## Abstract

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES).

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## Funding

This work was supported in part by the National Science Foundation under award numbers DMS-1907684 (DMA), NSF DMS-1352353 & DMS-1909035 (JLM), DMS-1716560 (JW) and DMS-1910824 (RC & RM); by the Office of Naval Research under award number ONR N00014-18-1-2490 (RC & RM); and by the Department of Energy, Office of Science, Applied Scientific Computing Research, under award number DE-AC02-05CH11231 (JW). JLM wishes to thank the Mathematical Sciences Research Institute for hosting him while a portion of this work was completed.

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## Appendices

### Appendix A. Verification of the HLS equations

In Section 2.2, we proposed evolving only *P**𝜃* via (2.17) and constructing *P*_{0}*𝜃*, *s*_{α} and *ζ*(*α*) from *P**𝜃* via (2.13) and (2.16). Here we show that both equations of (2.10) hold even though *P*_{0}*𝜃* and *s*_{α} are computed algebraically rather than by solving ODEs, and that these equations, in turn, imply that the curve kinematics are correct, i.e., (*ξ*_{t},*η*_{t}) = *U***n** + *V* **t**.

From (2.13), we have \(S_{t} = P_{0}\left [({\cos \limits } P\theta )(P\theta )_{t}\right ]\), \(C_{t} = -P_{0}\left [({\sin \limits } P\theta )(P\theta )_{t}\right ]\), and

In the last step, we used (2.15) and the fact that *P* is self-adjoint. Combining (2.17) and (A.1), we obtain

We must show that the second term is zero. This follows from *V*_{α} = *P*(*𝜃*_{α}*U*) in (2.11). Indeed,

where the integrals are from 0 to 2*π* and we used (2.15). Similarly, we have

Using *V*_{α} = *P*(*𝜃*_{α}*U*) again, we find that

Combining this with (A.3), we obtain *s*_{αt} = −*P*_{0}[*𝜃*_{α}*U*], as claimed.

As for the second assertion that (*ξ*_{t},*η*_{t}) = *U***n** + *V* **t**, note that the equations of (2.10) are equivalent to

By equality of mixed partials, the left-hand side equals \(\partial _{\alpha }\left [\zeta _{t}\right ]\), so we have *ζ*_{t} = (*V* + *i**U*)*e*^{i𝜃} up to a constant that could depend on *t* but not *α*. However, to enforce *ξ*_{t}(0) = *∂*_{t}0 = 0 in (2.16), we choose *V* (0) in (5.1) so that the real part of (*V* + *i**U*)*e*^{i𝜃} is zero at *α* = 0. We conclude that *ζ*_{t} − (*V* + *i**U*)*e*^{i𝜃} = *i**a*, where *a* is real and could depend on time but not *α*. We need to show that *a* = 0. Note that

The divergence theorem implies that \({\int \limits }_{\Gamma } U ds=0\). This is because ∇*ϕ* is single-valued and divergence free in Ω; \(U=\nabla \phi \cdot \hat {\mathbf n}\) on Γ; \(\nabla \phi \cdot \hat {\mathbf n}=0\) on the solid boundaries; and \((\nabla \phi \cdot \hat {\mathbf n})\vert _{x=2\pi }=-(\nabla \phi \cdot \hat {\mathbf n})\vert _{x=0}\) since ∇*ϕ* is periodic while \(\hat {\mathbf n}\) changes sign. From (2.16), \({\int \limits }_{0}^{2\pi }\eta \xi _{\alpha } d\alpha =0\) for all time. Differentiating, we obtain

Thus *a* = 0 and *ζ*_{t} = (*V* + *i**U*)*e*^{i𝜃}, as claimed.

### Appendix B. Variant specifying the stream function on the solid boundaries

The integral equations of Section 3.2 are tailored to the case where *V*_{1}, *a*_{2}, …, *a*_{N} in the representation (3.1) for Φ are given and the constant values *ψ*|_{k} are unknown. If instead *ψ* is completely specified on Γ_{k} for 1 ≤ *k* ≤ *N*, then we would have to solve for *a*_{2}, …, *a*_{N} along with the *ω*_{j}. In this scenario, \(\varphi =\phi \vert _{{\Gamma }_{0}^{-}}\) is given on the free surface, from which we can extract *V*_{1} as the change in *φ* over a period divided by 2*π*. So we can write

where \(a_{j}[\omega ] = \langle \mathbf {1}_{j},\omega \rangle = \frac 1{2\pi }{\int \limits }_{0}^{2\pi }\omega _{j} d\alpha \) are now functionals that extract the mean from *ω*_{2}, …, *ω*_{N}. Instead of (3.13), we would define

