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1 Journal of Computatonal Physcs 30 (0) Contents lsts avalable at ScenceDrect Journal of Computatonal Physcs journal homepage: An teratve method to solve a regularzed model for strongly nonlnear long nternal waves Wooyoung Cho a,b,, Arnaud Goullet a, Tae-Chang Jo c a Department of Mathematcal Scence, Center for Appled Mathematcs and Statstcs, New Jersey Insttute of Technology, Newark, NJ , USA b Dvson of Ocean Systems Engneerng, Korea Advanced Insttute of Scence and Technology, Daejeon , South Korea c Department of Mathematcs, Inha Unversty, Incheon 40-75, South Korea artcle nfo abstract Artcle hstory: Receved 7 June 00 Receved n revsed form November 00 Accepted 30 November 00 Avalable onlne 3 December 00 Keywords: Internal waves Regularzed model Iteratve method We present a smple teratve scheme to solve numercally a regularzed nternal wave model descrbng the large ampltude moton of the nterface between two layers of dfferent denstes. Compared wth the orgnal strongly nonlnear nternal wave model of Myata [0] and Cho and Camassa [], the regularzed model adopted here suppresses shear nstablty assocated wth a velocty jump across the nterface, but the couplng between the upper and lower layers s more complcated so that an addtonal system of coupled lnear equatons must be solved at every tme step after a set of nonlnear evoluton equatons are ntegrated n tme. Therefore, an effcent numercal scheme s desrable. In our teratve scheme, the lnear system s decoupled and smple lnear operators wth constant coeffcents are requred to be nverted. Through lnear analyss, t s shown that the scheme converges fast wth an optmum choce of teraton parameters. After demonstratng ts effectveness for a model problem, the teratve scheme s appled to solve the regularzed nternal wave model usng a pseudo-spectral method for the propagaton of a sngle nternal soltary wave and the head-on collson between two soltary waves of dfferent wave ampltudes. Ó 00 Elsever Inc. All rghts reserved.. Introducton Recently nonlnear nternal soltary waves propagatng n densty stratfed oceans have attracted much attenton and the number of feld observatons s rapdly ncreasng [5]. These long nternal waves frequently observed n coastal regons are often strongly nonlnear as the wave ampltude s comparable to the characterstc vertcal length scale. Therefore, the wellknown weakly nonlnear models such as the Korteweg-de Vres (KdV) equaton for un-drectonal waves and the Boussnesq equatons for b-drectonal waves commonly used for long surface waves n a homogeneous layer have been found to fal to descrbe large ampltude nternal soltary waves [,8]. When the densty changes rather abruptly over a thn transton layer (called the pycnoclne), the stratfed ocean s often appromated by two flud layers of dfferent constant denstes for whch the hgher-order nonlnear dspersve effects can be ncorporated nto a relatvely smple system of nonlnear evoluton equatons descrbng the moton of the nterface between the two flud layers [,0]. Its soltary wave solutons have been found to agree well wth laboratory eperments and numercal solutons of the Euler equatons [,]. Snce ths strongly nonlnear model neglects the effects of vscosty and a Correspondng author at: Department of Mathematcal Scence, Center for Appled Mathematcs and Statstcs, New Jersey Insttute of Technology, Newark, NJ , USA. E-mal address: wycho@njt.edu (W. Cho) /$ - see front matter Ó 00 Elsever Inc. All rghts reserved. do:0.06/j.jcp

