t n+1 ), we solve the followng ¼ D ðt nþ1 Þ f ; where the surface (t n+1 ) remans unchanged n ths step. Usng the forward Euler method, we have ð11þ ð13þ ð14þ ð15þ ð16þ f nþ1 ¼½I þ DtD ðt nþ1 Þ Šf n : ð17þ We denote the soluton from ths dffuson equaton on the fxed surface n+1 by f ð nþ1 ; y nþ1 ; t nþ1 Þ¼D nþ1 þ;dt IA Dtf ð n ; y n ; t n Þ: ð18þ The operators I and A Dt are defned n the prevous secton. The superscrpt (n + 1) n the operator D represents the surface n+1. The subscrpt + reflects the plus sgn n the operator ½I þ DtD ðt nþ1 ÞŠ,.e. D nþ1 þ;dt ¼½I þ DtD ðt nþ1 Þ Š: ð19þ

7 2546 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Ths explct scheme can be easly mplemented but the correspondng CFL condton s restrctve: the tme step Dt = O(h 2 ), where h s the underlyng fxed grd sze (not the nterpartcle dstance) Implct schemes To relax the tme step restrcton, we wll dscuss several mplct schemes whch wll result n a CFL condton of Dt = O(h) whch solely depends on the advecton part. We consder agan the numercal dscretzaton of the surface Laplacan we descrbed n Secton 3.2. For smplcty, we consder here only the two dmensonal case, t s relatvely straghtforward to generalze the approach to hgher dmensons. For a gven surface n, the surface Laplacan (9) can be wrtten as a lnear combnaton of dervatves on the surface D nf ¼ a 1 f s þ a 2 f ss for some a s dependng only on the coeffcents of the local quadratc reconstructon ð20þ fðsþ ¼a 0 þ a 1 s þ a 2 s 2 : As descrbed n Secton 3.2, assocated to each actve grd pont, we locally approxmate the functon f by a local least squares quadratc polynomal f ðsþ ¼b 0 þ b 1 s þ b 2 s 2 : In partcular, gven the data (s,f ) for =1,...,m, these coeffcents b s n the local reconstructon are determned by solvng the normal equaton 0 m P P P 1 s s 2 b 0 f B P P P s s 2 s 3 CB C B P A@ b 1 A s f A; ð23þ P P P P s 2 s 3 s 4 b 2 s 2 f whch gves 0 P f b 0 b 1;1 b 1;2 b 1;3 P B C b 1 A b 2;1 b 2;2 b C 2;3 A s f B C b 2 b 3;1 b 3;2 P 3;3 s 2f A : Ths mples that coeffcents b depend lnearly on the functon values f. Moreover, assumng that we are computng the surface Laplacan at s = s wth s = s for some =1,...,m, we can express the surface Laplacan (9) as a lnear combnaton of f s ð21þ ð22þ ð24þ D nf ¼ X c f þ c f ; ð25þ for some c ¼ c ðs ; b k1 ;k 2 ; a k3 Þ and b k1 ;k 2 depends on s,.e., locatons of footponts. Ths lnear relatonshp s mportant for developng teratve methods for mplct schemes. We frst consder the followng mplct scheme based on a smple operator splttng. We follow the explct scheme we have just developed n the prevous secton, D nþ1 þ;dt IA Dt, and replace the dffuson part by a backward Euler scheme. Ths mples or ½I DtD ðt nþ1 Þ Šf nþ1 ¼ f n ; ð26þ f nþ1 ¼½I DtD ðt nþ1 Þ Š 1 f n : ð27þ In the operator form, we denote the resultng numercal scheme by f ð nþ1 ; y nþ1 ; t nþ1 Þ¼ðD nþ1 ;Dt Þ 1 IA Dt f ð n ; y n ; t n Þ: ð28þ Unfortunately, dscretzng Eq. (26) does not result n a dagonally domnant system of lnear equatons for arbtrary Dt. We demonstrate ths usng a very smple case where the surface s flat and the grd ponts are evenly dstrbuted wth unform spacng of Dx = 1. We frst approxmate the functon f(x) locally usng the least squares fttng, ths gves f(x)=b 0 + b 1 x + b 2 x 2 where b 2 ¼ 0:14286f 2 0:07143f 1 0:14286f 0 0:07143f 1 þ 0:14286f 2 : For suffcently small Dt, we have Dt > 4( )Dt. Ths mples that the matrx we use to approxmate the operator ½I DtD ðt nþ1 ÞŠ s dagonally domnant only condtonally n Dt. The resultng system of lnear equatons can stll be nverted usng smple Jacob/Gauss Sedel teraton for suffcently small Dt. In general, however, the condton wll not be guaranteed for arbtrary Dt or local geometry. ð29þ

