A non-stiff boundary integral method for 3D porous media flow with surface tension

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1 A non-stiff boundary integral method for 3D porous media flow with surface tension D. M. Ambrose 1 and M. Siegel 1 Department of Mathematics, Drexel University, Philadelphia, PA 191 Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 71 Abstract We present an efficient, non-stiff boundary integral method for 3D porous media flow with surface tension. Surface tension introduces high order i.e., high derivative terms into the evolution equations, and this leads to severe stability constraints for explicit time-integration methods. Furthermore, the high order terms appear in nonlocal operators, making the application of implicit methods difficult. Our method uses the fundamental coefficients of the surface as dynamical variables, and employs a special isothermal parameterization of the interface which enables efficient application of implicit or linear propagator time-integration methods via a small-scale decomposition. The method is tested by computing the relaxation of an interface to a flat surface under the action of surface tension. These calculations employ an approximate interface velocity to test the stiffness reduction of the method. The approximate velocity has the same mathematical form as the exact velocity, but avoids the numerically intensive computation of the full Birkhoff-Rott integral. The algorithm is found to be effective at eliminating the severe time-step constraint that plagues explicit time-integration methods. 1 Introduction Boundary integral numerical methods have been widely and very successfully used to simulate and analyze problems of interfacial fluid mechanics. The methods apply to problems involving the motion of interfaces in potential flow and in zero-reynolds-number or Stokes flow. Applications include the flow of inviscid, irrotational fluids separated by an interface across which there is a jump in velocity the Kelvin-Helmholtz instability or density Rayleigh-Taylor instability, and flow in a Hele-Shaw cell or porous medium in which there is a jump in viscosity or mobility across the interface [3, 1, 11]. When applicable, boundary integral methods are among the easier methods to implement and are highly accurate. They have been used to investigate phenomena that require very high accuracy to resolve, such as finite-time singularity formation or pinch-off in evolving fluid interfaces [, 11, 13, 15]. Recent reviews on the mathematical development of boundary integral methods and their application to problems in fluid dynamics are [9, 1, 17]. 1

2 Surface tension forces have an important effect in multiphase or interfacial flow. Surface tension typically enters the boundary conditions through the Laplace-Young jump condition, which states that the jump in pressure across a free surface is proportional to the sum of the principal curvatures of the interface. Recently, there have been significant efforts aimed at providing an accurate and efficient treatment of surface tension in boundary integral computations [9]. The main difficulties in achieving this are: 1 the introduction by curvature of a high number of spatial derivatives into the governing equations, which results in a high-order time-step constraint or stiffness when explicit time-stepping methods are used, and the presence of the curvature term within a nonlocal integral operator, which complicates the use of implicit time-stepping methods to remove the stiffness. In the context of D interfacial fluid flow, Hou, Lowengrub and Shelley [1] have overcome the difficulties arising from the presence of surface tension by isolating the source of stiffness through an analysis of the equations of motion at small length scales. Their method relies on use of the interface tangent angle θ and length L, rather than its x and y positions, to separate out the dominant high order i.e., high derivative term, which then appears linearly in Fourier space with a constant coefficient. Implicit or linear propagator methods can then be implemented in an efficient manner. While the method has had great success in numerous applications within interfacial fluid dynamics, it has not yet been generalized to the motion of surfaces in 3D flow although [8] presents a small scale decomposition for 1D elastic filaments in 3D flow. Related computations of interfaces in 3D flow without surface tension are [7]. We present an efficient boundary integral numerical method for the motion of interfaces with surface tension in 3D flow. Our method is based on analytical results given in [], and relies on using the first and second fundamental coefficients of the surface as the dynamical variables. We also employ a special isothermal parameterization of the interface Xα, β, t, in which X α X α = X β X β and X α X β =. Although this restricts our computations to surfaces in which the average arclength in the α and β directions are equal, a modification of the method allows the treatment of more general surfaces and will be the subject of future work. The isothermal parameterization is dynamically maintained during evolution by a special choice of the tangential surface velocities. The governing equations for 3D interfacial flow in porous media are presented in section. volution equations for the fundamental coefficients are derived in section 3. That section also presents a pair of elliptic equations for the tangential interface velocities V 1 and V ; the solution to this system maintains the isothermal parameterization during evolution. Section derives the small scale decomposition and reformulates the evolution equations for the fundamental coefficients to separate out the high order terms that are dominant at small scales. We also present in section a model equation that approximates the interface velocity. The model velocity has the same mathematical form as the exact velocity, and is therefore useful for testing the stiffness reduction of the method, but avoids the numerically intensive computation of the Birkhoff-Rott integral. Section 5 describes an efficient method for the generation of initial data with an isothermal parameterization, and gives numerical results for surface-tension-driven relaxation of an interface under the model velocity. Concluding remarks are given in section 6.

