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

D. M. Ambrose1 and M. Siegel2

1Department of Mathematics, Drexel University, Philadelphia, PA 191042Department of Mathematical Sciences and Center for Applied

Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102

Abstract

We present an efficient, non-stiff boundary integral method for 3D porous mediaflow 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 ex-plicit time-integration methods. Furthermore, the high order terms appear in nonlocaloperators, making the application of implicit methods difficult. Our method uses thefundamental coefficients of the surface as dynamical variables, and employs a specialisothermal parameterization of the interface which enables efficient application of im-plicit 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 un-der the action of surface tension. These calculations employ an approximate interfacevelocity to test the stiffness reduction of the method. The approximate velocity hasthe same mathematical form as the exact velocity, but avoids the numerically intensivecomputation of the full Birkhoff-Rott integral. The algorithm is found to be effectiveat eliminating the severe time-step constraint that plagues explicit time-integrationmethods.

1 Introduction

Boundary integral numerical methods have been widely and very successfully used to sim-ulate and analyze problems of interfacial fluid mechanics. The methods apply to problemsinvolving the motion of interfaces in potential flow and in zero-Reynolds-number or Stokesflow. Applications include the flow of inviscid, irrotational fluids separated by an inter-face 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 thereis a jump in viscosity or mobility across the interface [3, 10, 11]. When applicable, boundaryintegral methods are among the easier methods to implement and are highly accurate. Theyhave been used to investigate phenomena that require very high accuracy to resolve, suchas finite-time singularity formation or pinch-off in evolving fluid interfaces [4, 11, 13, 15].Recent reviews on the mathematical development of boundary integral methods and theirapplication to problems in fluid dynamics are [9, 14, 17].

1

Surface tension forces have an important effect in multiphase or interfacial flow. Surfacetension 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 theprincipal curvatures of the interface. Recently, there have been significant efforts aimed atproviding an accurate and efficient treatment of surface tension in boundary integral compu-tations [9]. The main difficulties in achieving this are: (1) the introduction by curvature of ahigh number of spatial derivatives into the governing equations, which results in a high-ordertime-step constraint or ‘stiffness’ when explicit time-stepping methods are used, and (2) thepresence of the curvature term within a nonlocal (integral) operator, which complicates theuse of implicit time-stepping methods to remove the stiffness.

In the context of 2D interfacial fluid flow, Hou, Lowengrub and Shelley [10] have overcomethe difficulties arising from the presence of surface tension by isolating the source of stiffnessthrough an analysis of the equations of motion at small length scales. Their method relieson 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 appearslinearly in Fourier space with a constant coefficient. Implicit or linear propagator methodscan then be implemented in an efficient manner. While the method has had great success innumerous applications within interfacial fluid dynamics, it has not yet been generalized tothe motion of surfaces in 3D flow (although [8] presents a small scale decomposition for 1Delastic filaments in 3D flow). Related computations of interfaces in 3D flow without surfacetension are [7].

We present an efficient boundary integral numerical method for the motion of interfaceswith surface tension in 3D flow. Our method is based on analytical results given in [2], andrelies on using the first and second fundamental coefficients of the surface as the dynamicalvariables. We also employ a special ‘isothermal’ parameterization of the interface X(α, β, t),in which Xα ·Xα = Xβ ·Xβ and Xα ·Xβ = 0. Although this restricts our computations tosurfaces in which the average arclength in the α and β directions are equal, a modification ofthe method allows the treatment of more general surfaces and will be the subject of futurework. The isothermal parameterization is dynamically maintained during evolution by aspecial choice of the tangential surface velocities.

The governing equations for 3D interfacial flow in porous media are presented in section2. Evolution equations for the fundamental coefficients are derived in section 3. That sectionalso presents a pair of elliptic equations for the tangential interface velocities V1 and V2; thesolution to this system maintains the isothermal parameterization during evolution. Section4 derives the small scale decomposition and reformulates the evolution equations for thefundamental coefficients to separate out the high order terms that are dominant at smallscales. We also present in section 4 a model equation that approximates the interface velocity.The model velocity has the same mathematical form as the exact velocity, and is thereforeuseful for testing the stiffness reduction of the method, but avoids the numerically intensivecomputation of the Birkhoff-Rott integral. Section 5 describes an efficient method for thegeneration of initial data with an isothermal parameterization, and gives numerical resultsfor surface-tension-driven relaxation of an interface under the model velocity. Concludingremarks are given in section 6.

2

2 Governing Equations

Consider the flow of two immiscible, incompressible fluids that are separated by a sharpinterface in a three-dimensional porous medium. The flow is assumed to be 2π-periodic inthe 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, whichproduces a constant fluid velocity V∞k as z → ±∞, where k is a unit vector in the z-direction. We denote the domain of the upper fluid by D1, the lower fluid by D2, and theinterface by S. Physical quantities associated with the upper or lower fluid are indicated bya subscript 1 or 2, respectively.