8 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 3. Dscretzaton of the surface Laplacan gves an nonsymmetrc matrx. See text for detals. Another numercal dffculty s that the above dscretzaton gves a nonsymmetrc matrx for general surfaces. Consder Fg. 3 where we have two foot-ponts (red square) on the nterface (black sold lne). We plot the local coordnates used n the local reconstructon usng an arrow and the dashed lnes. We draw the projecton of each foot-ponts to these two local coordnate systems usng blue crcle. Dependng on the nterface, s a wll generally be dfferent from s b. Snce the coeffcents of the local reconstructon depend on the values s n the stencl, the contrbuton from the functon value at the left foot-pont to the surface Laplacan at the rght foot-pont wll be dfferent from the vse-versa relaton. Ths mples that the matrx we are nvertng s n general nonsymmetrc. Numercally, we propose the followng two strateges n solvng for f n+1 n Eq. (26). As descrbed above, for suffcently small Dt, the operator ½I DtD ðt nþ1 ÞŠ s stll dagonally domnant even though nonsymmetrc. In ths case, we can stll nvert t usng smple Gauss Sedel teraton. To smplfy the notaton, we ntroduce the varable g = f n. For each foot-pont (usng the label ), we terate the followng expresson for f,k ð1 c DtÞf ;k ¼ g þ Dt X c f ; ð30þ for all foot-ponts and k =1,... The superscrpt denotes the most-updated value for f,.e. t represents ether k or k 1. The ntal guess of the teraton can be the soluton from the forward Euler operator so that f ;0 ¼D nþ1 þ;dt IA Dtf ð n ; y n ; t n Þ: ð31þ The teraton converges very rapdly and t usually takes less than 10 teratons. For other cases where we do not have a good ntal guess of the soluton, or the Gauss Sedel teraton does not converge, or we want an uncondtonally stable scheme for the dffuson part, we smply solve the system of lnear equatons usng the sngular value decomposton (SVD). Iteratve solvers ncludng Quas-Mnmal esdual method (QM) [14] mght gve faster convergence, but t s challengng to fnd a good precondtoner for such nonsymmetrc matrx and so ths approach wll not be studed here. Ths mplct scheme descrbed above gves a CFL condton Dt = O(h) but stll s only frst order accurate. To ncrease the accuracy n solvng the dffuson equaton, we can replace the backward Euler step by a Crank Ncolson method f nþ1 ¼ I Dt 1 2 D ðt nþ1 Þ I þ Dt 2 D ðt nþ1 Þ f n : ð32þ Numercally, h the mplementaton of ths teraton method s smlar to (30) except that we replace Dt by Dt/2 and also defne g by I þ Dt D 2 ðt nþ1 Þ f n. Even though the Crank Ncolson scheme s of hgher order, the overall scheme f ð nþ1 ; y nþ1 ; t nþ1 Þ¼ðD nþ1 ;Dt=2 Þ 1 D nþ1 þ;dt=2 IA Dtf ð n ; y n ; t n Þ s stll frst order accurate due to the smple operator splttng. To obtan hgher order accuracy for the overall scheme, we need to splt the operators more accurately. One way s to use the Strang splttng whch ncely cancels the frst order error due to the smple splttng. There are two resultng numercal schemes whch are symmetrc to each other by swtchng the correspondng operators, f ð nþ1 ; y nþ1 ; t nþ1 Þ¼ IA Dt=2 h nþ1=2 ðd ;Dt=2 Þ 1 D nþ1=2 þ;dt=2 IA Dt=2 f ð n ; y n ; t n Þ ð34þ ð33þ and h f ð nþ1 ; y nþ1 ; t nþ1 Þ¼ ðd nþ1 ;Dt=4 Þ 1 D nþ1 þ;dt=4 ½ IA Dt h Š ðd n ;Dt=4 Þ 1 D n þ;dt=4 f ð n ; y n ; t n Þ: ð35þ To speed up the computatons, we combne adjacent operators n the tme marchng methods and obtan the followng schemes,