3 Governing quations Consider the flow of two immiscible, incompressible fluids that are separated by a sharp interface in a three-dimensional porous medium. The flow is assumed to be π-periodic in the x and y directions, and of infinite extent in both the positive and negative z-direction. Motion of the fluids is driven by gravity and a prescribed far-field pressure gradient, which produces a constant fluid velocity V ˆk as z ±, where ˆk is a unit vector in the z- direction. We denote the domain of the upper fluid by D 1, the lower fluid by D, and the interface by S. Physical quantities associated with the upper or lower fluid are indicated by a subscript 1 or, respectively. The equations governing flow in a porous medium are the incompressible Darcy equations, V i = k i p i + ρ i gz 1 V =, for i = 1,, where V i x, y, z = u i x, y, z, v i x, y, z, w i x, y, z denotes the fluid velocity in region D i for i = 1,. Here we have introduced the gravitational constant g, pressures p i x, y, z, fluid mobilities k i, and densities ρ i, the latter two of which are constant in fluid region D i. Boundary conditions at the interface S are V 1 n = V n = U, 3 p 1 p = σ κ 1 + κ where U is the normal velocity on S, κ 1 and κ are the principal curvatures, and σ is the coefficient of surface tension. The curvature κ i at a point x on the surface is taken as positive when a curve on the intersection of S and the principal plane at x turns in the direction of the normal n, defined in 6 below. The interface S is parameterized by variables α = α, β, and we write the Cartesian coordinates of the surface as X α, t = x α, t, y α, t, z α, t. 5 The unit tangent and normal vectors to the surface are denoted by ˆt 1 = X α X α, ˆt = X β X β, ˆn = ˆt 1 ˆt. 6 We derive a boundary integral formulation of the moving boundary problem 1- by modifying the derivation of [1] to include surface tension. First define the velocity potential in each phase by φ i = k i p i + ρ i gz so that V i = φ i, and since the fluids are incompressible, it follows that φ i =. Introduce the surface density µ, which is defined as the difference of the potentials evaluated at the interface, µ = φ 1 φ = k 1 p 1 + k p + gzk ρ k 1 ρ 1 on S. 7 3

4 We also need the sum of the potentials at the interface: Solving 7 and 8 for p 1 and p gives φ 1 + φ = k 1 p 1 k p gzk 1 ρ 1 + k ρ on S. 8 on S. Using 9 to substitute for p 1 and p in gives µ = k 1 p 1 = µ ρ 1 gz φ 1 + φ k 1 k 1 p = µ ρ gz φ 1 + φ 9 k k 1 + k 1 σκ 1 + κ ρ 1 ρ k1 1 + k 1 gz + k 1 k φ 1 + φ 1 k 1 + k on S. An expression for the third term on the right hand side of 1 is provided by the Biot- Savart formula, which gives the velocity in terms of S and the density µ: where VX = 1 π P V S γx X X X X 3 dsx + V ˆk, 11 γ = µ α X β µ β X α X α X β 1. 1 The relations 11 and 1 are derived by Caflisch and Li [5] in the context of potential flow, but the same derivation applies for the Darcy flow considered here. The Birkhoff-Rott formula above incorporates a specific choice for the tangential interface velocity, and later this will be modified to enforce a special parameterization of the surface. Substituting 1 into 11 and using ds = X α X β dαdβ yields VX = 1 π P V µ α X β µ β X α X X X X 3 dα dβ + V ˆk. 13 Differentiating 7 with respect to α and β gives a system of second-kind integral equations for µ α, µ β, where µ α = µ β = 1 + k 1 σκ 1 + κ α ρ 1 ρ k1 1 + k k 1 σκ 1 + κ β ρ 1 ρ k1 1 + k 1 k 1 k 1 gz α + k 1 k k 1 + k φ 1 + φ α gz β + k 1 k k 1 + k φ 1 + φ β, 1 φ 1 + φ α = φ 1 + φ X α = X α V ˆt 1, 15 φ 1 + φ β = φ 1 + φ X β = X β V ˆt. 16

