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A Fast Integral Equation Method for the Two-Dimensional Navier-Stokes Equations

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Pith's one-line read A complete integral-equation flow solver for two-dimensional incompressible Navier-Stokes achieves 10th-order spatial and up to 4th-order temporal accuracy with near-linear time per step.

desk verdict A solid, honest, and genuinely useful fast integral equation solver for 2D Navier-Stokes; the 10th-order spatial claim is only directly proven for a benign stationary test, but the paper deserves serious peer review. read the letter →

arxiv 1908.07392 v1 pith:6SC2ZZOR submitted 2019-08-20 math.NA cs.NAphysics.flu-dyn

Ludvig af Klinteberg , Travis Askham , Mary Catherine Kropinski This is my paper · ORCID
classification math.NAcs.NAphysics.flu-dyn MSC 65N3865R2076D05
keywords Navier-StokesequationsintegralequationmethodsmodifiedStokespartitionofunityextensionkernel-splitquadraturefastmultipolemethodspectraldeferredcorrectioncomplexdomains
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the remaining practical obstacles to integral equation methods for the two-dimensional incompressible Navier-Stokes equations—accurate evaluation of layer potentials close to the boundary and smooth extension of the forcing onto a bounding grid—can both be overcome, and that the assembled solver is usable on complex smooth domains. Each semi-implicit time step reduces the Navier-Stokes equations to a modified Stokes equation, whose solution is split into a particular part computed by Fourier transform from a smoothly extended forcing and a homogeneous part represented as a modified-Stokes double layer potential. Because the divergence-free condition is built into the representation and the boundary conditions are enforced by a well-conditioned second-kind integral equation, no projection step or artificial pressure boundary conditions are needed. Numerical experiments show 10th-order spatial convergence and up to 4th-order temporal convergence, with per-step cost $O(N_B \log N_B + N_{\partial \Omega})$, placing the method in the low-to-moderate Reynolds number regime. If correct, this makes high-order, geometrically flexible flow solvers substantially easier to assemble.

What carries the argument

The load-bearing machinery is the representation of each time step as a particular-plus-homogeneous solution of the modified Stokes equations, with three specialized components making it fast and accurate. First, partition-of-unity extension (PUX) builds a compactly supported, high-regularity extension $F^e$ of the forcing on a uniform grid over the bounding box, so the particular solution is a single Fourier multiplier evaluated by FFT and non-uniform FFT. Second, the homogeneous correction is a modified-Stokes double layer potential $D[\mu]$, whose kernel is the stresslet constructed from the modified biharmonic Green's function $G(x,y) = -(2\pi\alpha^2)^{-1}(\log\|x-y\| + K_0(\alpha\|x-y\|))$; the density solves the second-kind integral equation with a rank-one nullspace correction $W[\mu] = \hat n (\int_{\partial\Omega} \mu\cdot\hat n\, dS)/(\int_{\partial\Omega} dS)$. Third, kernel-split quadrature rewrites the stresslet as smooth factors times explicit singularities $\log\|r\|$, $r_k/\|r\|^2$, and $r_i r_j r_k/\|r\|^4$, evaluated by complex interpolatory quadrature with recursive formulas, and the FMM uses stabilized expansions in $Q_n(\alpha r)$, $K_n(\alpha r)$, $P_n(\alpha r)$, and $r^{|n|}$ that remain stable for all $\alpha$. A fast direct solver precomputes the compressed inverse of the fixed boundary system, and semi-implicit spectral deferred correction raises the temporal order.

What would settle it

Run the manufactured problem of Section 5.3 on a smooth domain with a different curvature profile, keeping the same smooth forcing and $\alpha=10$, across the full range of grid sizes $N=40$ to $N=1300$. If the relative $\ell^2$ or $\ell^\infty$ errors stop following the 10th-order slope, or if the error at $N=800$ rises substantially above the reported level while the layer-potential quadrature is unchanged, then the PUX regularity assumption is the limiting mechanism. A sharper version is to lower the PUX partition radius below the heuristic range and watch whether the convergence order collapses.