9 2548 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) and f ð nþ1 ;y nþ1 ;t nþ1 Þ¼ IA Dt=2 h ðd nþ1=2 ;Dt=2 Þ 1 D nþ1=2 þ;dt=2 h f ð nþ1 ;y nþ1 ;t nþ1 Þ¼ ðd nþ1 ;Dt=4 Þ 1 D nþ1 þ;dt=4 ½ IA Dt h ½IA Dt Š ðd 3=2 ;Dt=2 Þ 1 D 3=2 þ;dt=2 h ½IA Dt Š ðd 1=2 h h Š ðd n ;Dt=2 Þ 1 D n þ;dt=2 ðd 2 ;Dt=2 Þ 1 D 2 þ;dt=2 ½ ;Dt=2 Þ 1 D 1=2 þ;dt=2 IA Dt IA Dt=2 f ð 0 ;y 0 ;t 0 Þ ð36þ h Š ðd 0 ;Dt=4 Þ 1 D 0 þ;dt=4 f ð 0 ;y 0 ;t 0 Þ: ð37þ A fully mplct Crank Ncolson scheme In the prevous Secton, we have ntroduced varous mplct schemes based on operator splttng, ncludng the explct scheme (18), the backward Euler scheme (28), the Crank Ncolson scheme (33) and also two schemes based on Strang splttng (36) and (37). It s also desrable to develop a Crank Ncolson scheme wth no operator splttng, e.g. f nþ1 f n Dt ¼ 1 h 2 D ðt n Þf n þ D ðt nþ1 Þ f nþ1 : ð38þ Note that dfferent sets of foot-ponts are used to sample the two surfaces (t n ) and (t n+1 ). To solve f on the same set of footponts, we propose the followng strategy. We ntroduce a new varable F = D f whch s advected and s nterpolated at the new foot-ponts as descrbed n Secton 3.1. The resultng scheme can be wrtten as Fð n ; y n ; t n Þ¼D ðt n Þf ð n ; y n ; t n Þ; f ð nþ1 ; y nþ1 ; t nþ1 Þ¼ðD nþ1 ;Dt=2 Þ 1 IA Dt f ð n ; y n ; t n Þþ Dt 2 IA DtFð n ; y n ; t n Þ : ð39þ 3.4. Smoothng For motons dependng on hgh order dervatves of geometrcal quanttes, such as the surface Laplacan of the mean curvature, t may be necessary to apply smoothng to flter out hgh frequency modes n the soluton durng the evoluton to mantan stablty of the numercal scheme. One smple way s to smooth usng a Gaussan-type flter (e.g. [24]). Gven f ð~x Þ wth ~x s the coordnate n the (n 0 ) \ -drecton. We compute the weghted average quantty on the pont ~x ¼ ~x by P f ð~x Þ¼ wð~x; ~x Þf ð~x Þ P wð~x; ; ð40þ ~x Þ where wð~x; ~x Þ¼exp ð~x ~x Þ 2 2r 2! ð41þ and r = O(h) Surface dffuson flow and Wllmore flow In ths secton, we consder nterface moton that depends on hgh order geometry quanttes. We wll consder surface dffuson (e.g. [27,7,34]) and the related Wllmore flow [42]. For surface dffuson, we have a famly of compact, closed hypersurfaces (t) satsfyng the evoluton equaton v n ¼ D ðtþ H ð42þ wth (0) = 0, the ntal surface, and v n s the normal velocty defned on the surface (t). The quantty H s the total curvature of (t). Let A(t) and V(t) be the surface area of the surface (t) and ts enclosed volume, respectvely, then 1 2 Z daðtþ ¼ dt Z dvðtþ ¼ dt ðtþ ðtþ Z Z v n Hds ¼ HD ðtþ Hds ¼ jr ðtþ Hj 2 ds 6 0; ðtþ ðtþ v n ds ¼ 0; where r (t) s the surface gradent. Ths mples that the surface area of the surface (t) s non-ncreasng n tme whle the enclosed volume remans unchanged. A related geometrc flow s the Wllmore flow [42], whch nvolves a fourth order nonlnear evoluton equaton. The Wllmore energy for a compact closed hypersurface s ð43þ