5 The velocity of the interface is characterized by its normal and tangential velocities, i.e., X t = U ˆn + V 1ˆt 1 + V ˆt. 17 The normal velocity is specified by 13, 1 U = VX ˆn = ˆn π P V µ α X β µ β X α X X X X 3 dα dβ + V ˆk ˆn. 18 S The tangential velocities V 1 and V are chosen to enforce a specific parameterization of the interface, as discussed in section 3 below. We now nondimensionalize lengths by l, the length of the periodic box, and velocities by U c, which may be chosen as U c = V or U c = k i σ/l. Then the dimensionless version of 1 is µ α = Bκ α W z α + A T X α V ˆt 1 µ β = Bκ β W z β + A T X β V ˆt, 19 where A T = k 1 k σ, B = k 1 + k k1 1 + k 1 U c l, W = ρ 1 ρ g k1 1 + k 1, U c and κ = κ 1 + κ / is the mean curvature. quations are the desired boundary integral formulation, with µ α, µ β determined from the system of equations 19. Our goal is to derive and evaluate a non-stiff numerical method for evolving interfaces with surface tension in 3D flow. In the remainder of this paper we set A T = W = in 19 and consider motion driven by surface tension only, which is the simplest situation in which such methods apply. 3 The Fundamental Forms and Choice of Parameterization Given a parameterized surface Xα, β, we define the components of the first fundamental form [1] to be = X α X α, F = X α X β, G = X β X β, where subscripts denote differentiation. The components of the second fundamental form are L, M, and N, where L = X αα ˆn = X α ˆn α, M = X αβ ˆn = X α ˆn β = X β ˆn α, N = X ββ ˆn = X β ˆn β. We will study surfaces which have an isothermal parameterization; this means that = G and F =. This restricts our study to surfaces in which the average arclength in the α and β directions are equal, i.e., π π X α α, β dα dβ = 5 π π X β α, β dα dβ.

6 A modification of the parameterization allows this restriction to be relaxed, and is the subject of future work. Our approach to maintaining an isothermal parameterization is to start with a surface which has an isothermal parameterization, and to maintain the parameterization by appropriate choice of artificial tangential velocities. The mean curvature of the surface will be important for the problem we study, since surface tension enters through the Laplace-Young jump condition. In general, the mean curvature is κ = N + GL F M. G F With an isothermal parameterization, this reduces to 3.1 Geometric Identities κ = L + N. 1 Using the definition of the second fundamental form, the isothermal parameterization, and the definition of the normal vector, we have the following identities: ˆt 1 α ˆn = ˆn α ˆt 1 = L, ˆt 1 β ˆn = ˆn β ˆt 1 = ˆt α ˆn = ˆn α ˆt = M, ˆt β ˆn = ˆn β ˆt = N. We also have, using only the definitions of the tangent vectors and the isothermal parameterization, the following identities: ˆt 1 α ˆt = ˆt α ˆt 1 = β, ˆt 1 β ˆt = ˆt β ˆt 1 = α. These identities can be combined to yield the following formulas for second derivatives of the surface: X αα = α ˆt 1 β ˆt + Lˆn, X αβ = β ˆt 1 + α ˆt + M ˆn, X ββ = α ˆt 1 + ˆt + N ˆn. Furthermore, we can use these identities to get formulas for the first derivatives of the unit tangent and normal vectors: β ˆt 1 α = X αα α ˆt 1 = β ˆt + L ˆn, 6