2 0 W. Cho et al. / Journal of Computatonal Physcs 30 (0) thn densty transton layer between the two layers, the tangental veloctes are dscontnuous across the nterface and ts ntal value problem s known to be ll-posed [6,7,9]. To overcome ths dffculty, the model has been regularzed [3] by changng the short-wave behavor that s nsgnfcant n the long-wave dynamcs and, n fact, s modeled ncorrectly n the long wave asymptotc model. An attempt to solve the regularzed model numercally was made n Cho et al. [3], but t was found desrable to develop a more effcent numercal method due to a more complcated couplng between the upper and lower layers compared wth the orgnal strongly nonlnear model. In ths paper, we ntroduce an teratve scheme to solve the regularzed model of Cho et al. [3] and dscuss ts convergence. After presentng the regularzed model n Secton, we descrbe the teratve scheme and the result of convergence analyss n Secton 3. Wth choosng an optmum set of teraton parameters, we solve the regularzed model numercally for the propagaton of a sngle soltary wave and the asymmetrc head-on collson between two soltary waves to demonstrate the effectveness of the teratve scheme n Secton 4.. A regularzed strongly nonlnear model A regularzed strongly nonlnear nternal wave model for a system of two constant-densty layers bounded by the upper and lower rgd boundares can be wrtten [3], n terms of the local layer thcknesses g and the horzontal veloctes u evaluated at the rgd boundares n the upper ( = ) and lower ( = ) layers, as u 6 g u ; ¼ 0; ð: g ;t þ g u ;t þ u u ; þ gf ¼ P þ q g u ;t þ u u ; u ; : ð: Here the subscrpts and t represent partal dfferentaton wth respect to space and tme, respectvely, and the local layer thcknesses are defned by g ¼ h f; g ¼ h þ f; ð:3 where h are the undsturbed layer thcknesses and f s the dsplacement of the nterface. We remark that the orgnal strongly nonlnear long wave model of Myata [0] and Cho and Camassa [] wrtten n terms of the depth-averaged horzontal veloctes suffers from Kelvn Helmholtz (KH) nstablty due to a velocty dscontnuty across the deformed nterface. By ntroducng the horzontal veloctes at the upper and lower walls, u, t was shown n Cho et al. [3] that the model gven by (.) and (.) suppresses the KH nstablty when t s lnearzed. Snce the system s wrtten n a conserved form, t has the followng obvous conservaton laws d dt Z fd ¼ 0; d dt Z u d ¼ 0: We remark that, for numercal computatons, t s convenent to rewrte Eq. (.) as q u ðg u ; t ¼ P q u þ gf ðg u u ; þ 6 g u ;ðg 3 u ; ð:4 ; ð:5 where (.) has been used for g,t. Obvously, the last term on the rght-hand sde s a asymptotcally hgher-order term than the remanng terms and could be dropped for consstency. Nevertheless, t s kept here snce we would lke to solve (.) eactly. In terms of the well-posedness of the lnearzed system, the last term makes no dfference snce t s a nonlnear term and, therefore, t can be dropped wthout affectng the well-posedness f one wants to have a model asymptotcally equvalent to (.). The system of four equatons gven by (.) and (.5) for = and can be reduced to a system of two tme evoluton equatons. Choosng (.) for = and subtractng (.5) for = and = to elmnate P yelds the followng two evoluton equatons for f and V f t ¼ g u 6 g u ; ; ð:6 V t ¼ X ð q u þ gf ðg u u ; þ 6 g u ;ðg 3 u ; ¼ ¼ ; ð:7 where V(,t) s defned by V ¼ X ð q u ðg u ; : ð:8

3 W. Cho et al. / Journal of Computatonal Physcs 30 (0) Once the evoluton equatons gven by (.6) and (.7) are ntegrated n tme for f and V, the horzontal veloctes u can be found by solvng the followng system of coupled dfferental equatons wth varable coeffcents g u 6 g u ; þ g u 6 g u ; ¼ C; ð:9 q u ðg u ; q u ðg u ; ¼ V: ð:0 Notce that g n (.9) and (.0) are known snce f s already computed at a new tme step. Here, whle the second equaton s nothng but (.8), the frst Eq. (.9) s obtaned by addng (.) for = and to elmnate f t and ntegratng once n wth an ntegraton constant C, whch