10 EðÞ ¼ 1 2 Z H 2 ds: In addton to ts ntrnsc mathematcal nterest, ths model has been used to characterze the energy of vescles (e.g. [16,33]). Any crtcal pont of ths functonal s called a Wllmore surface, whch satsfes the equaton D H þ 2HðH 2 KÞ ¼0; where K s the Gaussan curvature. To determne such a surface, one may evolve a compact surface to the Wllmore surface va Wllmore flow ðtþ 0 ¼ reðþ whch mples the moton by the followng normal velocty v n ¼ D H þ 2HðH 2 KÞ: Both of these flows depend on the surface Laplacan of the mean curvature. To numercally approxmate ths quantty at each foot-pont, we frst nterpret the mean curvature as a functon defned on the nterface. The correspondng numercal quantty at each foot-pont can be approxmated from the local reconstructon. For nstance, n two dmensons for example, the mean curvature at the foot-pont can be approxmated by the local quadratc reconstructon f(s) usng jðs f 00 ðsþ Þ¼ : ð48þ f1 þ½f 0 ½sŠŠ 2 g 3=2 s¼s For more detals on computng ths quantty at all foot-ponts, we refer nterested readers to [20]. Wth the mean curvature defned at all foot-ponts, we can then compute the surface Laplacan as n (9) by takng f = H. Snce the underlyng PDE s fourth order, solvng the resultng evoluton equaton usng an explct scheme leads to a CFL tme step restrcton of Dt = O(h 4 ). Here we descrbe a sem-mplct method to solve the surface dffuson moton and the Wllmore flow. We consder the followng relatons between the normal n and the tangent vector t on the nterface t s ¼ jn; n s ¼ jt: The fourth dervatve of the nterface s therefore gven by S. Leung et al. / Journal of Computatonal Physcs 230 (2011) ð44þ ð45þ ð46þ ð47þ ð49þ y ssss ¼ t sss ¼ð jnþ ss ¼ð j s n j 2 tþ s ¼ð j ss þ j 3 Þn 3jj s t ¼ð j ss j 2 y ss nþn 3jj s t: ð50þ Ths mples that the velocty nduced by surface dffuson can be wrtten as v ¼ j ss n ¼ y ssss j 3 n þ 3jj s t ¼ y ssss þ j 2 y ss þ v t: ð51þ Snce the tangental velocty on the nterface wll not change the correspondng shape, we smply advect the foot-ponts numercally usng the velocty v ¼ y ssss þ j 2 y ss : In partcular, we compute j 2 explctly usng the curvature on the nterface n at t = t n,.e. (j n ) 2, and treat y ssss and y ss mplctly as explaned n Secton 3.3.2, where s s the old metrc defned on the nterface at t = t n. Ths gves h y nþ1 y n ¼ Dt y nþ1 ssss þðjn Þ 2 y nþ1 ss ; " # 1 y n : ð53þ y nþ1 ¼ 4 Dtðjn Þ 2 ð52þ 4. Applcaton to flows wth local nextensblty In ths secton, we wll consder modelng only a two dmensonal flow wth a local nextensblty constrant gven by [18,30]

11 2550 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) ðv t Þ s jv n ¼ 0; where s s the arclength parametrzaton, j s the sgned curvature and v n and v t are the normal and the tangental veloctes, respectvely. It s possble to generalze ths approach to three dmensons. For nstance, preservng the surface area, we have dv s u ¼ 0; where dv s =trr s s the surface dvergence. Ths generalzes the local nextensblty condton from two dmensons (54) to dv s v t Hv n ¼ 0; where v t s the tangental velocty and H s the total curvature [29]. To mpose ths constrant n the flow nduced by the orgnal energy E o (), we ntroduce a locally defned Lagrange multpler K so that the new energy E n () we are mnmzng s gven by (e.g. [35]) Z E n ðþ ¼E o ðþþ K ðs s 0 Þd; ð57þ where s 0 and s are the arclength parametrzaton of the ntal nterface 0 and the current nterface, respectvely. Let