Primary Results on Simulation of a New Kind Traveling Wave. of Permanent Form

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1 Primary Results on Simulation of a New Kind Traveling Wave of Permanent Form Min CHEN y Abstract This talk concerns the nonlinear dispersive waves in a water channel. Results from the study of a new version of the classical Boussinesq model system for the two-way propagation of water waves will be presented. The talk will feature presentations about theoretical work, numerical analysis, implementation ofnumerical schemes and the use of these schemes to better understand the models, reports on laboratory eperiments and comparisons of model predictions with real-world data. Some interesting simulations will be presented, which include a new set of traveling waves of permanent form, the generation of tidal waves and head-on collision of tidal waves. 1 Introduction In this paper, we study the propagation of waves in a uniform horizontal channel of length L lled to a depth h with an incompressible perfect uid. Assuming the wave motionis generated irrotationally and that it is uniform across the width of the channel, the twodimensional Euler equations are the full equations of motion. Since the numerical simulation of Euler equations are rather challenging and costly (cf. [1]), especially when a relative long time period is involved, further approimations are often made in practice. Assuming that the maimum deviation a of the free surface from its undisturbed position is small relative to h (small-amplitude waves), that the typical wavelength is large relative to h (long waves), and that the Ursell number S = a 2 =h is of order one, the Euler equations may be formally approimated by (1) t, 1 6 t =,u, (u) ; u t, 1 6 u t =,, uu ; for (,t) 2 =[;L] [;T]; where p L = L =h, is distance along the channel scaled by h, t is elapsed time scaled by h=g with g being the acceleration of gravity. The dependent variable is such that h(1 + (; t)) is the total depth at location at time t, andu = u(; t) is the horizontal velocity scaled by c = p gh at the water level time t. q 2 h at the location along the channel at As pointed out in [4], the system (1) is formally equivalent and correct to the Boussinesq's original system (cf. [6]) through rst order with regard to the small parameter This work was supported in part by NSF grant DMS and DMS y Department of Mathematics, Penn State University, University Park, PA, 1682

2 = a=h, and has the same formal status as the famous KdV equation which describe waves moving in one direction. Among a class of formally equivalent systems studied in [4], system (1) is particularly interesting because that the dispersive relation is stabilizing for all wave numbers and the natural initial-boundary-value problems that arise in laboratory eperiments are well-posed, which is to say that when (1) is associated with the initial and boundary conditions (2) (;t)=h 1 (t); (L; t) =h 2 (t); (; ) = h (); u(;t)=v 1 (t); u(l; t) =v 2 (t); u(; ) = v (); which satisfy the consistency requirements h 1 () = h (); h 2 () = h (L); v 1 () = v (); v 2 () = v (L); there eists an unique solution over a positive time interval and the solution is as regular as the initial and boundary data aords. 2 Numerical Scheme In this section, we present the scheme which was shown in [2] to be fourth-order accurate both in time and in space, unconditional stable, and have optimal computational compleity, which istosay the operation cost per time step is of order M with M being the number of grid points in the spatial discretization. The numerical scheme is based on the integral equations of (1). Inverting the operator (1, 1 ) subject to the boundary conditions in (2) and integrating the right-hand side by parts, one nds () with t = u t = Z L Z L K(; s) = 1 12 K(; s)(u + u)ds + S(L, )h 1 + S()h 2; K(; s)( u2 )ds + S(L, )v 1 + S()v 2;, S(L,, s)+sign(, s)s(l,j, sj) and S() = sinh(=p 6) sinh(l= p 6) : Let t be the step-size for the temporal discretization and the length of the spatial discretization; let (M + 1) be the number of spatial mesh points, so that M = L. The equations in () are rst discretized in space via numerical quadrature; the resultant system of ordinary dierential equations are then integrated forward in time by a nite-dierence, predictor-corrector method. Spatial Discretization The spatial discretization is eected by approimating ( i ) = R L K( i;s)y(s)ds; i =; 1; ;M. Using the trapezoidal rule with boundary corrections