depends on the boundary condtons. Physcally (.9) mples that the volume flu s ndependent of. When dscretzed, the system gven by (.9) and (.0) could be computatonally epensve to solve, n partcular, for twodmensonal problems unless an effcent numercal scheme s found. Even for one-dmensonal problem, f one uses a pseudo-spectral method, the operator becomes a full matr whch s tme-consumng to nvert at every tme step snce the operator depends on tme. In fact, n Cho et al. [3], a drect nverson method wth a second-order fnte dfference scheme to appromate the operators was adopted to solve (.9) and (.0) and found numercally neffectve. 3. An teratve scheme and ts convergence To solve the system gven by (.9) and (.0) effcently, we rewrte (.9) as g U þ g U ¼ R ; q U q U ¼ R ; ð3: ð3: where U and R are gven by U ¼ u a h u ;; ð3:3 R ¼ C þ X 6 g3 u ; a g h u ; ; ð3:4 ¼ R ¼ V X ¼ ð q ðg u ; a h u ; : ð3:5 In (3.3) (3.5), notce that terms wth constant a are ntroduced to wrte (3.) and (3.) as a system of equatons for U from whch u can be easly obtaned by nvertng the lnear operators wth constant coeffcents gven by (3.3). Then, as descrbed n Append A, a are determned for a new teratve scheme to converge fast and are assumed to be postve so that the operators to be nverted to fnd u from known U are postve defnte. Once the rght-hand sdes R are evaluated wth the results from the prevous teraton step, (3.) and (3.) are the lnear equatons for U at the new teraton step, whose epressons n terms of R can be found analytcally. Then, u can be computed by nvertng ndependently for = and = the lnear operators wth constant coeffcents defned n (3.3), as mentoned prevously. It s convenent to nvert these constant operators, for eample, when a pseudo-spectral method s used. On the other hand, f one uses a fnte dfference method for spatal dscretzaton, the coeffcents do not have to be constant. Therefore, a h n (3.3) and (3.5) and a g h n (3.4) can be replaced by a g and a g 3, respectvely, and an teratve scheme descrbed below can be appled wthout any modfcaton. A smlar teratve scheme can be used for a system gven by (.9) and (.0), but, then, u have to be solved smultaneously snce the terms n the square brackets that appear on the left-hand sdes of (.9) and (.0) are all dfferent. 3.. Iteratve scheme For gven g, the solutons of (3.) and (3.) can be obtaned by usng the followng teratve scheme g U ðnþ þ g U ðnþ ¼ R ðn ; ð3:6 q U ðnþ q U ðnþ ¼ R ðn ; ð3:7 where U ðnþ are U at the (n + )th teraton step and R ðn from the nth teraton step. Then, U ðnþ U ðnþ are the rght-hand sdes of (3.) and (3.) evaluated wth the results can be found analytcally as ¼ q Rðn þ g R ðn ; U q g þ q g ðnþ ¼ q Rðn g R ðn : ð3:8 q g þ q g

4 04 W. Cho et al. / Journal of Computatonal Physcs 30 (0) Now the equatons for the upper and lower layers are decoupled and u can be found ndependently for = and =as U u ðnþ ¼ a ðnþ : ð3:9 The operator (3.9) can be easly nverted numercally. For eample, for the second-order central dfference scheme, a trdagonal matr solver can be used and, for a pseudo-spectral method, t s smple dvson snce the operator s ndependent of space by constructon. We emphasze that the operators are rearranged wth ntroducng a before any numercal dscretzaton of the orgnal operators s made so that the epressons for U can be found analytcally, as shown n (3.8). Then, the resultng lnear systems wth smple operators gven by (3.9) are nverted to fnd u wthout any operator splttngs, whch are often requred for the well-known teratve schemes such as Jacob or Gauss Sedel methods. Ths teratve scheme s appled repeatedly untl the followng condton s met u ðnþ u ðn ma u ðn < ; ð3:0 ma where we adopt = 0 for our computatons presented later. Alternatvely, we could use a crteron based on the resduals of the dscretzed lnear system gven by (3.) and (3.) wth whch we contnue the teraton process untl the resduals reach a certan tolerance level. As wll be dscussed n Secton 3.3, the crteron gven by (3.0) s found more satsfactory and s therefore adopted n ths paper. In the followngs, we derve a condton under whch the teratve scheme converges for fed physcal parameters (h and q ), and show how to choose a such that fastest convergence s acheved. 