v o n and v o t be the normal and the tangental veloctes whch decrease the orgnal energy E o () usng gradent descent. The velocty that decreases E n () s therefore gven by u ¼ðv o n þ KjÞn þðv o t þ K sþt: Thus K plays the role of an nterfacal tenson. The functon K defned on the nterface s now determned by requrng that u satsfes the local nextensblty condton ðu tþ s jðu nþ ¼0: Ths yelds the followng Helmholtz equaton for K K ss þ j 2 K ¼ðv o t Þ s jv o n : After we solve ths equaton, we update the nterface accordng to Eq. (58). It can be shown that ths modfed flow (58) stll mnmzes the orgnal energy E o (). For nstance, we have de o Z dt ¼ ðv o n n þ v o tþud t Z h Z ¼ ðv o n þðv Þ2 o t Þ2 d v okj n þ v ok t s d Z h Z ¼ ðv o n þðv Þ2 o t Þ2 d þ K 2 j 2 þ K 2 s d: Now, snce Z Z K 2 j 2 þ K 2 s d ¼ we obtan Z K 2 j 2 þ K 2 s And ths mples that de o dt 6 0; Z d 6 v okj n þ v ok Z h Z t s d 6 ðv o n þðv 1=2 1=2 Þ2 o t Þ2 d K 2 j 2 þ K 2 s d ; ð62þ h ðv o n þðv Þ2 o t Þ2 d: and the equalty holds f v o ¼ Kj n and v o ¼ K t s. In the case when the gradent descent of the orgnal energy E o () satsfes the local nextensblty condton, we have ðv o t Þ s jv o n ¼ 0; ð65þ and so K ss þ j 2 K ¼ 0: Ths equaton has a trval soluton K = 0. In partcular, f the steady state soluton s a crcle, ths trval soluton to K s a soluton to (60). Indeed, the soluton s not unque snce one can have the crcle rotatng n a constant but arbtrary speed such that ðv 0 t Þ s ¼ 0. In ths paper however, we wll not study the exstence and unqueness to the Eq. (60) for general nterfaces. Unlke those equatons we obtaned n developng mplct schemes for the advecton dffuson equaton, ths Helmholtz equaton s nhomogeneous and, even more challengng, the coeffcent n the Helmholtz operator, j 2, has a very large range ð54þ ð55þ ð56þ ð58þ ð59þ ð60þ ð61þ ð63þ ð64þ ð66þ

12 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) for general shapes of the nterface. The smple Gauss Sedel type teraton has dffculty n convergng to a steady state soluton. In the current mplementaton, we solve Eq. (60) usng the SVD. 5. Example 5.1. Advecton equaton on a crcle The frst example s a two-dmensonal case where a crcle centered at (0.5,0.75) wth radus 0.15 s rotated by the velocty u ¼ 2ð0:5 yþ; v ¼ 2ðx 0:5Þ: ð67þ The crcle rotates about (0.5, 0.5) wth perod T = p. On the crcle, we solve the advecton equaton where Df Dt ¼ 0; D s the materal dervatve along the trajectory of a partcle on the nterface. The ntal condton defned on the crcle s gven by f ðx; y; 0Þ ¼x: The exact soluton to ths nterface problem s gven by f ðxðtþ; yðtþ; TÞ ¼x 0 ; where x(t)=x(0) = x 0 and y(t)=y(0). Fg. 4 shows the rate of convergence of our method appled to solve ths smple advecton equaton on an evolvng surface, whch s approxmately thrd order (2.9453) Surface dffuson on a sphere We next smulate the dffuson of a materal along a movng surface. We frst repeat a smlar two-dmensonal example as n the prevous test where we agan consder the rotaton of a crcle as defned above. However, to match the scalng of the dffuson tme, we modfy the rgd body rotaton by ð68þ ð69þ ð70þ ð71þ u ¼ 200ð0:5 yþ; v ¼ 200ðx 0:5Þ: ð72þ The crcle now rotates about (0.5, 0.5) wth perod T = p/100. On the movng crcle, we solve the followng advecton dffuson equaton Fg. 4. Convergence of solvng advecton equaton on a rotatng crcle.