The equations governing flow in a porous medium are the incompressible Darcy equations,

Vi = −ki∇(pi + ρigz) (1)

∇ ·V = 0, (2)

for i = 1, 2, where Vi(x, y, z) = (ui(x, y, z), vi(x, y, z), wi(x, y, z)) denotes the fluid velocityin region Di for i = 1, 2. Here we have introduced the gravitational constant g, pressurespi(x, y, z), fluid mobilities ki, and densities ρi, the latter two of which are constant in fluidregion Di. Boundary conditions at the interface S are

V1 · n = V2 · n = U, (3)

p1 − p2 = σ (κ1 + κ2) (4)

where U is the normal velocity on S, κ1 and κ2 are the principal curvatures, and σ is thecoefficient of surface tension. The curvature κi at a point x on the surface is taken as positivewhen a curve on the intersection of S and the principal plane at x turns in the direction ofthe 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

t1 =Xα

|Xα| , t2 =Xβ

|Xβ| , n = t1 × t2. (6)

We derive a boundary integral formulation of the moving boundary problem (1)-(4) bymodifying the derivation of [1] to include surface tension. First define the velocity potentialin each phase by

φi = −ki(pi + ρigz)

so thatVi = ∇φi,

and since the fluids are incompressible, it follows that ∆φi = 0. Introduce the surface densityµ, which is defined as the difference of the potentials evaluated at the interface,

µ = φ1 − φ2 = −k1p1 + k2p2 + gz(k2ρ2 − k1ρ1) on S. (7)

3

We also need the sum of the potentials at the interface:

φ1 + φ2 = −k1p1 − k2p2 − gz(k1ρ1 + k2ρ2) on S. (8)

Solving (7) and (8) for p1 and p2 gives

p1 = − µ

2k1

− ρ1gz − φ1 + φ2

2k1

p2 =µ

2k2

− ρ2gz − φ1 + φ2

2k2

(9)

on S. Using (9) to substitute for p1 and p2 in (4) gives

µ = − 2

k−11 + k−1

2

σ(κ1 + κ2)− 2ρ1 − ρ2

k−11 + k−1

2

gz +k1 − k2

k1 + k2

(φ1 + φ2) (10)

on S.An expression for the third term on the right hand side of (10) is provided by the Biot-

Savart formula, which gives the velocity in terms of S and the density µ:

V(X) =1

4πPV

∫ ∫

S

γ(X′)× (X−X′)|X−X′|3 ds(X′) + V∞k, (11)

whereγ = (µαXβ − µβXα)(Xα ×Xβ)−1. (12)

The relations (11) and (12) are derived by Caflisch and Li [5] in the context of potentialflow, but the same derivation applies for the Darcy flow considered here. The Birkhoff-Rottformula above incorporates a specific choice for the tangential interface velocity, and laterthis will be modified to enforce a special parameterization of the surface. Substituting (12)into (11) and using ds = |Xα ×Xβ| dαdβ yields

V(X) =1

4πPV

∫ ∞

−∞

∫ ∞

−∞(µαXβ − µβXα)× (X−X′)

|X−X′|3 dα′dβ′ + V∞k. (13)

Differentiating (7) with respect to α and β gives a system of second-kind integral equationsfor µα, µβ,

µα = − 2

k−11 + k−1

2

σ(κ1 + κ2)α − 2ρ1 − ρ2

k−11 + k−1

2

gzα +k1 − k2

k1 + k2

(φ1 + φ2)α

µβ = − 2

k−11 + k−1

2

σ(κ1 + κ2)β − 2ρ1 − ρ2

k−11 + k−1

2

gzβ +k1 − k2

k1 + k2

(φ1 + φ2)β, (14)

where

(φ1 + φ2)α = ∇(φ1 + φ2) ·Xα = 2|Xα|V · t1, (15)

(φ1 + φ2)β = ∇(φ1 + φ2) ·Xβ = 2|Xβ|V · t2. (16)

4

The velocity of the interface is characterized by its normal and tangential velocities, i.e.,

Xt = U n + V1t1 + V2t

2. (17)

The normal velocity is specified by (13),

U = V(X) · n = n · 1

4πPV

∫

S

(µαXβ − µβXα)× (X−X′)|X−X′|3 dα′dβ′ + V∞k · n. (18)

The tangential velocities V1 and V2 are chosen to enforce a specific parameterization of theinterface, as discussed in section 3 below.