3.. Convergence For Fourer analyss, we frst lnearze the system of (3.) and (3.) wth assumng that f/h h a u ðnþ þ h a u ðnþ ¼ C þ X 6 a h 3 uðn ; ; ¼ q a u ðnþ q a u ðnþ ¼ V X ð q a ¼ By substtutng the followng epresson nto (3.) and (3.) u ðn h uðn ; : ð3: ð3: ¼ e k ; ð3:3 where k s the wave number, we have h ð þ a k h aðnþ þ h ð þ a k h aðnþ ¼ C X q ð þ a k h aðnþ q ð þ a k h aðnþ ¼ V þ X ¼ ¼ 6 a k h 3 aðn ; ð3:4 ð q a k h aðn ; ð3:5 where f represents the Fourer transform of f. For convergence of our teratve scheme, the behavor for large k s crucal (see Append A) and, as k?, (3.4) and (3.5) can be appromated to a h 3 a h 3 ðnþ a ¼ a 3 6 h a 3 6 h ðn a a q h a q h a ðnþ q a h q a ; ð3:6 h whch can be re-wrtten as a ðnþ ¼ Aða a ðnþ ; a ; ð3:7 where matr A depends on a for fed physcal parameters (h and q ). For convergence, the absolute values of two egenvalues of matr A, k,, have to be smaller than jk ða ; a j < ; whch determne the ranges of a. For smplcty, the two flud denstes are assumed to be close to each other so that q /q (relevant for oceanc applcatons), the matr A becomes 0 h a þ h a 6 a h. 3 ð3a h B C A h þ h h 3 ð3a h h a 6 þ h a A; ð3:9 a ð3:8

5 W. Cho et al. / Journal of Computatonal Physcs 30 (0) whose two egenvalues are gven by h = k ; ¼ a ð3h þ h þa ðh þ 3h ða ð3h þ h a ðh þ 3h þ 6a a h h : ð3:0 a a ðh þ h a a ðh þ h Then, for convergence of our teratve scheme, we choose a for whch the mamum of the two egenvalues has to be less than. Ths happens when the frst two terms n (3.0) vansh and the absolute values of the two egenvalues are the same and less than. Ths requres a to satsfy the followng relatonshp ð3h a þ h a ¼ ðh þ h a ðh þ 3h : From (3.), snce a > 0, as mentoned prevously, t can be seen that a satsfes the followng nequalty a > h þ 3h ðh þ h > 0; ð3: ð3: under whch t can be shown that the condton gven by (3.8) s always fulflled. Fnally, for optmum values of a, we need to choose a whch makes k as small as possble and the results are a ¼ h þ 3h 6ðh þ h ; a ¼ 3h þ h 6ðh þ h : ð3:3 Then, the teratve scheme s epected to converge fastest and jk j s gven by = 4h h jk j¼jk j¼ ; ð3:4 ð3h þ h ðh þ 3h whch s less than /. Fg. 3.(a) shows jk j, for varyng a, gven by (3.0) wth a negatve sgn for the depth rato of h /h = 3 whle Fg. 3.(b) shows a regon n the (a,a )-plane where the teratve scheme converges. In the shaded regon, both ja jand ja j are less than whle ja j > n the non-shaded regon. For the depth rato of h /h = 3, the optmum values of a ndcated by the dot are a = 5/ and a = /4 from (3.3) and the absolute values of k are jk j = /5. For the case of arbtrary densty ratos, the optmum values of a are gven n Append B wth ncludng the effect of fnte wave ampltude a, whch requres one to change the local thcknesses h and h to h a and h + a, respectvely A smple test for convergence To test the teratve scheme, we consder a model problem smlar to (.9) and (.0) wth h =, h =3, q =, and q =.003 Fg. 3.. (a) jk j gven by (3.0) wth a negatve sgn for h /h = 3. (b) The teratve scheme converges for a n a shaded regon where jk j <. On the dashed lne, the absolute values of two egenvalues are dentcal (jk j = jk j). The mnmum egenvalues occur at optmum values of a whch are gven by (3.3) and ndcated by the dot n the fgure.