The right-hand side *b* in (3.15) would become \(b_{0}(\alpha )=\left [\varphi (\alpha )-V_{1}\xi (\alpha )\right ]\) and *b*_{k}(*α*) = [*ψ*(*ζ*_{k}(*α*)) − *V*_{1}*η*_{k}(*α*)], where *φ* and \(\psi \vert _{{\Gamma }_{k}}\) are given. The latter would usually be constant functions, though a nonzero flux through the cylinder boundaries can be specified by allowing \(\psi \vert _{{\Gamma }_{k}}\) to depend on *α*. However, we still require \(\psi \vert _{{\Gamma }_{k}}\) to be periodic (since the stream function is single-valued in our formulation), so the net flux out of each cylinder must be zero.

We now prove invertibility of this version of \(\mathbb {A}\), which maps *ω* to the restriction of the real or imaginary parts of \(\check {\Phi }(z)\) to the boundary. We refer to these real or imaginary parts as the “boundary values” of \(\check {\Phi }\). In the same way, \(\mathbb {B}\) maps *ω* to the boundary values of \(\tilde {\Phi }\) in (3.10). Note that \(\mathbb {A}\) differs from \(\mathbb {B}\) by a rank *N* − 1 correction in which a basis for \(\mathcal {V}=\ker \mathbb {B}\) is mapped to a basis for the space \(\mathcal {R}_{\text {cyl}}\) of boundary values of \(\operatorname {span}\{{\Phi }_{\text {cyl}}(z-z_{j})\}_{j=2}^{N}\). From Section 6.1, we know that \(\dim \left (\operatorname {coker}(\mathbb {B})\right )=N-1\), so we just have to show that \(\mathcal {R}_{\text {cyl}}\cap \operatorname {ran}(\mathbb {B})=\{0\}\). Suppose the boundary values of \({\Phi }_{c}(z)={\sum }_{j=2}^{N} a_{j}{\Phi }_{\text {cyl}}(z-z_{j})\) belong to \(\operatorname {ran}(\mathbb {B})\). Then there are dipole densities *ω*_{j} such that the corresponding sum of Cauchy integrals \(\tilde {\Phi }(z)={\sum }_{j=0}^{N}{\Phi }_{j}(z)\) has these same boundary values. The imaginary part, \(\tilde \psi \), satisfies the Laplace equation in Ω, has the same Dirichlet data as *ψ*_{c} on Γ_{1}, …, Γ_{N}, and the same Neumann data as *ψ*_{c} on Γ_{0} (due to *∂*_{n}*ψ* = *∂*_{s}*ϕ*). Since solutions are unique, \(\tilde \psi =\psi _{c}\). But the conjugate harmonic function to \(\tilde \psi \) is single-valued while that of *ψ*_{c} is multiple-valued unless all the *a*_{j} = 0. We conclude that \(\mathcal {R}_{\text {cyl}}\cap \operatorname {ran}(\mathbb {B})=\{0\}\), as claimed.

### Appendix C. Cauchy integrals, layer potentials and sums over periodic images

In this section we consider the connection between Cauchy integrals and layer potentials and the effect of summing over periodic images and renormalization. As is well-known [60], Cauchy integrals are closely related to single- and double-layer potentials through the identity

where *ζ* − *z* = *r**e*^{i𝜃}. We adopt the sign convention of electrostatics [24, 49] and define the Newtonian potential as \(N(\zeta ,z)=-(2\pi )^{-1}\log |\zeta -z|\). The double-layer potential (with normal **n**_{ζ} pointing left from the curve *ζ*, as in Section 3 above) has the geometric interpretation

For a closed contour in the complex plane, we have

so, if *ω* is real-valued, the real part of a Cauchy integral is a double-layer potential with dipole density *ω* while the imaginary part is a single-layer potential with charge density − *d**ω*/*d**s*. In the spatially periodic setting, the real part of the two formulas in (3.3) may be written

and

Equation (C.5) follows from Euler’s product formula \({\sin \limits } w \! =\! w{\prod }_{m=1}^{\infty }(1-(w/m \pi )^{2})\), which gives

where \(c_{0}=-\frac 1{2\pi }\log 2\) and \(c_{m}=-\frac 1{2\pi }\log |2\pi m|\) if *m*≠ 0. It was possible to drop the terms *c*_{m} in (C.5) and (C.6) since \(\omega _{j}^{\prime }(\alpha )\) is integrated over a period of *ω*_{j}(*α*). However, these terms have to be retained to express (C.6) as a principal value integral,