7 ˆt 1 β = X αβ β ˆt 1 = α ˆt + M ˆn, ˆt α = X αβ α ˆt = β ˆt 1 + M ˆn, ˆt β = X ββ β ˆt = α ˆt 1 + N ˆn. Using the definition of ˆn and cross-product identities, with the above or, alternatively, using the definition of the second fundamental form, we get ˆn α = L ˆt 1 M ˆt, ˆn β = M ˆt 1 N ˆt. 3. volution quations for the Fundamental Coefficients We derive evolution equations for the six components of the first and second fundamental coefficients under the assumption of an isothermal parameterization. These equations have two main advantages for numerical computation: 1 high order or dominant terms at small scales have a particularly simple form which permits efficient implementation of implicit methods, and the interface shape can be recovered from knowledge of the six fundamental coefficients [1]. We start by differentiating the evolution equation with respect to α and β. Using the geometric identities, we find β X αt = V 1α + V U L ˆt 1 β + V α V 1 U M ˆt L + U α + V 1 + M V ˆn, X βt = α V 1β V U M ˆt 1 α + V β + V 1 U N + ˆt From these we can easily find evolution equations for, F, and G. U β + V 1 M + V N ˆn. 3 t = X αt X α = X αt ˆt 1 = V 1α + V β UL, G t = X βt X β = X βt ˆt = V β + V 1 α UN, F t = X αt ˆt + X βt ˆt 1 = V α + V 1β V 1 β + V α UM. 5 7

8 We now want to give formulas for L t, M t, and N t. Since the definition of the second fundamental form involves ˆn, we first need a formula for ˆn t. First, notice that since ˆn is a unit vector, ˆn t ˆn =. Then, we have ˆn t = ˆt 1 ˆt t = X αt ˆt + ˆt 1 X βt t ˆn Xαt = ˆn ˆn ˆt + ˆt 1 Xβt ˆn ˆn. 6 Using cross-product identities and the above formulas for X αt and X βt, we have Uα ˆn t = + V 1L + V M ˆt 1 Uβ + V 1M + V N ˆt. We now differentiate the definition L = X α ˆn α, which gives L t = X αt ˆn α X α ˆn αt = X αt ˆt 1 ˆt 1 ˆn α X αt ˆt ˆt ˆn α ˆt 1 ˆn αt = X αt ˆt 1 ˆt 1 ˆn α X αt ˆt ˆt ˆn α ˆt 1 ˆn t α + ˆt 1 α ˆt ˆt ˆn t. 7 Using the above calculations in 7 we find the following: L t = V 1α + V β UL L + V α V 1 β + Uα α + V 1L + V M + UM M β Uβ + V 1M We perform the same type of calculations to deduce M t. At first we find + V N 8 M t = X α ˆn β t = X αt ˆn β X α ˆn βt = X αt ˆt 1 ˆt 1 ˆn β X αt ˆt ˆt ˆn β ˆt 1 ˆn βt = X αt ˆt 1 ˆt 1 ˆn β X αt ˆt ˆt ˆn β ˆt 1 ˆn t β + ˆt 1 β ˆt ˆt ˆn t. 9 As before, use some of the identities derived previously to find the following formula for M t : M t = V 1α + V β UL M + V α V 1 β + Uα β + V 1L + V M We perform the same steps now to calculate N t : α UM N Uβ + V 1M + V N. 3 N t = X β ˆn β t = X βt ˆn β X β ˆn βt = X βt ˆt 1 ˆt 1 ˆn β X βt ˆt ˆt ˆn β ˆt ˆn t β + ˆt β ˆt 1 ˆt 1 ˆn t. 31 8

9 Once again we use identities derived earlier to get an evolution equation for N : N t = V 1β V α UM M + V β + V 1 α + β Uβ + V 1M 3.3 Tangential Velocities + V N + UN N α Uα + V 1L + V M. 3 We now find a pair of elliptic equations for the tangential velocities. If we start with a surface that has an isothermal parameterization, and choose the tangential velocities so that they satisfy these elliptic equations, then the isothermal parameterization will be maintained at positive times. The elliptic equations are simply found by requiring t = G t and F t =. Using the formulas for t, F t, and G t from the previous section, we have V 1α V β V 1 α V β = UL N, This can be rewritten as Finally, we can write this as V1 V αα αα V 1β + V α V 1 β + V α V1 + + α V1 β V1 V ββ ββ V + β = V α = UM. UL N, = UM. UL N = UM = α α UM + UL N β β,. 33 Clearly, given, U, and the second fundamental coefficients, we can solve these equations for V 1 / and V /, and thus for V 1 and V themselves. Small scale decomposition Following [] we rewrite the Birkhoff-Rott integral to factor out the dominant contribution at small scales or high wavenumbers. The dominant contribution comes from high order terms, i.e., those with the highest number of derivatives, which are embedded in the Birkhoff-Rott 9