We now nondimensionalize lengths by l, the length of the periodic box, and velocities byUc, which may be chosen as Uc = V∞ or Uc = kiσ/l2. Then the dimensionless version of (14)is

µα = −Bκα −Wzα + AT |Xα|V · t1

µβ = −Bκβ −Wzβ + AT |Xβ|V · t2, (19)

where

AT =2(k1 − k2)

k1 + k2

, B =4σ

(k−11 + k−1

2 )Ucl2, W =

2(ρ1 − ρ2)g

(k−11 + k−1

2 )Uc

,

and κ = (κ1 + κ2)/2 is the mean curvature. Equations (17)-(18) are the desired boundaryintegral formulation, with µα, µβ determined from the system of equations (19).

Our goal is to derive and evaluate a non-stiff numerical method for evolving interfaceswith surface tension in 3D flow. In the remainder of this paper we set AT = W = 0 in (19)and consider motion driven by surface tension only, which is the simplest situation in whichsuch methods apply.

3 The Fundamental Forms and Choice of Parameteri-

zation

Given a parameterized surface X(α, β), we define the components of the first fundamentalform [12] to be

E = Xα ·Xα, F = Xα ·Xβ, G = Xβ ·Xβ,

where subscripts denote differentiation. The components of the second fundamental formare L, M, and N, where

L = Xαα · n = −Xα · nα,

M = Xαβ · n = −Xα · nβ = −Xβ · nα,

N = Xββ · n = −Xβ · nβ.

We will study surfaces which have an isothermal parameterization; this means that E = Gand F = 0. This restricts our study to surfaces in which the average arclength in the α andβ directions are equal, i.e.,

∫ 2π

0

∫ 2π

0

|Xα(α′, β′)| dα′dβ′ =∫ 2π

0

∫ 2π

0

|Xβ(α′, β′)| dα′dβ′. (20)

5

A modification of the parameterization allows this restriction to be relaxed, and is the subjectof future work. Our approach to maintaining an isothermal parameterization is to start witha surface which has an isothermal parameterization, and to maintain the parameterizationby appropriate choice of artificial tangential velocities.

The mean curvature of the surface will be important for the problem we study, sincesurface tension enters through the Laplace-Young jump condition. In general, the meancurvature is

κ =EN + GL− 2FM

2(EG− F 2).

With an isothermal parameterization, this reduces to

κ =L + N

2E. (21)

3.1 Geometric Identities

Using the definition of the second fundamental form, the isothermal parameterization, andthe definition of the normal vector, we have the following identities:

t1α · n = −nα · t1 =

L√E

,

t1β · n = −nβ · t1 = t2

α · n = −nα · t2 =M√E

,

t2β · n = −nβ · t2 =

N√E

.

We also have, using only the definitions of the tangent vectors and the isothermal para-meterization, the following identities:

t1α · t2 = −t2

α · t1 = −Eβ

2E,

t1β · t2 = −t2

β · t1 =Eα

2E.

These identities can be combined to yield the following formulas for second derivativesof the surface:

Xαα =Eα

2√

Et1 − Eβ

2√

Et2 + Ln,

Xαβ =Eβ

2√

Et1 +

Eα

2√

Et2 + M n,

Xββ = − Eα

2√

Et1 +

Eβ

2√

Et2 + N n.

Furthermore, we can use these identities to get formulas for the first derivatives of the unittangent and normal vectors:

t1α =

Xαα√E− Eα

2Et1 = −Eβ

2Et2 +

L√E

n,

6

t1β =

Xαβ√E− Eβ

2Et1 =

Eα

2Et2 +

M√E

n,

t2α =

Xαβ√E− Eα

2Et2 =

Eβ

2Et1 +

M√E

n,

t2β =

Xββ√E− Eβ

2Et2 = −Eα

2Et1 +

N√E

n.

Using the definition of n and cross-product identities, with the above (or, alternatively, usingthe definition of the second fundamental form), we get

nα = − L√E

t1 − M√E

t2,

nβ = − M√E

t1 − N√E

t2.

3.2 Evolution Equations for the Fundamental Coefficients

We derive evolution equations for the six components of the first and second fundamentalcoefficients under the assumption of an isothermal parameterization. These equations havetwo main advantages for numerical computation: (1) high order or dominant terms at smallscales have a particularly simple form which permits efficient implementation of implicitmethods, and (2) the interface shape can be recovered from knowledge of the six fundamentalcoefficients [12].

We start by differentiating the evolution equation with respect to α and β. Using thegeometric identities, we find

Xαt =

(V1α + V2

Eβ

2E− U

L√E

)t1 +

(V2α − V1

Eβ

2E− U

M√E

)t2

+

(Uα + V1

L√E

+ MV2√E

)n, (22)

Xβt =

(V1β − V2

Eα

2E− U

M√E

)t1 +

(V2β + V1

Eα

2E− U

N√E

)t2

+

(Uβ + V1

M√E

+ V2N√E

)n. (23)

From these we can easily find evolution equations for E, F, and G.