13 2552 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 5. Maxmum value of f(h,t) defned on the crcle under the rgd body rotaton and ts error usng the explct scheme wth the underlyng unform mesh of resoluton (frst row) and (second row). The crcles on the left subfelds are the computed soluton. The sold lne s the exact soluton. Fg. 6. Convergence of varous numercal schemes for solvng the heat equaton on an evolvng surface wth moton governed by (a) the smple rgd body rotaton and (b) the outward normal expanson.

14 Df Dt ¼ D Sf ; S. Leung et al. / Journal of Computatonal Physcs 230 (2011) where D/Dt s the materal dervatve along the trajectory of a partcle on the nterface. The ntal condton defned on the crcle s gven by ð73þ f ðh; 0Þ ¼snðnhÞ; where n =2,h = tan 1 [(y 0.75)/(x 0.5)]. The exact soluton to ths problem s gven by f ðxðtþ; YðtÞ; tþ ¼exp n2 t f ðxð0þ; Yð0Þ; 0Þ; r 2 ð74þ ð75þ where r s the radus of the crcle, and (X(t),Y(t)) s the trajectory satsfyng dxðtþ ¼ u; dt dyðtþ ¼ v: dt The maxmum value of f(h,t) on the nterface s then gven by max f ðh; tþ ¼exp n2 t : ð77þ h r 2 In Fg. 5, we plot the computed maxmum value of f(h,t) on the crcle at varous tmes usng the explct scheme wth an underlnng grd of resoluton (a), and also ther correspondng errors (b). We also plot the correspondng quanttes usng an underlnng grd of resoluton n Fg. 5 on the second row. ð76þ Fg. 7. Maxmum value of f(h,/,t) defned on the sphere and ts error usng the underlyng unform mesh of resoluton The crcles on the second row are the computed soluton. The sold lne s the exact soluton.

15 2554 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) In Fg. 6, we demonstrate the convergence of varous numercal schemes for solvng the heat equaton on an evolvng surface. In Fg. 6(a), we show the convergence results of the rgd body rotaton demonstrated above. In Fg. 6(b), we solve the surface heat equaton on an ntal crcle of radus r 0 = 0.1 centered at (0.5,0.5) expandng n the normal drecton wth speed 200,.e. v n = 200, for tme t f = p/4000. The ntal condton of a functon f defned on the nterface s gven by f(h;t = 0) = sn (nh) wth n = 2. The exact soluton on the crcle at the fnal tme can be found explctly, n 2 t f f ðh; t ¼ t f Þ¼exp snðnhþ: ð78þ r 0 ðr 0 þ v n t f Þ Fg. 8. ate of convergence for solvng the dffuson equaton on a sphere. Fg. 9. Evoluton of an ntal 7-fold symmetrc star shaped nterface by surface dffuson usng the explct method wth Dx = 1/128 and (upper row) Dt =16Dx 4 and (bottom row) Dt =32Dx 4.

16 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) As expected, the explct scheme gves approxmately frst order convergence whle the fully mplct CN scheme (39), and the Strang splttng schemes (36) and (37) all gve approxmately second order convergence. An nterestng observaton s that the CN scheme based on smple splttng exhbts a dfferent convergence behavor for these two types of moton. For the rgd rotaton, the underlyng metrc remans unchanged,.e. the surface Laplacan operator D ðt n Þ ¼ D ðt nþ1 Þ. Thus the smple splttng n fact concdes wth that from the fully mplct CN scheme (39), whch s why the methods (33) and (39) gve smlar results n Fg. 6(a). We repeat a smlar test n three-dmensons but wthout an appled velocty. We consder a sphere of radus 0.25 centered at (0.5,0.5,0.5) wth a functon defned on the surface ntally gven by a lnear combnaton of two sphercal harmonc functons f ðh; /; t ¼ 0Þ ¼sn 5 h cosð5/þþsn 4 h cos h cosð4/þ: The exact soluton to the heat equaton on ¼ D sf s gven by f ðh; /; tþ ¼exp 30t f ðh; /; 0Þ: r 2 0 ð79þ ð80þ ð81þ We show the maxmum value of f(h,/,t) on the surface at dfferent tmes and error n Fg. 7 usng a mesh of In Fg. 8, we have plotted the error defned as E Dx ¼ exp 0:3 max f ðh; /; 0Þ max f ðh; /; 0:01Þ; ð82þ r 2 0 p wth max f ðh; /; 0Þ ¼ 5 þ ffffffffffff 4= pffffffffffff p þ ffffffffffff 3=2 41. The fgure shows that the rate of convergence s approxmately 2 (1.8303). Fg. 10. Evoluton of an ntal 7-fold symmetrc star shaped nterface by surface dffuson usng the explct method wth Dx = 1/256 and (upper row) Dt =16Dx 4 and (bottom row) Dt =32Dx 4.