10 integral. Assume X α is sufficiently regular and that X α X α for α α. By Taylor s expansion, Xα, β Xα, β Xα, β Xα, β = X αα, β α α + X β α, β β β + O α α 3 [α, β ] 3 α α Define and rewrite the Birkhoff-Rott integral 13 as j = µ α X β µ β X α, 35 VX = 1 π P V { X j α α α + X β β β + X X 3 α α 3 X X 3 X αα α + X β β } β dα dβ, 36 3 α α 3 where primed functions are evaluated at α, β and all other functions at α, β. quation 36 can be represented in terms of Riesz transforms, which are defined by H 1 fα, β = 1 π P V H fα, β = 1 π P V The symbols of the Riesz transforms are [16] where the Fourier coefficients ˆfk are defined by fα, β α α dα dβ, α α 3 37 fα, β β β dα dβ. α α 3 38 Ĥ 1 fk = i k 1 k ˆf k, Ĥ fk = i k k ˆf k, 39 ˆfk = 1 π e ik 1α+k β fα, β dαdβ, and k = k 1, k is the vector wavenumber. The Riesz transform of a vector f = f 1, f is defined as the vector whose components are the tranforms of f i. In terms of Riesz transforms, equation 36 takes the form VX = H 1 j Xα 3 j Xβ + H + Jα, β, t 1 3 where J denotes the principle value integral associated with the second and third terms within braces in 36. The dominant terms at small scales in 1 are the two Riesz transforms. To see this, note that at next order in the Taylor s expansion 3, the terms are of the form gα, β α α i β β j α α 3 1

11 where g depends on first and second derivatives of X and i and j are nonegative integers with i + j =. The Taylor s expansion of J therefore has highest order terms of the form G i,j hα, β = where h = j g. The symbol of G i,j is given by hα, β α α i β β j α α 3 dα dβ Ĝ i,j k = πki 1k j k 3 ĥ k. 3 For sufficiently smooth X it follows that Ĝi,jk H = O b i k for k >> 1, where Ĥik for k i = 1, are Fourier coefficients of the Riesz tranforms in 1. Thus the Riesz transforms are the dominant terms in 1 at high wavenumber, and J is smoother or lower order of degree 1. We proceed to simplify the Riesz transforms in 1 to better isolate the high order or dominant behavior at small scales. Introduce the notation f g to indicate that the difference between f and g is smoother than g. We also need that, at small scales, smooth functions can be passed through Riesz transforms, i.e., H i [fg] = gh i [f] + [f] where the commutator is a smoothing operator on f []. Calculate the cross product in 1 using the relation X α X β = n X α X β, pass n through the Riesz tranforms incurring a commutator and take the inner product with n to obtain the intermediate result U = 1 { H 1 µα 1 + H µβ 1 } + J 1 α, β, t, 5 Here J 1, which contains the original Birkhoff-Rott integral, is lower order than the Riesz transform terms contained within the braces. We now specialize to the case A T = W =, which describes the relaxation of the interface under surface tension. Define Y = L + N and note from 19, 1, that Then from 5, U = B 3 µ = BY. 6 {H 1 Y α + H Y β } + J 1 α, β, t + J α, β, t, 7 where we have used that is lower order than Y and passed some factors through the Riesz transform, incurring a commutator. The term J = 1 { } µα µβ H 1 + H 1 B {H Y α + H Y β } 8 is lower order than the Riesz transform terms in 7. quation 7 describes the behavior of the normal velocity U at small scales and is the main result of this section. 11