Et = 2Xαt ·Xα = 2√

E(Xαt · t1) = 2√

EV1α + V2Eβ√E− 2UL, (24)

Gt = 2Xβt ·Xβ = 2√

E(Xβt · t2) = 2√

EV2β + V1Eα√E− 2UN,

Ft =√

E(Xαt · t2 + Xβt · t1) =√

E(V2α + V1β)− V1Eβ + V2Eα

2√

E− 2UM. (25)

7

We now want to give formulas for Lt, Mt, and Nt. Since the definition of the secondfundamental form involves n, we first need a formula for nt. First, notice that since n is aunit vector, nt · n = 0. Then, we have

nt = (t1 × t2)t =Xαt√

E× t2 + t1 × Xβt√

E− Et

En

=

(Xαt√

E· n

)(n× t2) + (t1 × n)

(Xβt√

E· n

). (26)

Using cross-product identities and the above formulas for Xαt and Xβt, we have

nt = −(

Uα√E

+V1L

E+

V2M

E

)t1 −

(Uβ√E

+V1M

E+

V2N

E

)t2.

We now differentiate the definition L = −Xα · nα, which gives

Lt = −Xαt · nα −Xα · nαt

= −(Xαt · t1)(t1 · nα)− (Xαt · t2)(t2 · nα)−√

E (t1 · nαt)

= −(Xαt · t1)(t1 · nα)− (Xαt · t2)(t2 · nα)−√

E (t1 · nt)α +√

E (t1α · t2)(t2 · nt). (27)

Using the above calculations in (27) we find the following:

Lt =

(V1α +

V2Eβ

2E− UL√

E

)(L√E

)+

(V2α − V1Eβ

2E− UM√

E

)(M√E

)

+√

E∂α

(Uα√E

+V1L

E+

V2M

E

)+

(Eβ

2√

E

)(Uβ√E

+V1M

E+

V2N

E

)(28)

We perform the same type of calculations to deduce Mt. At first we find

Mt = −(Xα · nβ)t = −Xαt · nβ −Xα · nβt

= −(Xαt · t1)(t1 · nβ)− (Xαt · t2)(t2 · nβ))−√

E (t1 · nβt)

= −(Xαt · t1)(t1 · nβ)− (Xαt · t2)(t2 · nβ))−√

E (t1 · nt)β +√

E (t1β · t2)(t2 · nt). (29)

As before, use some of the identities derived previously to find the following formula for Mt :

Mt =

(V1α +

V2Eβ

2E− UL√

E

)(M√E

)+

(V2α − V1Eβ

2E− UM√

E

)(N√E

)

+√

E∂β

(Uα√E

+V1L

E+

V2M

E

)−

(Eα

2√

E

)(Uβ√E

+V1M

E+

V2N

E

). (30)

We perform the same steps now to calculate Nt :

Nt = −(Xβ · nβ)t = −Xβt · nβ −Xβ · nβt

= −(Xβt · t1)(t1 · nβ)− (Xβt · t2)(t2 · nβ)−√

E (t2 · nt)β +√

E (t2β · t1)(t1 · nt). (31)

8

Once again we use identities derived earlier to get an evolution equation for N :

Nt =

(V1β − V2Eα

2E− UM√

E

) (M√E

)+

(V2β +

V1Eα

2E− UN√

E

)(N√E

)

+√

E∂β

(Uβ√E

+V1M

E+

V2N

E

)+

(Eα

2√

E

)(Uα√E

+V1L

E+

V2M

E

). (32)

3.3 Tangential Velocities

We now find a pair of elliptic equations for the tangential velocities. If we start with a surfacethat has an isothermal parameterization, and choose the tangential velocities so that theysatisfy these elliptic equations, then the isothermal parameterization will be maintained atpositive times. The elliptic equations are simply found by requiring Et = Gt and Ft = 0.Using the formulas for Et, Ft, and Gt from the previous section, we have

V1α − V2β − V1Eα − V2Eβ

2E=

U(L−N)√E

,

V1β + V2α − V1Eβ + V2Eα

2E=

2UM√E

.

This can be rewritten as(

V1√E

)

α

−(

V2√E

)

β

=U(L−N)

E,

(V1√E

)

β

+

(V2√E

)

α

=2UM

E.