3 and taking account of the fact that K(; s) isdiscontinuous at = s, one has that ( i ) 1 2 K i ()y() + (K i (i, )+K i (i + ))y(i)+k i (L)y(L) + M,1 X j=1;j6=i K i (j)y(j) (K i (s)y(s)) +, (K i (s)y(s)) i, +(K i (s)y(s)) i +,(K i (s)y(s)) M, where K i (s) =K( i ;s). The derivatives y (i), i =; ;M,may be replaced by the second-order nite dierences. Therefore ( i )may be approimated by a function i (y) where y =(y ; ;y M )withy i = y(i). The semi-discrete algorithm of () is then to nd vectors n =(n (t); ;n M (t)) and w =(w (t); ;w M (t)) which are approimations to and u respectively, such that for i =1; ;M, 1, (4) (n i ) t = i (w + n w)+s(l, i )h 1 + S( i)h 2; n = h 1 ; n M = h 2 ; (w i ) t = i (n w w)+s(l, i)v 1 + S( i)v 2; w = v 1 ; w M = v 2 : The symbol n w denotes the component-wise product of n and w, which is to say n w =( u ; ; M u M ). If M +1 is the number of spatial mesh points, a direct evaluation of i (y), i = ; 1; ;M,willinvolve on the order of M 2 operations. To reduce the computation to order M operations, an acceleration technique which was put forward in [] and[2] is used in the computation. Temporal Discretization dierential equations (5) The system (4) may be written as a system of ordinary d u = f(t; u); dt where u (n; w) T. The Adams fourth-order predictor-corrector scheme (P 4 EC 4 E)isused for the integration of (5) in time. The numerical scheme for u is ~u l+1 = u l + t 24 u l+1 = u l + t 24 55f l (u l ), 59f l,1 (u l,1 )+7f l,2 (u l,2 ), 9f l, (u l, ) ; 9f l+1 (~u l+1 )+19f l (u l ), 5f l,1 (u l,1 )+f l,2 (u l,2 ) ; where f l (u l )=f(lt; u l ). In case the eact values of the boundary terms h 1 ;h 2 ;v 1 and v 2 in (4) are not available, they are calculated via the fourth-order central dierence formula. Eistence of oscillatory waves of permanent form and their stability. In paper [5], an eact oscillatory traveling-wave solution u e (; t; )= 15 2 sech2 p (, t) ; (6) e (; t; )= 15 2sech 2 p (, t), sech 4 p 1 (, 5 2 t) ;

4 Eact solution subjected to an ocsilatary disturbance Fig. 1. Overview of the wave prole which consists a big trough in the middle, a smaller crest in front the trough R and another +1 after the trough, was found. Solution (6) represents a zero-volume, namely,1 ed =, traveling wave of permanent form. In this eperiment, we subjected the eact solution to two kinds of disturbances and observed the evolution of the wave for a relatively long time interval. In both cases, wetookl = 8, = t = 1=64, and conducted the simulation for t 2 [; 18]. In the rst case, the perturbation on the eact solution is oscillatory and has a volume,:27. More precisely, () =(1+:1 sin()) e (; ; 24); u () =(1+:1 cos())u e (; ; 24); was used as initial condition. An overview of the evolution is plotted in gure 1, which shows that a leading steady traveling wave followed by a small tail were developed as time evolves. In gure 2, we compared the leading wave of the solution at t = 18 with the eact solution (6). The trough of the leading wave has a depth,:544 which is not as deep as theonein(6),andthecrestshave a height 1:265 which is about 82:1% of that in (6). The velocity prole of the leading wave isvery close to that in (6). Based on the numerical data, we found that the phase velocity of the leading wave is approimately 2:4688 which is slight smaller than 2:5, the phase velocity of the eact solution. It is worth to note that the leading wave has a volume which equals to,:41. Since our numerical computation preserves the mass conservation, which was conrmed throughout the computation, the tail part of the solution has a volume :41, :27 = :827. Acloserlookofthetail(bystretching the y-ais) can be found in gure which shows that the tail consists a leading solitary wave followed by a dispersive tail of an even smaller amplitude. The leading solitary wave can be understood by the positive volume of the tail, which isequalto:827.

5 2 Wave profile at t=18 compared with eact solution, h=ht=1/64 8 Velocity profile at t=18 compared with eact solution, h=ht=1/ Fig. 2. The leading wave prole of (; u) at t =18(solid line) compared with (6) (dash line).1 Wave profile at t=18, h=ht=1/64.1 Velocity profile at t=18, h=ht=1/ Maimum value = 1.265; Minimum value= Maimum value = Fig.. Wave and velocity proles at t =18 (7) In the second case, solution (6) was subjected to a volume-free disturbance, namely ()=(1+:1) e (; ; 24); u () =u e (; ; 24); was used as initial data. Although the disturbance in this case is rather dierent from the one in the rst case, it is worth to note that the solutions at t =18arevery similar. The leading oscillatory wave developed in this case is almost identical to that in the rst case. References [1] J. Thomas Beale, Thomas Y. Hou and John Lowengrub, Convergence ofboundary integral methods for water waves, SIAM J. Numer. Anal,, 5 (1996), pp [2] J. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, to appear in Physica D (1998). [] J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves, Phil. Trans. Royal Soc. London A, 2 (1981), pp [4] J. L. Bona, J. -C. Saut and J. F. Toland, Boussinesq equations for small-amplitude long wavelength water waves, preprint. [5] M. Chen, Eact Solutions of Various Boussinesq systems, to appear in Appl. Math. Lett. (1998). [6] G. B. Whitham, Linear and nonlinear waves, John Wiley & Sons: New York, (1974).

is large when compared to the undisturbed water depth h, and that the Stokes

is large when compared to the undisturbed water depth h, and that the Stokes EXACT TRAVELING-WAVE SOLUTIONS TO BI-DIRECTIONAL WAVE EQUATIONS Min CHEN Department of Mathematics, Penn State University University Park, PA 1680 U.S.A. Abstract. In this paper, we present several systematic