17 2556 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Moton by surface dffuson and Wllmore flow In ths secton, we smulate hgh order geometrc moton due to surface dffuson and Wllmore flow. Fgs show a smple case where an ntal 7-fold symmetrc star shaped nterface evolves by surface dffuson. In Fgs. 9 and 10 we plot the solutons at varous tmes durng the evoluton computed by the smple explct scheme usng an underlyng mesh wth Dx = 1/128 and Dx = 1/256, respectvely. Hgh order geometrc nformaton,.e. the surface Laplacan of the curvature, s computed by frst treatng the curvature as a functon defned at the foot-ponts, then computng the correspondng dervatves of the local least squares reconstructon of ths resultng functon. Gaussan smoothng of the curvature as descrbed n Secton 3.4 s appled before nterpolaton and dfferentaton. For a relatvely large Dt =32Dx 4, the solutons show nstablty n the evoluton. The Dt = O(Dx 4 ) constrant makes the computaton neffcent for small Dx. Fg. 11 shows the solutons at the fnal tme t f computed usng the sem-mplct scheme from Eq. (53) descrbed n Secton 3.5 usng varous Dt s. In the upper row, Fg. 11(a) and (b), we use an underlyng mesh wth Dx = 1/128. The tmestep Dt s chosen to be 128Dx 4 and 512Dx 4, respectvely. These tmesteps are sgnfcantly larger than the tmestep restrcton n the explct scheme. For the case where Dt 512Dx 4, the soluton s n fact obtaned by dong only four steps of tme marchng. Ths explans why the soluton n Fg. 11(b) s not accurate at all. Smlarly n Fg. 11(c) and (d), we have shown the soluton at the tme t f usng varous Dt s wth a fner mesh. All evolutons are stable but the accuracy n the soluton decreases when we ncrease the tme step. We have also shown the soluton to the Wllmore flow wth the same ntal condton n Fg. 12. As n the surface dffuson case, the sem-mplct scheme effectvely removes the restrctve tme step constrant Dt = O(Dx 4 ). The GBPM can be easly extended to three dmensons as descrbed n Secton 3.2. To demonstrate ths, n Fgs , the evoluton of a torus and a dumbbell shaped nterface under surface dffuson are shown. The topologcal changes n Fgs. 13 and 14 are captured ncely usng the GBPM. The dea s to deactvate any grd pont and ts assocated foot-pont whose Lagrangan nformaton at the correspondng foot-pont conflcts wth any of ts neghbor. emoval and addton of meshless samplng ponts s smple for the GBPM. Fg. 11. Soluton of an ntal 7-fold symmetrc star shaped nterface by surface dffuson at t f = usng the sem-mplct method wth (a) Dx =1/ 128 and Dt = 128 Dx 4 = Dx 2 /128 t f /16, (b) Dx = 1/128 and Dt = 512Dx 4 = Dx 2 /32 t f /4, (c) Dx = 1/256 and Dt = 512Dx 4 = Dx 2 /128 t f /64, and (d) Dx =1/ 256 and Dt = 2048Dx 4 = Dx 2 /32 t f /16.