Finally, we can write this as

(V1√E

)

αα

+

(V1√E

)

ββ

=

(U(L−N)

E

)

α

+

(2UM

E

)

β

,

(V2√E

)

αα

+

(V2√E

)

ββ

=

(2UM

E

)

α

−(

U(L−N)

E

)

β

. (33)

Clearly, given E, U, and the second fundamental coefficients, we can solve these equationsfor V1/

√E and V2/

√E, and thus for V1 and V2 themselves.

4 Small scale decomposition

Following [2] we rewrite the Birkhoff-Rott integral to factor out the dominant contribution atsmall 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

integral. Assume X(~α) is sufficiently regular and that X(~α) − X(~α′) 6= 0 for ~α 6= ~α′. ByTaylor’s expansion,

X(α, β)−X(α′, β′)|X(α, β)−X(α′, β′)|3 =

Xα(α′, β′)(α− α′) + Xβ(α′, β′)(β − β′)

[E(α′, β′)]32 |~α− ~α′|3

+ O(|~α− ~α′|−1). (34)

Definej = µαXβ − µβXα, (35)

and rewrite the Birkhoff-Rott integral (13) as

V(X) =1

4πPV

∫ ∞

−∞

∫ ∞

−∞j′ ×

{X′

α(α− α′) + X′β(β − β′)

E ′ 32 |~α− ~α′|3

+X−X′

|X−X′|3

− X′α(α− α′) + X′

β(β − β′)

E ′ 32 |~α− ~α′|3

}dα′dβ′, (36)

where primed functions are evaluated at (α′, β′) and all other functions at (α, β). Equation(36) can be represented in terms of Riesz transforms, which are defined by

H1f(α, β) =1

2πPV

∫ ∞

−∞

∫ ∞

−∞

f(α′, β′)(α− α′)

|~α− ~α′|3 dα′dβ′, (37)

H2f(α, β) =1

2πPV

∫ ∞

−∞

∫ ∞

−∞

f(α′, β′)(β − β′)

|~α− ~α′|3 dα′dβ′. (38)

The symbols of the Riesz transforms are [16]

H1f(k) = −ik1

|k| fk, H2f(k) = −ik2

|k| fk, (39)

where the Fourier coefficients f(k) are defined by

f(k) =1

4π2

∫ ∞

−∞

∫ ∞

−∞e−i(k1α+k2β)f(α, β) dαdβ, (40)

and k = (k1, k2) is the (vector) wavenumber. The Riesz transform of a vector f = (f1, f2) isdefined as the vector whose components are the tranforms of fi. In terms of Riesz transforms,equation (36) takes the form

V(X) = H1

(j×Xα

2E32

)+ H2

(j×Xβ

2E32

)+ J(α, β, t) (41)

where J denotes the principle value integral associated with the second and third terms withinbraces in (36). The dominant terms at small scales in (41) are the two Riesz transforms. Tosee this, note that at next order in the Taylor’s expansion (34), the terms are of the form

g(α′, β′)(α− α′)i(β − β′)j

|~α− ~α′|3

10

where g depends on first and second derivatives of X and i and j are nonegative integerswith i + j = 2. The Taylor’s expansion of J therefore has highest order terms of the form

Gi,jh(α, β) =

∫ ∞

−∞

∫ ∞

−∞

h(α′, β′)(α− α′)i(β − β′)j

|α− α′|3 dα′dβ′ (42)

where h = j× g. The symbol of Gi,j is given by

Gi,j(k) = −2πki1k

j2

|k|3 hk. (43)

For sufficiently smooth X it follows that Gi,j(k) = O(bHi(k)|k| ) for |k| >> 1, where Hi(k) for

i = 1, 2 are Fourier coefficients of the Riesz tranforms in (41). Thus the Riesz transformsare the dominant terms in (41) at high wavenumber, and J is smoother or lower order ofdegree −1.

We proceed to simplify the Riesz transforms in (41) to better isolate the high orderor dominant behavior at small scales. Introduce the notation f ∼ g to indicate that thedifference between f and g is smoother than g. We also need that, at small scales, smoothfunctions can be passed through Riesz transforms, i.e.,

Hi[fg] = gHi[f ] + E[f ] (44)

where the commutator E is a smoothing operator on f [2]. Calculate the cross product in(41) using the relation Xα×Xβ = n|Xα×Xβ|, pass n through the Riesz tranforms (incurringa commutator) and take the inner product with n to obtain the intermediate result

U = −1

2

{H1

(µα

E12

)+ H2

(µβ

E12

)}+ J1(α, β, t), (45)

Here J1, which contains the original Birkhoff-Rott integral, is lower order than the Riesztransform terms contained within the braces.