6 06 W. Cho et al. / Journal of Computatonal Physcs 30 (0) Table 3. Comparson of the numercal solutons of (3.5) and (3.6) wth the eact solutons gven by (3.7). Here N teraton s the number of teratons wth tolerance =0 and e are defned as e = j u eact u numercal j ma /ju eact j ma. Notce that the optmum values are a = 5/ and a = /4 and a second-order fnte dfference scheme has been used for spatal dscretzaton. For a = 5/4 and a = /8, the teratve scheme faled snce jk j >. a a N teraton e 0 4 e 0 4 5/ / / / / / / / / / / / / / / / /4 /8 faled Table 3. Error estmates for varyng D wth the optmum values of a gven by (3.3). a a D N teraton e 0 4 e 0 4 e D m 5/ /4 p/ / /4 p/ m = / /4 p/ m = h u h 6 u ; q u h u ; þ h u h 6 u ; q u h u ; ¼ r ð; ¼ r ð; ð3:5 ð3:6 where r () and r () are gven by substtutng the followng eact solutons nto the left-hand sdes of (3.5) and (3.6), respectvely u eact ð ¼snðþcosð; u eact ð ¼ cosðþ3 snð; for p: ð3:7 To solve (3.5) and (3.6) usng the teratve scheme, we use a second-order fnte dfference method wth D = p/00 for spatal dscretzaton and =0.InTable 3., the numercal solutons are compared wth the eact solutons gven by (3.7). Clearly the least number of teratons s requred wth the optmum values of a gven by (3.3) and the teratve scheme fals wth a n a dvergent regon (the non-shaded regon n Fg. 3.). Ths ndcates that our analyss for k? presented n Secton 3. s vald although the mamum wave number resolved n our computatons (whch s p/d) s large, but fnte. Snce we use a second-order dfference appromaton, the errors of our numercal solutons are proportonal to D, as shown n Table 3.. The mamum resduals of the dscretzed system of (3.5) and (3.6) for our numercal solutons are also computed as , , and for D =p/00, p/400, and p/800, respectvely. Snce the resduals are senstve to the choce of D, a crteron based on the resduals dscussed n Secton 3. seems to be less useful for our teratve scheme. 4. Numercal solutons of the regularzed model To solve the evoluton Eqs. (.6) and (.7) numercally, we use a fourth-order Runge Kutta method for tme ntegraton and a pseudo-spectral method for spatal dscretzaton. As descrbed n Secton, once f and V are updated at a new tme Table 4. Average number of teratons for varyng a for the propagaton of a soltary wave wth wave ampltude a 0 = 0.6 and physcal parameters (q,q, h,h,g)= (,.003,,3,). A pseudo-spectral method s used for spatal dscretzaton wth N = 9 and the total doman length s L = 00 whle a fourth-order Runge Kutta method s used for tme ntegraton wth Dt = 0.0 and t ma = 000. a N teraton wth a = Faled k k a N teraton wth a = Faled k k

7 W. Cho et al. / Journal of Computatonal Physcs 30 (0) Fg. 4.. Numercal solutons of the regularzed model for a sngle soltary wave n a reference frame movng wth constant speed c. The ntal wave profle s the soltary wave soluton of the model of Myata Cho Camassa and c s ts wave speed. (a) a = 0.4 and c = 0.05; (b) a/h = 0.8 and c = Fg. 4.. Numercal solutons of the regularzed model: (a) Head-on collson of two soltary waves of the Myata Cho Camassa model wth ampltudes a = 0.8 and a = 0.4. (b) Evoluton of an ntal profle gven by f(,0) = 0.6 sech (0.) wth zero horzontal velocty. step, the horzontal veloctes u and u can be found ndependently by nvertng the lnear operators wth constant coeffcents defned n (3.9). Ths nverson s a smple dvson n Fourer space such that the Fourer transform of u ðnþ can be found as ^u ðnþ ¼ðþa h k U b ðnþ. To test the teratve scheme, we consder the propagaton of a sngle soltary wave of ampltude a 0 = 0.6 of the orgnal model of Myata[0] and Cho and Camassa [] wth (q,q,h,h,g) = (,.003,,3,). It should be emphaszed agan that the propagaton of a sngle soltary wave cannot be smulated wth the orgnal model whch s llposed. For these physcal parameters, the optmum values for a are found, from (B.0) for fnte ampltude waves, as a = and a = In Table 4., for varyng a, we present the average number of teratons necessary for convergence at each substep of the Runge Kutta method wth D t = 0.0 satsfyng the CFL condton based on the lnear long wave speed for 0 6 t Here, the length of our total computatonal