12 .1 A model problem To illustrate our non-stiff boundary integral method we consider the model or approximate velocity obtained by setting J 1 = in 7. The model velocity has the same mathematical form as the exact velocity, in that the high order terms are embedded within a nonlocal operator and, additionally, there is a nonlocal lower order term. However, use of the model velocity avoids computation of the Birkhoff-Rott integral which is contained within J 1. Computation of the Birkhoff-Rott integral is numerically intensive; a naive or straightforward evaluation requires ON operations, where N is the number of node points in the discretization of the interface. This can be reduced to ON log N operations using the Fast Multipole Method [6], but at the expense of considerable programming effort. The nonlocal model velocity can be computed with spectral accuracy using the FFT, and allows us to test the efficacy of our non-stiff boundary integral method without the added effort necessary to implement a Fast Multipole Method. An additional advantage is that X α, t decouples from the evolution equations for the fundamental coefficients, and therefore the coefficients can be evolved without simultaneous calculation of the interface position. An implementation of the method with the exact velocity 7, using a fast summation algorithm to evaluate the Birkhoff-Rott integral for periodic surfaces, will be described in a future paper.. Reformulation of the evolution equations We reformulate the evolution equations for the fundamental coefficients to isolate the high order terms. First note that the equation for Y t can be written in the form which serves as a definition of F. From 7, U αα Y t = U + F α, β, t, 9 B 3/ {H 1Y ααα + H Y βαα }, 5 and similarly U ββ B {H 1Y 3/ αββ + H Y βββ }, 51 where we have used that is lower order than Y. Clearly, the highest order term in the evolution equation 9 for Y is U, which goes like a third derivative of Y. It follows that the equation for Y with the highest order terms factored out has a simple form: Y t = where from 5, 9 J 3 = 1 { H 1 µα B 3/ {H 1Y ααα + H Y βαα + H 1 Y αββ + H Y βββ } + J 3 α, β, t, 5 1 } µβ + H + F α, β, t 1 B 3/ {H 1Y ααα + H Y βαα + H 1 Y αββ + H Y βββ } 53 1

13 is lower order than the terms within braces in 5. If α, β, t were constant in space, the dominant first term on the right hand side of equation 5 would diagonalize under the Fourier transfrom, allowing efficient implementation of implicit integration methods. This motivates defining and writing where Y t = Rα, β, t = B 3/ m B 3/ m t = min α, β, t 5 α,β {H 1 Y ααα + H Y βαα + H 1 Y αββ + H Y βββ } + Rα, β, t 55 B 3/ m {H 1 Y ααα + H Y βαα + H 1 Y αββ + H Y βββ } + J It is important to note that R contains terms that are the same order as the terms within braces in 55. However, comparing 55 and 56, we see that the high order terms in R have a smaller magnitude coefficient. A frozen coefficient analysis shows that implicit differencing of the terms within braces in 55 and explicit treatment of R provides a stable non-stiff method. The main advantage of evolution equation 55 for Y is that it diagonalizes under the Fourier tranform, i.e., d = B k 3 Ŷ k + ˆR k, 57 dtŷk m 3/ so that an implicit time integration or linear propagator method can be efficiently implemented. The other fundamental coefficients are recovered by time integration of equations, 8, and 3. The highest order terms in these equations are contained in the normal velocity U, which by 7 depends only on Y through the Laplace-Young jump condition and lower order terms. Therefore, these equations can be integrated using an explicit method, and the stability will be slave to the stability of the time integration of 57 for Y. 5 Numerical Method 5.1 Initial data Initial data must be chosen to satisfy the conditions for an isothermal parameterization, i.e., = G and F =. Let z = Zx, y be a given initial interface shape satisfying. We construct an isothermal parameterization by evolving an initially flat interface with parameterization xα, β, = α, yα, β, = β, zα, β, =. The interface is evolved according to 17, with the normal velocity U chosen to be proportional to the difference between the prescribed data and the current interface position, i.e., U α = Q [Zx α, t, y α, t z α, t] ˆk ˆn, 58 13