We now specialize to the case AT = W = 0, which describes the relaxation of the interfaceunder surface tension. Define Y = L + N and note from (19), (21), that

µ = −BY

2E. (46)

Then from (45),

U =B

4E32

{H1(Yα) + H2(Yβ)}+ J1(α, β, t) + J2(α, β, t), (47)

where we have used that E is lower order than Y and passed some factors through the Riesztransform, incurring a commutator. The term

J2 = −1

2

{H1

(µα

E12

)+ H2

(µβ

E12

)}− B

4E32

{H1(Yα) + H2(Yβ)} (48)

is lower order than the Riesz transform terms in (47). Equation (47) describes the behaviorof the normal velocity U at small scales and is the main result of this section.

11

4.1 A model problem

To illustrate our non-stiff boundary integral method we consider the model or approximatevelocity obtained by setting J1 = 0 in (47). The model velocity has the same mathematicalform as the exact velocity, in that the high order terms are embedded within a nonlo-cal operator and, additionally, there is a nonlocal lower order term. However, use of themodel velocity avoids computation of the Birkhoff-Rott integral which is contained withinJ1. Computation of the Birkhoff-Rott integral is numerically intensive; a naive or straight-forward evaluation requires O(N2) operations, where N is the number of node points in thediscretization of the interface. This can be reduced to O(N log N) operations using the FastMultipole Method [6], but at the expense of considerable programming effort. The nonlocalmodel velocity can be computed with spectral accuracy using the FFT, and allows us to testthe efficacy of our non-stiff boundary integral method without the added effort necessary toimplement a Fast Multipole Method. An additional advantage is that X(~α, t) decouples fromthe evolution equations for the fundamental coefficients, and therefore the coefficients canbe evolved without simultaneous calculation of the interface position. An implementation ofthe method with the exact velocity (47), using a fast summation algorithm to evaluate theBirkhoff-Rott integral for periodic surfaces, will be described in a future paper.

4.2 Reformulation of the evolution equations

We reformulate the evolution equations for the fundamental coefficients to isolate the highorder terms. First note that the equation for Yt can be written in the form

Yt = ∆U + F (α, β, t), (49)

which serves as a definition of F . From (47),

Uαα ∼ B

4E3/2{H1(Yααα) + H2(Yβαα)} , (50)

and similarly

Uββ ∼ B

4E3/2{H1(Yαββ) + H2(Yβββ)} , (51)

where we have used that E is lower order than Y . Clearly, the highest order term in theevolution equation (49) for Y is ∆U , which goes like a third derivative of Y . It follows thatthe equation for Y with the highest order terms factored out has a simple form:

Yt =B

4E3/2{H1(Yααα) + H2(Yβαα) + H1(Yαββ) + H2(Yβββ)}+ J3(α, β, t), (52)

where from (45), (49)

J3 = −1

2∆

{H1

(µα

E12

)+ H2

(µβ

E12

)}+ F (α, β, t)

− B

4E3/2{H1(Yααα) + H2(Yβαα) + H1(Yαββ) + H2(Yβββ)} (53)

12

is lower order than the terms within braces in (52).If E(α, β, t) were constant in space, the dominant (first) term on the right hand side of

equation (52) would diagonalize under the Fourier transfrom, allowing efficient implementa-tion of implicit integration methods. This motivates defining

Em(t) = minα,β

E(α, β, t) (54)

and writing

Yt =B

4E3/2m

{H1(Yααα) + H2(Yβαα) + H1(Yαββ) + H2(Yβββ)}+ R(α, β, t) (55)

where

R(α, β, t) =

(B

4E3/2− B

4E3/2m

){H1(Yααα) + H2(Yβαα) + H1(Yαββ) + H2(Yβββ)}+ J3. (56)

It is important to note that R contains terms that are the same order as the terms withinbraces in (55). However, comparing (55) and (56), we see that the high order terms in R havea smaller magnitude coefficient. A frozen coefficient analysis shows that implicit differencingof the terms within braces in (55) and explicit treatment of R provides a stable non-stiffmethod.

The main advantage of evolution equation (55) for Y is that it diagonalizes under theFourier tranform, i.e.,

d

dtYk = −B|k|3

4E3/2m

Yk + Rk, (57)

so that an implicit time integration or linear propagator method can be efficiently imple-mented. The other fundamental coefficients are recovered by time integration of equations(24), (28), and (30). The highest order terms in these equations are contained in the normalvelocity U , which by (47) 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., E = G and F = 0. Let z = Z(x, y) be a given initial interface shape satisfying (20).We construct an isothermal parameterization by evolving an initially flat interface withparameterization

x(α, β, 0) = α, y(α, β, 0) = β, z(α, β, 0) = 0.