Through (C.7), we can regard \(\log |\sin \limits (w/2)|\) as a renormalization of the divergent sum of the Newtonian potential over periodic images in 2D. Setting aside these technical issues, it is conceptually helpful to be able to interpret *ϕ*_{0}(*z*) and *ϕ*_{j}(*z*) from (3.3) as double and single layer potentials with dipole and charge densities *ω*_{0}(*α*) and \(\omega _{j}^{\prime }(\alpha )/s_{\alpha }\), respectively, over the real line or over the periodic array of obstacles. Of course, it is more practical in 2D to work directly with the formulas involving complex cotangents over a single period, but (C.4) and (C.5) are a useful starting point for generalization to 3D.

### Appendix D. Alternative derivation of the vortex sheet strength equation

In this appendix, we present an alternative derivation of (4.14) that makes contact with results reported elsewhere [7, 14] in the absence of solid boundaries. As in Section 3, the velocity potential is decomposed into \(\phi (z)=\tilde \phi (z) + \phi _{\text {mv}}(z)\) where \(\tilde \phi (z)\) is the sum of layer potentials and *ϕ*_{mv}(*z*) is the multi-valued part. We also define **W** as in (4.10), where the component Birkhoff-Rott integrals **W**_{0j} are given in complex form by

The Plemelj formulas (4.3) imply that when the interface is approached from the fluid region,

Recall that *φ*(*α*,*t*) = *ϕ*(*ζ*(*α*,*t*),*t*) is the restriction of the velocity potential to the free surface as it evolves in time, and note that \(\varphi _{\alpha }=s_{\alpha }\nabla \phi \cdot \hat {\mathbf {t}}\). Solving for *γ*_{0}, then, we have

Differentiating with respect to time, we get

In Section 4.2, we avoided directly taking time derivatives of *γ*_{0}(*α*,*t*), **W**(*α*,*t*) and *φ*(*α*,*t*), which lead to more involved calculations here due to the moving boundary. We know that \(\hat {\mathbf {t}}_{t}=\theta _{t}\hat {\mathbf {n}},\) and that *𝜃*_{t} = (*U*_{α} + *V* *𝜃*_{α})/*s*_{α}. We substitute these to obtain

We now work on the equation for *φ*_{αt}. As was done in [7], the convective derivative (3.25) together with the Bernoulli equation gives

We write \(\mathbf {W} =U\hat {\mathbf {n}}+(\mathbf {W} \cdot \hat {\mathbf {t}})\hat {\mathbf {t}}\), substitute (D.2) into (D.5), and use \(\mathbf {W} \cdot \mathbf {W} =U^{2}+(\mathbf {W} \cdot \hat {\mathbf {t}})^{2}\):

We differentiate with respect to *α*:

We substitute (D.6) into (D.4), noticing that the *U**U*_{α} terms cancel:

We group this as follows:

The quantity in square brackets simplifies considerably using the equations *V*_{α} = *s*_{αt} + *𝜃*_{α}*U*, \(U=\mathbf {W} \cdot \hat {\mathbf {n}},\) and \(\hat {\mathbf {t}}_{\alpha }=\theta _{\alpha }\hat {\mathbf {n}}\). Together with the boundary condition for the pressure (the Laplace-Young condition), we obtain

This agrees with the equation for *γ*_{0,t} as found in [7] if one assumes (*s*_{α})_{α} = 0. The calculation of [7] has no solid boundaries and a second fluid above the first, which we take to have zero density when comparing to (D.7).

Our final task is to compute \(s_{\alpha }\mathbf {W}_{t}\cdot \hat {\mathbf {t}} = (\mathbf {W}_{00,t} + {\cdots } + \mathbf {W}_{0N,t})\cdot (s_{\alpha } \hat {\mathbf t})\) in the right-hand side of (D.7). Differentiating (D.1) with respect to time for 1 ≤ *j* ≤ *N* gives

Here, as above, a prime denotes *∂*_{α} and we note that the solid boundaries remain stationary in time. Suppressing *t* in the arguments of functions again, we conclude that for 1 ≤ *j* ≤ *N*,

where *ζ*_{t} is treated as the vector (*ξ*_{t},*η*_{t}) in the dot product. When *j* = 0, we regularize the integral

and then differentiate both sides with respect to time

Observing that \(\zeta _{t}^{\prime }W_{00}^{*}=\left ([\zeta _{t}W_{00}^{*}]_{\alpha }-\zeta _{t}W_{00,\alpha }^{*}\right )\), we find that

Finally, setting **W**_{mv} = ∇*ϕ*_{mv}(*ζ*(*α*,*t*)), we compute

When (D.8), (D.9) and (D.10) are combined and substituted into (D.7), several of the terms cancel:

Also, in (D.9), *ζ*_{t} ⋅**W**_{00} cancels the *ζ*_{t}(*α*) term in the integrand, leaving behind a principal value integral. Including the other terms of (D.7), moving the unknowns to the left-hand side, and dividing by 2, we obtain (4.14).