Watch

Extended reading notes

Core claim

The central claim is that a complete, fast integral equation solver for the incompressible Navier-Stokes equations in two-dimensional bounded domains can be built by combining a partition-of-unity function extension, an FFT-based volume potential, a modified-Stokes double layer potential, kernel-split quadrature, an FMM, and a fast direct solver. The velocity at each substep is written $u = u_P + u_H$, where $u_P$ solves the inhomogeneous modified Stokes equations $(\alpha^2 - \Delta)u_P + \nabla p_P = F$ with $\alpha^2 = \mathrm{Re}/\delta t$, via the periodic stokeslet multiplier $\hat S_{jl}(k) = (\delta_{jl} - \hat k_j \hat k_l)/(\alpha^2 + |k|^2)$ acting on the Fourier coefficients of the extended forcing $F^e$, and $u_H = D[\mu]$ is a double layer potential whose density satisfies $\tfrac12 \mu + D[\mu] + W[\mu] = f - u_P$ on the boundary. The paper contributes the quadrature machinery that makes the layer potentials accurate at arbitrary target points, including points arbitrarily close to the boundary and for large $\alpha$, and an overall complexity showing linear or near-linear scaling per time step. The reported experiments—convergence tests on a starfish domain, flow past obstacles, viscous spin-down, and vortex shedding—support 10th-order spatial and up to 4th-order temporal accuracy, with a stability condition $\delta t\,\mathrm{Re} \lesssim O(1)$ and successful simulations through $\mathrm{Re}=200$, leading the authors to state that the method is suited to low and moderate Reynolds numbers.

Load-bearing premise

The load-bearing premise is that the partition-of-unity extension really produces a compactly supported extension of the forcing with enough smoothness across the whole bounding box; the paper picks the extension parameters by heuristics and observes convergence rather than proving or adaptively controlling that regularity, so on a different geometry or forcing the claimed high-order spatial accuracy could be lost.

Editorial extensions

If this is right

  • On smooth domains, the solver delivers 10th-order spatial accuracy in both the velocity and its gradient, so a fixed uniform grid can resolve smooth flows to fine accuracy; the experiments show the gradient error reaching about $10^{-9}$.
  • The boundary floats over a uniform grid, so complex geometries are handled without body-conforming meshes, and the no-slip condition is imposed exactly through the boundary integral correction.
  • Because the boundary matrix is fixed from step to step, a precomputed compressed inverse makes each new time step cost $O(N_B \log N_B + N_{\partial \Omega})$ after the one-time setup.
  • The stability restriction $\delta t \,\mathrm{Re} \lesssim O(1)$ matches explicit treatment of advection, and it defines the practical regime as low-to-moderate Reynolds numbers, up to the hundreds or low thousands.
  • The same spatial machinery supports at least 4th-order temporal accuracy through semi-implicit spectral deferred correction, so the method is not tied to first-order IMEX stepping.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the particular-solution half of the pipeline—PUX extension plus FFT volume potential—is independent of the boundary-integral machinery, so it could be reused in other forced elliptic or parabolic solvers on complex domains, such as forced heat or unsteady Stokes.
  • Editorial inference: the near-boundary error floors in the stationary tests suggest that extension regularity, not the kernel-split quadrature, sets the accuracy ceiling for smooth problems; an adaptive criterion on the extension's smoothness or support is a natural upgrade.
  • Editorial inference: because the stability condition forces $\delta t \sim 1/\mathrm{Re}$ while the needed spatial resolution grows with $\alpha$, the method is unlikely to reach very high Reynolds numbers without a different treatment of advection, such as implicit or characteristic-based advection.
  • Editorial inference: the fast direct solver precomputation exploits a fixed geometry and fixed time step, so moving boundaries, deforming drops, or changing step sizes would forfeit the main efficiency gain and likely require a time-dependent unsteady Stokes formulation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper presents a complete integral-equation-based solver for the two-dimensional incompressible Navier-Stokes equations on smooth bounded domains. The solver combines a partition-of-unity extension (PUX) of the forcing function, an FFT-based periodic volume potential, a modified-Stokes double-layer representation with a fast direct solver, kernel-split quadrature for near-boundary evaluation, an FMM for off-boundary evaluation, and SISDC time stepping. The central claims are that the method achieves 10th-order spatial accuracy and up to 4th-order temporal accuracy, with linear or near-linear cost per time step, and that it is well suited to moderate Reynolds numbers. Validation includes 16th-order convergence for the homogeneous stationary problem, 10th-order convergence for an inhomogeneous modified-Stokes problem, expected SISDC convergence orders in a spin-down test, a heuristic stability condition, and qualitatively plausible vortex shedding for Re=25 to 200. The authors provide open-source code.