The interface is evolved according to (17), with the normal velocity U chosen to beproportional to the difference between the prescribed data and the current interface position,i.e.,

U(~α) = Q [Z(x(~α, t), y(~α, t))− z(~α, t)] k · n, (58)

13

02

46 0

24

6−0.1

−0.05

0

0.05

0.1

yx

z

Figure 1: The isothermal surface parameterization corresponding to (60).

where (x(~α, t), y(~α, t), z(~α, t)) is the current interface position and Q is a positive constantwhich determines the rate at which the evolving surface approaches the specified initial data.The tangential components of velocity at the interface, V1 and V2, are determined by solvingthe elliptic system (33) so that the surface parameterization remains isothermal throughoutthe evolution. The evolution is stopped when

maxα,β

| Z(x(~α, t), y(~α, t))− z(~α, t) |≤ εd (59)

where εd is a given tolerance.The evolution problem (17) with velocity (58) is not stiff, and is integrated using an

explicit 2nd 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 arecalculated from the definition (6) via differentiation of X. The elliptic system (33) fortangential velocities V1, V2 is solved with spectral accuracy using the FFT.

Figure 1 shows the isothermal parameterization corresponding to the data

Z(x, y) = 0.1 ∗ cos(x) cos(y). (60)

The calculation used Np = 128 equally spaced node points in α and β, and we set ∆t =2.0× 10−3 and εd = 10−12. Only every other grid line is shown in the figure.

Figure 2 plots log10 |E(~α)−G(~α)| (left) and log10 |F (~α)| (right) for the surface in figure1. The figure shows that |E − G| and |F | are less than 10−11, indicating that the interfaceparameterization is effectively isothermal.

5.2 Time evolution

We use a linear propagator method [10] to integrate equation (57). Linear propagator meth-ods are based on factoring out the dominant (linear) term at high wavenumbers, and providestable and potentially high-order methods for diffusive problems. If the dominant term hasa constant coefficient, these methods propagate the highest order modes exactly.

14

0

2

4

6

0

2

4

6−18

−17

−16

−15

−14

−13

−12

−11

xy

log|

E−

G|

0

2

4

6

0

2

4

6−18

−17

−16

−15

−14

−13

−12

−11

xy

log|

F|

Figure 2: log10 |E −G| and log10 |F | versus x and y.

Use an integrating factor to rewrite equation (57) as

∂

∂tψk(t) = exp

(B|k|3

4

∫ t

0

1

E3/2m (t′)

dt′)

Rk, (61)

where

ψk(t) = Yk exp

(B|k|3

4

∫ t

0

1

E3/2m (t′)

dt′)

. (62)

Equation (61) is discretized using a second order Adams-Bashforth method. In terms of Yk

the result is

Y n+1k = Y n

k ek(tn, tn+1) +∆t

2

(3Rn

kek(tn, tn+1)− Rn−1k ek(tn−1, tn+1)

), (63)

where

ek(t1, t2) = exp

(−B|k|3

4

∫ t2

t1

1

E3/2m (t′)

dt′)

. (64)

The linear propagator method removes the stiffness by propagating the linear or high orderterm in (57) from tn to tn+1 at the exact exponential rate. We obtain second order accuracyby replacing the continuous integrals with their trapezoidal rule approximations:

ek(tn, tn+1) = exp

(−B|k|3∆t

8

[1

(Enm)3/2

+1

(En+1m )3/2

])(65)

ek(tn−1, tn+1) = exp

(−B|k|3∆t

4

[1

2(En−1m )3/2

+1

(Enm)3/2

+1

2(En+1m )3/2

]). (66)

The discretization propagates the exact exponential decay of the high order terms by asecond order accurate approximation. Recall that that En+1 is computed explicitly from(24).

Equations (24), (28), and (30) for E, L and M are discretized explicitly in time usinga second order Adams-Bashforth method. The highest order terms in these equations arecontained in the normal velocity U , see (47). The stability of these explicit discretizations are‘slave’ to the stability of the discretization (63) for Y , so they do not lead to any additionaltime-step constraint. We use N = Y −L to recover N from the solutions for Y and L. Riesz

15

transforms are computed with spectral accuracy in space using the symbols (39), and theelliptic problem for the tangential velocities V1, V2 is solved using the FFT.

As noted in [10], near equilibrium the second-order linear propagator method requires∆t < C(B), i.e. for stability the time step is independent of the spatial discretization. Moregenerally 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 minimizedby reducing the time step.

5.3 Numerical results

The stability constraints of the linear propagator method are compared with an explicitAdams-Bashforth discretization of evolution equations (24), (28), (30), and (32) for thefundamental coefficients. We consider the relaxation under surface tension of the interfaceshown in figure 1, with surface tension coefficient B = 1.0. For Np equally spaced nodepoints in the α and β variables, a frozen coefficient analysis of equation (57) reveals the timestep constraint for stability of an explicit method to be

∆t ≤ CE3/2m /(BN3

p ), (67)

where C is a constant independent of Np.