14 .1.5 z.5.1 x 6 y 6. Figure 1: The isothermal surface parameterization corresponding to 6 where x α, t, y α, t, z α, t is the current interface position and Q is a positive constant which determines the rate at which the evolving surface approaches the specified initial data. The tangential components of velocity at the interface, V 1 and V, are determined by solving the elliptic system 33 so that the surface parameterization remains isothermal throughout the evolution. The evolution is stopped when max α,β Zx α, t, y α, t z α, t ɛ d 59 where ɛ d is a given tolerance. The evolution problem 17 with velocity 58 is not stiff, and is integrated using an explicit nd order Runge-Kutta method. The method is spectrally accurate in space, i.e., derivatives are calculated using the FFT, and the unit normal and tangent vectors are calculated from the definition 6 via differentiation of X. The elliptic system 33 for tangential velocities V 1, V is solved with spectral accuracy using the FFT. Figure 1 shows the isothermal parameterization corresponding to the data Zx, y =.1 cosx cosy. 6 The calculation used N p = 18 equally spaced node points in α and β, and we set t =. 1 3 and ɛ d = 1 1. Only every other grid line is shown in the figure. Figure plots log 1 α G α left and log 1 F α right for the surface in figure 1. The figure shows that G and F are less than 1 11, indicating that the interface parameterization is effectively isothermal. 5. Time evolution We use a linear propagator method [1] to integrate equation 57. Linear propagator methods are based on factoring out the dominant linear term at high wavenumbers, and provide stable and potentially high-order methods for diffusive problems. If the dominant term has a constant coefficient, these methods propagate the highest order modes exactly. 1

15 log G log F y x y x 6 Figure : log 1 G and log 1 F versus x and y. Use an integrating factor to rewrite equation 57 as t ψ B k 3 t 1 kt = exp m 3/ t dt R k, 61 where ψ k t = Ŷk exp B k 3 t 1 m 3/ t dt. 6 quation 61 is discretized using a second order Adams-Bashforth method. In terms of Ŷk the result is Ŷ n+1 k = Ŷ k n e k t n, t n+1 + t 3 ˆR ke n n 1 k t n, t n+1 ˆR k e k t n 1, t n+1, 63 where e k t 1, t = exp B k 3 t 1 t 1 m 3/ t dt. 6 The linear propagator method removes the stiffness by propagating the linear or high order term in 57 from t n to t n+1 at the exact exponential rate. We obtain second order accuracy by replacing the continuous integrals with their trapezoidal rule approximations: [ ] e k t n, t n+1 = exp B k 3 t 1 65 e k t n 1, t n+1 = exp 8 B k 3 t m + 1 [ n 3/ 1 m n / m n+1 3/ n m 3/ + 1 m n+1 3/ ]. 66 The discretization propagates the exact exponential decay of the high order terms by a second order accurate approximation. Recall that that n+1 is computed explicitly from. quations, 8, and 3 for, L and M are discretized explicitly in time using a second order Adams-Bashforth method. The highest order terms in these equations are contained in the normal velocity U, see 7. The stability of these explicit discretizations are slave to the stability of the discretization 63 for Y, so they do not lead to any additional time-step constraint. We use N = Y L to recover N from the solutions for Y and L. Riesz 15

