Removing the Stiffness from Interfacial Flows with Surface Tension

322 HOU, LOWENGRUB, AND SHELLEYwhere t. = n At, andand soS t2 dt' "]ek(tl, tz) = exp -- ~ (2zr ]kl) 3 ft, za(t')J" (70)The use of "linear propagator" is now clear; 0 at the n thtime step is propagated forward to the (n + 1 )th timestep atthe exact exponential rate associated with the linear term. IfN-0, this yields the exact solution to the linear problem.Of course, the factor e(tl, t2) still has a continuous timedependence. We retain second order by replacing theintegrals with their trapezoidal rule approximations, i.e.,1ek(t., tn+l)=exp (-S~t(21r lkl)3[(~ ~ (Ln~- 1)3])SAtek(tn_l, tn+l)=exp -----~-(2/l:lkl) 3 (71)- 1X _2( Ln _ 1 ) 3Thus, in the numerical scheme, 0 is propagated by a secondorderapproximation of the exact exponential rate. Recallthat L" +1 is computed explicitly by Eq. (65).Remarks. 1. Near equilibrium, the second-order linearpropagator method is unconditionally stable. That is, werequire only d t ~< C(S) independently of the wavenumber k.More generally, however, there may be a CFL conditionarising from the transport term hidden in N (see Eqs. (53)and (54)). This term is not seen in the near equilibriumanalysis. The long time simulations are always performedwith a CFL condition in mind. The exponential dampingfactors also tend to suppress instabilities such as thosearising from aliasing errors or underresolution. It has alsobeen observed that, in the presence of singular behavior ofthe interface, this method tends to oversmooth the solution[ 19]. This difficulty is overcome by reducing the timestep. We believe that it may also be ameliorated by usinghigher order time discretizations. If the solution is smooth,then this difficulty is not noticeable.2. An advantage of a linear propagator method is thatit is easy to derive higher order time discretizations. This isclear if there is no time-dependence in the coefficient of thelinear term. On the other hand, if there is time dependence(as is the case here), then a time integral appears in theintegrating factor (see Eq. (68)). An explicit quadrature ofthis integral is avoided by introducing a new independentvariable corresponding exactly to the integral. LetI= I[L](t) be given byI(t)= fo1dt'L3(t,) (72)1I t = L--- 3, I(0) = 0. (73)Then, we supplement the original system (59) and (67) andsolveL, = M[L, I, ~k](t) (74)1I, =~--~ (75)~k, = eZ(S/2)~z~lkl)3b~[ L, L ~b](k, t). (76)It is now straightforward to discretize this system to highorder. Fourth-order Runge-Kutta is currently beingimplemented.Crank-Nicholson DiscretizationAnother stable scheme is obtained by using theCrank-Nicholson discretization on the leading order stiffterm and leapfrog on the nonlinearity. This gives thesecond-order integrationO°+l(k)--O° l(k)2At+ A?"(k). (77)Since L n+l, L n, and L n-1 are known, Eq. (77) gives anexplicit expression for O n + l(k).Remarks. I. A near equilibrium analysis shows thatthis method is unconditionally stable. Again, there may bea CFL condition from the transport term. Our computationsconfirm stability for short times, but they also indicatethat for long times and, in the fully nonlinear regime, thatthe Crank-Nicholson method is susceptible to an aliasinginstability. The time of onset of this instability increases asthe spatial resolution is increased. However, we find that wecan control this instability and retain accuracy, even overlong times, by the application of a high-order Fourier filter(25 th order). We will discuss this further when we give thespatial discretizations and numerical results.2. The Crank-Nicholson discretization may also be usefulto compute in other frames of reference (i.e., using otherchoices of T). Of course, s~ is no longer constant in ~, butthe general small scale decompositions (54) and (55), (56)still apply. The PDE for s, is not expected to be stiff and theleading order terms in the 0 (and y) equations are still linearin 0 (and y), albeit with variable coefficients. Thus by