doman s taken to be L = 00 and the number of grd ponts (Fourer modes) s N = (5/ ) 9 among whch 9 Fourer modes are meanngful snce the rest s used to elmnate alasng errors wth consderng that the regularzed model s the fourth-order nonlnear equatons. From Table 4., we notce that the fastest convergence s acheved wth the optmum values of a estmated from (B.0), but the teratve scheme s so effectve that a small number of teratons s requred even for non-optmum teraton parameters as long as the absolute values of the correspondng egenvalues are less than one. Fg. 4. shows the long-term numercal solutons of the regularzed model (.) and (.) ntalzed wth the soltary wave soluton of the orgnal strongly nonlnear model of Myata [0] and Cho and Camassa [] for 0 6 t and wth Dt = 0. and N = (5/) 0. The ntal wave ampltudes are a = 0.4 and a = 0.8 for whch the optmum teraton parameters are, from (B.0), (a,a ) = (0.7509,0.30) and (a,a ) = (.33,0.704), respectvely. For these numercal computatons, the average number of teraton per each tme ntegraton s appromately N teraton = 9 for a = 0.4 and N teraton = 0 for a = 0.8. Compared wth the numercal scheme used n Cho et al. [3] where the drect nverson of a lnear system s requred, the new teratve scheme s found more than 4 tmes faster. The elapsed computng tmes for a = 0.8 are measured 0.35 s and.54 s per tme step for the new and old numercal schemes, respectvely. Notce that small dspersve waves are shed downstream snce the ntal wave profle s close to, but not the soluton of the regularzed model. To further test the new teratve scheme for more general tme-dependent problems, we consder two dfferent ntal condtons wth N = (5/) 9 and Dt = 0.0. Fg. 4.(a) shows the asymmetrc head-on collson between two soltary waves

8 08 W. Cho et al. / Journal of Computatonal Physcs 30 (0) of the Myata Cho Camassa model of ampltudes 0.8 and 0.4. Durng the collson, the peak ampltude reaches about.63. Here a are computed based on the ntal ampltude of the larger soltary wave whch s much less than the local mamum ampltude of.63 durng ths computaton, but the teratve scheme shows no sgn of dvergence. Fg. 4.(b) shows the tme evoluton of an ntal profle gven by f(,0) = 0.6sech (0.) located at the center of the computatonal doman wth perodc boundary condtons. Wth u =0att = 0, the ntal profle s splt nto two soltary waves, whch propagate n the opposte drectons to collde symmetrcally at the boundares of the computatonal doman. Here we compute a based on the ntal mamum dsplacement of the nterface so that (a,a ) = (0.938,0.9). 5. Concludng remarks We propose an effectve teratve method to solve a system of coupled nonlnear evoluton equatons, regularzed to suppress shear nstablty, for large ampltude long nternal waves n a two-layer system. Through lnear analyss, a condton for teraton parameters for convergence s provded and tested. It s found that the teratve scheme wth the optmum values for fastest convergence s effectve n solvng the regularzed model even when the wave ampltude s large. Although the one-dmensonal model s consdered n ths paper, a smlar teratve model can be used to solve the two-dmensonal regularzed model [4] wth bottom topography. Acknowledgements WC and AG gratefully acknowledge support from the Offce of Naval Research through Grant N and the WCU (World Class Unversty) program through the Natonal Research Foundaton of Korea funded by the Mnstry of Educaton, Scence and Technology (R ). The work of TJ was supported by Basc Scence Research Program through the Natonal Research Foundaton of Korea funded by the Mnstry of Educaton, Scence and Technology ( ). Append A. Iteratve operator nverson Consder a smple equaton for u() gven by L½uŠ u ¼ f ð; ða: where f() s a known slowly-varyng functon. Although (A.) can be solved analytcally, we attempt to solve t numercally to eplan our teratve scheme. Snce u() s epected to be a slowly varyng functon and the second term on the left-hand sde of (A.) s supposed to be small compared wth the frst term, we assume to use the followng smple teratve scheme. u ðnþ ¼ f ðþu ðn ; where u (n) s the estmate at the nth teraton step and u (0) s an ntal guess. In Fourer space, by wrtng u (n) = a (n) e k, (A.) yelds an equaton for a (n+) a ðnþ ðk ¼ f ðk k ¼ h f ðk k f ðk k a ðn ¼ ð k nþ f ðkþð k nþ a ð0 ðk: ð k For large k, a (n+) can be appromated by a ðnþ ðk ð k n f ðkþð k nþ a ð0 ðk; ða: ða:3 ða:4 whch mples that a (n+) (k) for large k grows as the number of teraton n ncreases and the teraton scheme would fal no matter how small a (0) (k) s. Ths concluson also holds even for the case of a (0) (k) = 0 whch s dentcal to an teraton scheme gven by u ðnþ ¼ u ðn wth u (0) = f(). To overcome ths dffculty, we frst replace the teraton scheme, after subtractng au from both sdes of (A.), by u ðnþ au ðnþ ¼ f ðþð au ðn ; where a s a constant to be determned, and then a (n+) can be found as ða:5 a ðnþ ðk ¼ ½ða k =ð þ ak Š nþ ½ða k =ð þ ak Š " # nþ f ðk ða k þ a ð0 ðk; ða:6 þ ak ð þ ak whose behavor for large k s found to be nþ a ð0 ðk: ða:7 a ðnþ ðk ¼ a a

9 W. Cho et al. / Journal of Computatonal Physcs 30 (0) Then, for convergence, we have the followng condton for a a < ; a whch yelds a >. Notce that the case of a = corresponds to drect nverson of operator L. ða:8 Append B. Iteraton parameters for arbtrary wave ampltudes and densty ratos We frst lnearze the system of (3.) and (3.) about the mamum dsplacement from the mean nterface (or about f = a) g a u ðnþ þ g a u ðnþ ¼ C þ X 6 g a h g u ðn ; ðb: ; ¼ q a u ðnþ q a u ðnþ ¼ V X ð q g a h ¼ u ðn ; ; where g and g should be understood as h a and h + a. Then, by substtutng nto the lnearzed system u ðn we have ¼ e k ; ðb:3 g ð þ a k h aðnþ þ g ð þ a k h aðnþ ¼ C X q ð þ a k h aðnþ q ð þ a k h aðnþ ¼ V þ X ¼ ¼ ðb: 6 g a h k g ; ðb:4 ð q g a h k ; ðb:5 where f represents the Fourer transform of f. For convergence of the teratve scheme, the behavor for large k s crucal and, as k?, ths system can be appromated to 0 a g h a g h ðnþ a g 6 g a h g 6 g a h B C a q h a q h a ðnþ q g a h q g a A aðn ; ðb:6 h whch can be wrtten as a ðnþ ¼ Aða a ðnþ ; a ; ðb:7 where matr A dependng on a and physcal parameters such as h, q, and the characterstc wave ampltude a s gven by 0 h A ¼ a h q g g a h þ q g 6 g a h q h g 3 a =3 B h K q h g 3 a =3 a h q g g a h þq g 6 g a A; ðb:8 h wth K ¼ a a h h ðg q þ g q. The egenvalues of matr A can be found as k ; ¼ a h g ð3q g þ q g þa h g ðg q þ 3g q a a h hðg q þ g q h = ða h g ð3q g þ q g a h g ðq g þ 3q g þ 6a a q q h h g 3 g 3 a a h hðg q þ g q : ðb:9 The mnmum of ma {,} jk j occurs when a ¼ g ðq g þ 3q g 6h ðq g þ q g ; a ¼ g ðq g þ 3q g 6h ðq g þ q g ; ðb:0 and the correspondng egenvalue becomes jk j¼ 4q q g g = ; ðb: ðq g þ 3q g ðq g þ 3q g

10 030 W. Cho et al. / Journal of Computatonal Physcs 30 (0) whch can be shown to be less or equal to /. Therefore, the teratve scheme should converge wth the values of a gven by (B.0) for any wave ampltudes and densty ratos. As a? 0 and q /q?, notce that (B.0) and (B.) can be reduced to (3.3) and (3.4), as epected. References [] R. Camassa, W. Cho, H. Mchallet, P. Rusas, J.K. Sveen, On the realm of valdty of strongly nonlnear asymptotc appromatons for nternal waves, J. Flud Mech. 549 (006) 3. [] W. Cho, R. Camassa, Fully nonlnear nternal waves n a two-flud system, J. Flud Mech. 396 (999) 36. [3] W. Cho, R. Barros, T. Jo, A regularzed model for strongly nonlnear nternal soltary waves, J. Flud Mech. 69 (009) [4] A. Goullet, R. Barros, W. Cho, Two-dmensonal evoluton of nternal soltary waves: a model and ts numercal solutons, 00, n preparaton. [5] K.R. Helfrch, W.K. Melvll, Long nonlnear nternal waves, Ann. Rev. Flud Mech. 38 (006) [6] T.-C. Jo, W. Cho, Dynamcs of strongly nonlnear soltary waves n shallow water, Stud. Appl. Math. 09 (00) [7] T.-C. Jo, W. Cho, On stablzng the strongly nonlnear nternal wave model, Stud. Appl. Math. 0 (008) [8] C.G. Koop, G. Butler, An nvestgaton of nternal soltary waves n a two-flud system, J. Flud Mech. (98) 5 5. [9] R. Lska, L. Margoln, B. Wendroff, Nonhydrostatc two-layer models of ncompressble flow, Comput. Math. Appl. 9 (995) [0] M. Myata, Long nternal waves of large ampltude, n: H. Horkawa, H. Maruo (Eds.), Proceedngs of the IUTAM Symposum on Nonlnear Water Waves, 988, pp