16 transforms are computed with spectral accuracy in space using the symbols 39, and the elliptic problem for the tangential velocities V 1, V is solved using the FFT. As noted in [1], near equilibrium the second-order linear propagator method requires t < CB, i.e. for stability the time step is independent of the spatial discretization. More generally there may be a CFL condition condition resulting from the transport term in R. The exponential damping factors also tend to smooth the solution, but this can be minimized by reducing the time step. 5.3 Numerical results The stability constraints of the linear propagator method are compared with an explicit Adams-Bashforth discretization of evolution equations, 8, 3, and 3 for the fundamental coefficients. We consider the relaxation under surface tension of the interface shown in figure 1, with surface tension coefficient B = 1.. For N p equally spaced node points in the α and β variables, a frozen coefficient analysis of equation 57 reveals the time step constraint for stability of an explicit method to be t C 3/ m /BN 3 p, 67 where C is a constant independent of N p. Figure 3 shows the Fourier transform log 1 ˆL k versus k = k, l calculated by the explicit method at t =.1 for the two resolutions N p = 3 top and N p = 6 bottom. Fourier coefficients are shown for k N p / and l N p 1, and coefficients for the other N p / 1 k-values are determined from the real-valuedness of L. Note that modes with l = N p / + 1 to l = N p actually correspond to negative wavenumber modes l = N p / + 1 to l = 1. The time steps for the lower resolution plots at top are t =.5 1 left plot and t =.5 1 right plot. There are no spectral or Fourier filters [11] used in the calculation, so that we may assess the stability of high-wavenumber modes generated by round-off error. Instability in a spectral method is typically observed as a rapid and unphysical growth in the amplitudes of the high wavenumber modes. High wavenumber growth and instability are clearly exhibited in the calculation at the top left in figure 3. The time step here, t =.5 1, is above but close to the stability threshold, and the explicit method is found to be stable at t =.1 when the time step is decreased by a factor of, as shown in the plot at top right. Increasing the number of node points to N p = 6 gives instability at the time step t = bottom, left and stability at t = bottom, right. The approximate factor of 8 reduction in time step required to achieve stability is expected from the constraint 67. Results from the linear propagator method, using a time step t = , are shown in figure. It is clear from the spectra that the method is stable, even at this relatively large time step, for N p = 6 and N p = 18. Indeed, we find the method is stable for time steps as large as t =.1 at N p = 18. This example indicates that the small-scale decomposition presented here is effective at removing the stiffness due to surface tension in 3D porous media flow. 16

17 5 5 log ˆLk 1 log ˆLk l 1 5 k l 1 5 k log ˆLk 1 log ˆLk l 1 k l 1 k 3 Figure 3: Plot of log 1 ˆL k versus k for the explicit method. Top left: N p = 3, t =.5 1. Top right: N p = 3, t =.5 1. Bottom left: N p = 6, t = Bottom right: N p = 6, t = log ˆLk 1 log ˆLk l 1 k l 5 k 6 8 Figure : Plot of log 1 ˆL k versus k for the linear propagator method with t = Left: N p = 6. Right: N p =

18 6 Conclusions We have presented a non-stiff boundary integral method for 3D porous media flow with surface tension. Our method uses the first and second fundamental coefficients of the surface as dynamical variables, and employs a special isothermal parameterization of the interface. This allows high order terms to be extracted from the evolution equations so that they diagonalize under the Fourier transform, enabling the efficient application of implicit or linear propagator time-integration methods. Our method includes an efficient algorithm for the generation of initial data with an isothermal parameterization by evolving a flat interface toward a prescribed initial surface shape. The method is tested by computing the relaxation of an interface to a flat surface under the action of surface tension. These calculations employ an approximate interface velocity to test the stiffness reduction of the method. The approximate velocity has the same mathematical form as the exact velocity, but avoids the numerically intensive computation of the full Birkhoff-Rott integral. The algorithm is found to be effective at eliminating the severe time-step constraint that plagues explicit time-integration methods. The use of an isothermal parameterization limits our method to surfaces in which the average arclength in the α and β directions are equal. A modification of the algorithm enables computations to be performed for more general surfaces. This will be the subject of a future paper in which we also implement the method with the exact velocity 7, using a fast summation algorithm to evaluate the Birkhoff-Rott integral for periodic surfaces, which has been recently developed by the authors. Acknowledgements This work was supported by the NSF under Grant Nos. DMS and DMS-3556 MS and DMS DMA. Simulations were conducted on the NJIT computing cluster, supported by the NSF/MRI under Grant No. DMS-59. References [1] D. M. Ambrose. Well-posedess of two-phase Darcy flow in 3D. Quart. Appl. Math., 65:189 3, 7. [] D. M. Ambrose and N. Masmoudi. Well-posedness of 3D vortex sheets with surface tension. Commun. Math. Sci., 5:391 3, 7. [3] G. R. Baker, D. I. Meiron, and S. A. Orszag. Generalized vortex methods for free-surface flow problems. J. Fluid Mech., 13:77 51, 198. [] G.R. Baker, R.. Caflisch, and M. Siegel. Singularity formation during Rayleigh-Taylor instability. J. Fluid Mech., 5:51 78, [5] R.. Caflisch and X.-F. Li. Lagrangian theory for 3D vortex sheets with axial or helical symmetry. Trans. Th. Stat. Phys., 1: ,

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