Significance. If the claims hold, this is a valuable demonstration that state-of-the-art FIEM components can be assembled into a robust Navier-Stokes solver for complex geometries. The paper's genuine strengths are its careful validation against exact solutions, the treatment of nearly singular quadrature with explicit dependence on the parameter alpha, the use of a fast direct solver for the fixed boundary system, and the availability of the implementation. The physical experiments, especially the vortex-shedding transition between Re=25 and Re=50, are credible qualitative demonstrations. The main caveat is that the headline spatial-order claim is established only for a stationary modified-Stokes test with an entire forcing function, not for the full time-dependent Navier-Stokes solver.

major comments (2)
  1. [Section 5.3 and Section 5.4; Eq. (6), Eq. (73)] The 10th-order spatial convergence claim is validated only for the stationary inhomogeneous modified Stokes problem (9) with the entire forcing F(x)=(-1,2)cos(x1+x2) in (129), whose PUX extension is unusually benign. The full Navier-Stokes solver instead feeds F = alpha^2 u - Re (u·grad)u from Eqs. (6) and (73) into the same PUX+FFT particular-solution pipeline, and the advective term of a moderate-Re flow develops boundary-layer structure near the boundary. The only full-NS convergence experiment, Section 5.4, refines Delta t at a fixed 500x500 spatial grid and therefore cannot certify spatial order for the coupled method. Since the abstract and conclusion state that the solver is 10th-order accurate in the spatial grid spacing, this support gap is load-bearing. A full-system h-refinement study at fixed Delta t, or an explicit restriction of the 10th-order claim to the stationary modified Stokes step, is needed.
  2. [Section 3.1.1 and Section 5.3] The PUX extension parameters are chosen via heuristics from [23] (epsilon=2, partition radius R spanning 20 to 50 grid points), with no adaptive control of extension regularity or a posteriori error estimate. The observed 10th-order rate is attributed to PUX in Section 5.3, but the test forcing is entire; for forcing arising from the nonlinear term at moderate Re, the required C^q regularity of the extension is not guaranteed. The plot in Figure 7 already indicates that particular-solution errors concentrate near the boundary, and the alpha-dependence study in Figure 6 shows accuracy degrading for large alpha. The authors should either add a test with a non-entire or boundary-layer-like forcing to demonstrate that the PUX-based particular solution maintains high order for realistic NS forcing, or qualify the accuracy claim accordingly.
minor comments (4)
  1. [Appendix A, Eq. (A.4)] The compatibility condition is written as an integral over Omega, but it should be an integral over the boundary dOmega, and the equality to zero is missing. The intended statement is presumably int_{dOmega} g · n dS = 0.
  2. [Section 5.3, paragraph after Figure 6] The sentence 'due to the scheme outlined in section 4.5.4. 4.5.5 Specifically' contains a formatting error; the '4.5.5' should not be a standalone fragment.
  3. [Figure 3 caption] The caption says 'The splits T^S_1 - T^L_3', which is ambiguous; this should be clarified, for example as the pairs T^S_i, T^L_i, i=1,2,3.
  4. [Section 5.7] The vortex-shedding results are reported only qualitatively. Adding a quantitative comparison, such as the Strouhal number versus Reynolds number or a drag estimate, would strengthen the evidence that the method is well suited to moderate-Re unsteady flows.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all accuracy claims are checked against independent manufactured/exact solutions.