### Appendix E. Treating the bottom boundary as an obstacle

The conformal map *w* = *e*^{−iz} maps the infinite, 2*π*-periodic region \({\Omega }_{1}^{\prime }\) below the bottom boundary to a finite domain, with \(-i\infty \) mapped to zero. Let \(w_{j}=e^{-iz_{j}}\) denote the images of the points *z*_{j} in (3.2), which are used to represent flow around the obstacles via multi-valued velocity potentials. We also define the curves

which traverse closed loops in the *w*-plane, parameterized clockwise. The image of the fluid region lies to the right of Υ_{0}(*α*) and to the left of Υ_{j}(*α*) for 1 ≤ *j* ≤ *N*. The terms *V*_{1}*z* and *a*_{j}Φ_{cyl}(*z* − *z*_{j}) appearing in (3.2) all have a similar form in the new variables,

We can think of *V*_{1}*z* as a multiple-valued complex potential on the 2*π*-periodic domain of logarithmic type with center at \(z_{1}=-i\infty \). It maps to \(V_{1}i\log (w-w_{1})\) in the *w*-plane, where *w*_{1} = 0. From (3.5), we see that the *n* th sheet of the Riemann surface for Φ_{cyl}(*z*(*w*) − *z*_{j}) is given by − *i* Log(1 − *w*_{j}/*w*) + 2*π**n*, which has a branch cut from the origin to *w*_{j}. When traversing the curve *w* = Υ_{k}(*α*) with *α* increasing, the function Φ_{cyl}(*z*(*w*) − *z*_{j}) decreases by 2*π* if *k* = *j*, increases by 2*π* if *k* = 1, and returns to its starting value for the other boundaries, including the image of the free surface (*k* = 0). This is done so that only the *V*_{1}*z* term has a multiple-valued real part on Γ_{0}, which simplifies the linear systems analyzed in Section 6.1–6.2 above.

The cotangent-based Cauchy integrals Φ_{j}(*z*) in (3.3) transform to (1/*w*)-based Cauchy integrals in the new variables, aside from an additive constant in the kernels [24]. In more detail,

For 1 ≤ *j* ≤ *N*, we then have

with a similar formula for Φ_{0}(*z*(*w*)), replacing *i**ω*_{j}(*α*) by *ω*_{0}(*α*). The second term is a constant function of *w* that prevents **1**_{1} from being annihilated by \(\mathbb {B}\) in Section 3.2. This is the primary way in which the bottom boundary differs from the other obstacles in the analysis of Sections 6.1–6.2.

We note that \(\tilde {\Phi }(z(w))={\sum }_{j=0}^{N}{\Phi }_{j}(z(w))\) is analytic at *w* = 0, which allows us to conclude that if its real or imaginary part satisfies Dirichlet conditions on \({\Gamma }_{1}^{-}\), it is zero in \({\Omega }_{1}^{\prime }\). A similar argument using *w* = *e*^{iz} works for the region \({\Omega }_{0}^{\prime }\) above the free surface, which was needed in Section 6.2 above.

### Appendix F. Evaluation of Cauchy integrals near boundaries

In this section we describe an idea of Helsing and Ojala [42] to evaluate Cauchy integrals with spectral accuracy even if the evaluation point is close to the boundary. We modify the derivation to the case of a 2*π*-periodic domain, which means the \(\frac 1z\) Cauchy kernels in [42] are replaced by \(\frac 12\cot \frac z2\) kernels here. The key idea is to first compute the boundary values of the desired Cauchy integral *f*(*z*). The interior values are expressed in terms of these boundary values. From the residue theorem, we have

where \(\partial {\Omega }=\cup _{k=0}^{N} {\Gamma }_{k}\). Multiplying the second equation by *f*(*z*) and subtracting from the first, we obtain

The integrand is a product of two analytic functions of *z* and *ζ*, namely \(\frac {\zeta -z}2\cot \frac {\zeta -z}2\) and the divided difference \(f[\zeta ,z]=\left (f(\zeta )-f(z)\right )/(\zeta -z)= {{\int \limits }_{0}^{1}} f^{\prime }(z+(\zeta -z)\alpha ) d\alpha \). In particular, \(f[\zeta ,\zeta ]=f^{\prime }(\zeta )\) is finite, and the *k* th partial derivative of *f*[*ζ*,*z*] with respect to *ζ* is bounded, uniformly in *z*, by \(\max \limits _{w\in {\Omega }}|f^{{(k+1)}}(w)|/(n+1)\). Thus, the integrand is smooth and the integral can be approximated with spectral accuracy using the trapezoidal rule,

Solving for *f*(*z*) gives

In (5.5), we interpret this as a quadrature rule for evaluating the first integral of (F.1) that maintains spectral accuracy even if *z* approaches or coincides with a boundary point *ζ*_{k}(*α*_{km}).