18 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 12. Soluton of an ntal 7-fold symmetrc star shaped nterface under Wllmore flow at t f = usng the sem-mplct method wth (a) Dx =1/ 128 and Dt = 128Dx 4 = Dx 2 /128 t f /16, (b) Dx = 1/128 and Dt = 512Dx 4 = Dx 2 /32 t f /4, (c) Dx = 1/256 and Dt = 512Dx 4 = Dx 2 /128 t f /64, and (d) Dx =1/ 256 and Dt = 2048Dx 4 = Dx 2 /32 t f /16. Fg. 13. Moton of a torus by surface dffuson. The radus at the ntal tme s The topologcal change s ncely captured by the GBPM. We have also performed analogous smulatons for moton by the Wllmore flow; the results are shown n Fgs As observed prevously (e.g. [26]) the dumbbell does not pnch off under Wllmore flow but rather evolves to a sphere Local nextensble flows Consder frst a case n whch a crcle of radus r 0 s evolved by a flow n the normal drecton wth v 0 n ¼ 1 and v 0 t ¼ 0. Ths velocty feld does not satsfy the nextensblty constrant. In partcular, the arclength grows lnearly wth tme gven by 2p(r 0 + t). Imposng the local nextensble constrant gven by Eq. (60), we have

19 2558 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 14. Moton of a dumbbell shape by surface dffuson. The radus of the mddle cylndrcal part at the ntal tme s The topologcal change s ncely captured by the GBPM. Fg. 15. Moton of a dumbbell shape by surface dffuson up to t =10 4. The radus of the mddle cylndrcal part at the ntal tme s The topologcal change s ncely captured by the GBPM. Fg. 16. Moton of a torus by Wllmore flow up to t = The radus at the ntal tme s K hh þ K ¼ r: Assumng the soluton K s ndependent of h, we have K = r and therefore Eq. (58) gves u = 0 whch means that the ntal crcle should stay crcle wth the same radus. In Fg. 19(a) we have shown the change n the total arclength of a crcle wth an ntal radus r 0 = 0.1. The blue crcles are the computed total arclength under the flow v 0 n ¼ 1 wthout the local nextensblty constrant. The computed total length of the nterface when the local nextensble constrant s appled s shown n red. ð83þ

20 S. Leung et al. / Journal of Computatonal Physcs 230 (2011) Fg. 17. Moton of a dumbbell shape by Wllmore flow up to t = The radus of the mddle cylndrcal part at the ntal tme s Fg. 18. Moton of a dumbbell shape by Wllmore flow up to t = The radus of the mddle cylndrcal part at the ntal tme s A slghtly more complcated case s the moton by mean curvature. Wepstart ffffffffffffffffffffffffffffffffffffffffffff wth an ntal crcle of radus r 0 = 0.15,.e. the curvature j =1/r 0. The total arclength of the crcle s gven by rðtþ ¼ 0:15 2 2t for 0 6 t /2. In Fg. 19(b) we show the computed arclength of the evoluton of the crcle wth and wthout the local nextensble constrant. Smlar to the prevous case, the crcle stays unchanged wth nextensblty. The next example s the Wllmore flow (46). In two dmensons, the correspondng normal velocty reduces to v n ¼ D j 1 2 j3 : ð84þ And, therefore, an ntally crcular nterface grows n the outward drecton wth radus gven by rðtþ ¼ð2t þ r 4 0 Þ1=4.In Fg. 19(c), we show the computed arclength of an ntal crcle of radus r 0 = 0.15 under Wllmore flow wth and wthout the local nextensble constrant. The results are smlar to the prevous cases. We next consder a more extreme case for Wllmore flow wth an ntal 4-fold symmetrc nterface profle r ¼ r 0 ½1 þ cosð4hþš where r 0 = 0.25 and = In Fg. 20, we show the evoluton of the ntal nterface under Wllmore flow wth (a) and wthout (b) the local nextensble constrant. In both of these fgures, we plot the ntal condton as the red dashed lne. The nterface at dfferent tmes s plotted usng blue sold lnes. Even though both flows (the constraned and the unconstraned flows) tend to a crcle, the rad of the crcles are dfferent snce the constraned flow preserves the nterface length. ð85þ 6. Concluson and future work In ths work, we have developed numercal algorthms for solvng advecton dffuson PDEs on movng nterfaces usng the grd based partcle method (GBPM). We also demonstrated the ablty of the GBPM to smulate hgh order geometrc flows ncludng changes n the nterface topology. To solve the equatons effcently, we developed sem-mplct methods to overcome the severe tme step constrants requred for the stablty of explct methods. In the future, we plan to extend the algorthms to ncorporate global constrants such as the conservaton of enclosed volume. Because of the local representaton of the nterface n the GBPM ths s a challengng task. In addton, we plan to extend the GBPM to solve problems n whch there s couplng between bulk and surface processes.