Figure 3 shows the Fourier transform log10 |Lk| versus k = (k, l) calculated by the explicitmethod at t = .1 for the two resolutions Np = 32 (top) and Np = 64 (bottom). Fouriercoefficients are shown for 0 ≤ k ≤ Np/2 and 0 ≤ l ≤ Np − 1, and coefficients for the otherNp/2 − 1 k-values are determined from the real-valuedness of L. Note that modes withl = Np/2 + 1 to l = Np actually correspond to negative wavenumber modes l = −Np/2 + 1to l = −1. The time steps for the lower resolution plots at top are ∆t = 4.5 × 10−4 (leftplot) and ∆t = 2.25× 10−4 (right plot). There are no spectral or Fourier filters [11] used inthe calculation, so that we may assess the stability of high-wavenumber modes generated byround-off error.

Instability in a spectral method is typically observed as a rapid and unphysical growthin the amplitudes of the high wavenumber modes. High wavenumber growth and instabilityare clearly exhibited in the calculation at the top left in figure 3. The time step here,∆t = 4.5 × 10−4, is above but close to the stability threshold, and the explicit method isfound to be stable at t = 0.1 when the time step is decreased by a factor of 2, as shownin the plot at top right. Increasing the number of node points to Np = 64 gives instabilityat the time step ∆t = 5.0 × 10−5 (bottom, left) and stability at ∆t = 2.5 × 10−5 (bottom,right). The approximate factor of 8 reduction in time step required to achieve stability isexpected from the constraint (67).

Results from the linear propagator method, using a time step ∆t = 5.0×10−3, are shownin figure 4. It is clear from the spectra that the method is stable, even at this relatively largetime step, for Np = 64 and Np = 128. Indeed, we find the method is stable for time steps aslarge as ∆t = 0.1 at Np = 128. This example indicates that the small-scale decompositionpresented here is effective at removing the stiffness due to surface tension in 3D porous mediaflow.

16

05

1015

20

0

10

20

30

40−20

−15

−10

−5

0

kl

log

|Lk|

05

1015

20

0

10

20

30

40−20

−15

−10

−5

0

kl

log

|Lk|

010

2030

40

0

20

40

60

80−20

−15

−10

−5

0

kl

log

|Lk|

010

2030

40

0

20

40

60

80−20

−15

−10

−5

0

kl

log

|Lk|

Figure 3: Plot of log10 |Lk| versus k for the explicit method. Top left: Np = 32, ∆t =4.5× 10−4. Top right: Np = 32, ∆t = 2.25× 10−4. Bottom left: Np = 64, ∆t = 5.0× 10−5.Bottom right: Np = 64, ∆t = 2.5× 10−5.

010

2030

40

0

20

40

60

80−20

−15

−10

−5

0

kl

log

|Lk|

020

4060

80

0

50

100

150−20

−15

−10

−5

0

kl

log

|Lk|

Figure 4: Plot of log10 |Lk| versus k for the linear propagator method with ∆t = 5 × 10−3.Left: Np = 64. Right: Np = 128.

17

6 Conclusions

We have presented a non-stiff boundary integral method for 3D porous media flow withsurface tension. Our method uses the first and second fundamental coefficients of the surfaceas 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 theydiagonalize under the Fourier transform, enabling the efficient application of implicit orlinear propagator time-integration methods. Our method includes an efficient algorithm forthe generation of initial data with an isothermal parameterization by evolving a flat interfacetoward a prescribed initial surface shape.

The method is tested by computing the relaxation of an interface to a flat surface underthe action of surface tension. These calculations employ an approximate interface velocityto test the stiffness reduction of the method. The approximate velocity has the same math-ematical form as the exact velocity, but avoids the numerically intensive computation of thefull Birkhoff-Rott integral. The algorithm is found to be effective at eliminating the severetime-step constraint that plagues explicit time-integration methods.

The use of an isothermal parameterization limits our method to surfaces in which theaverage arclength in the α and β directions are equal. A modification of the algorithm en-ables computations to be performed for more general surfaces. This will be the subject of afuture paper in which we also implement the method with the exact velocity (47), using afast summation algorithm to evaluate the Birkhoff-Rott integral for periodic surfaces, whichhas been recently developed by the authors.

AcknowledgementsThis work was supported by the NSF under Grant Nos. DMS-0708977 and DMS-0354560(MS) and DMS-0926378 (DMA). Simulations were conducted on the NJIT computing cluster,supported by the NSF/MRI under Grant No. DMS-0420590.

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