### Appendix G. Remarks on generalization to three dimensions

We anticipate that both methods of this paper generalize to 3D with some modifications. One aspect of the problem becomes easier in 3D, namely that the velocity potential is single-valued. However, one loses complex analysis tools such as summing over periodic images in closed form with the cotangent kernel and making use of the residue theorem to accurately evaluate layer potentials near the boundary.

The velocity potential method can be adapted to 3D by replacing constant boundary conditions for the stream function on the solid boundaries with homogeneous Neumann conditions for the velocity potential. This entails using a double-layer potential on the free surface and single-layer potentials on the remaining boundaries. In her recent PhD thesis [48], Huang shows how to do this in an axisymmetric HLS framework. She implemented the method to study the dynamics of an axisymmetric bubble rising in an infinite cylindrical tube. One of the biggest challenges was finding an analog of the Hilbert transform to regularize the hypersingular integral that arises for the normal velocity. Huang introduces a three-parameter family of harmonic functions involving spherical harmonics for this purpose. This method can handle background flow along the axis of symmetry, but many technical challenges remain for the non-axisymmetric case, e.g., for doubly periodic boundary conditions in the horizontal directions.

Analogues of the vortex sheet method in three dimensions have been developed previously in various contexts. Caflisch and Li [17] work out the evolution equations in a Lagrangian formulation of a density-matched vortex sheet with surface tension in an axisymmetric setting. Nie [61] shows how to incorporate the HLS method to study axisymmetric, density-matched vortex sheets. In his recent PhD thesis, Koga [51] studies the dynamics of axisymmetric vortex sheets separating a “droplet” from a density-matched ambient fluid. He develops a mesh refinement scheme based on signal processing and shows how to regularize singular axisymmetric Biot-Savart integrals with new quadrature rules. Koga implements these ideas using graphics processing units (GPUs) to accelerate the computations.

The non-axisymmetric problem with doubly periodic boundary conditions has been undertaken by Ambrose et al. [10]. They propose a generalized isothermal parameterization of the free surface, building on work of Ambrose and Masmoudi [8], which possesses several of the advantages of the HLS angle-arclength parameterization in 2D. The context of [10] is interfacial Darcy flow in porous media, which also involves Birkhoff-Rott integrals in 3D:

Here \(\vec {\alpha }=(\alpha ,\beta ),\) and the surface is given by \(\mathbf {X}(\vec {\alpha })=(\xi (\vec \alpha ), \eta (\vec \alpha ), \zeta (\vec \alpha ))\) with *ζ* now the *z*-coordinate instead of the complexified surface. In the integrand, the subscripts *α* and *β* represent derivatives with respect to these variables, and quantities without a prime are evaluated at \(\vec {\alpha }\) while quantities with a prime are evaluated at \(\vec {\alpha }^{\prime }\). The domain of integration is \(\mathbb {R}^{2}\). The quantity *ω* is, as in the 2D problem, the source strength in the double-layer potential.

The lack of a closed formula for the sum over periodic images in (G.1) contributes to the computational challenge of implementing the method in 3D. In [10], a fast method for calculation of this integral is introduced, based on Ewald summation. This involves splitting the calculation of the integral into a local component in physical coordinates and a complementary calculation in Fourier space; the method is optimized so that the two sums take similar amounts of work. We expect that the single layer potentials that occur at solid boundaries in the multiply connected case of the present paper could be computed similarly in 3D.

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### Cite this article

Ambrose, D.M., Camassa, R., Marzuola, J.L. *et al.* Numerical algorithms for water waves with background flow over obstacles and topography.
*Adv Comput Math* **48**, 46 (2022). https://doi.org/10.1007/s10444-022-09957-z

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DOI: https://doi.org/10.1007/s10444-022-09957-z

### Keywords

- Water waves
- Multiply connected domain
- Layer potentials
- Cauchy integrals
- Overturning waves
- Splash singularity
- GPU acceleration