full rationale

All central numerical claims are checked against closed-form external solutions: the homogeneous solver against sums of stokeslets (Eq. 128), the stationary inhomogeneous solver against the Fourier-mode solution (Eq. 130), and the time-stepping against the Bessel-series spin-down solution (Eq. 132). In no case is a parameter fitted to the measured error or a target quantity defined in terms of the output. PUX parameters (epsilon=2, R range, alpha*h <= 4.5) are heuristics of the component methods and are fixed before the convergence tests; the reported 10th-order rate is read off the measured slope against an exact solution, not imposed by construction. The cited component works ([5], [8], [23], [47]) are used as building blocks, but the paper's own exact-solution tests provide independent support, so none of these self-citations becomes load-bearing in a circular sense. The apparent gap that full Navier-Stokes spatial order is not separately grid-refined is a validation-support gap, not a circularity.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The central claim rests on established layer potential theory and on several numerically calibrated tunables (PUX parameters, quadrature thresholds, panel subdivision criterion). No new physical entities are introduced.

free parameters (7)
  • PUX RBF shape parameter epsilon = 2
    Set to 2 throughout (Section 3.1.1); controls the RBF interpolation matrix condition number and extrapolation accuracy, chosen by hand.
  • PUX partition radius R = 0.15 to 0.5 in experiments (spans 20-50 grid points)
    Determined by heuristics from [23] (Section 3.1.1); the size of the circular patches controls the regularity and accuracy of the function extension.
  • Kernel-split quadrature Bernstein radius thresholds = rho >= 3.5; sqrt(3.5) <= rho < 3.5; rho < sqrt(3.5)
    Empirical cutoffs deciding when direct 16 point Gauss-Legendre, 32 point upsampled, or kernel-split quadrature is applied (Section 4.5.3).
  • Panel subdivision criterion alpha*h <= 4.5 = 4.5
    Empirically determined maximum alpha times panel length for which kernel-split quadrature remains stable (Section 4.5.4).
  • FMM expansion order p_FMM = 40
    Chosen to achieve 12 digit accuracy (Section 3.2.1).
  • Gauss-Legendre nodes per panel n = 16
    Composite Gauss-Legendre panels with n=16 nodes used throughout (Section 3.2).
  • Power series switch thresholds for stresslet functions = z <= 1.5 for Ti, z <= 2 for T'i; 11 or 13 series terms
    Empirically determined in Appendix C to avoid cancellation errors for small arguments.
assumptions (4)
  • standard math The modified Stokes double layer potential with the nullspace correction W defines an invertible second kind integral equation (Lemma 4).
    Proved in the cited literature [14] and stated in Appendix A; the solver relies on it for unique solvability.
  • standard math The jump conditions for the modified Stokes layer potentials (Lemma 2) hold for the smooth boundaries used.
    Standard results from [40,52,14] invoked in Section 2.2 and Appendix A.
  • domain assumption PUX produces a compactly supported extension Fe with regularity high enough for the Fourier volume potential to converge at the claimed order.
    Section 3.1.1 relies on heuristics from [23] and observed 10th order convergence; no rigorous error bound is provided.
  • domain assumption The 1D advection-diffusion model (136) captures the stability behavior of the full 2D nonlinear solver.
    Section 5.5 derives delta_t Re <= 2 from the model and uses it to explain the observed C close to 1 condition; this is a model, not a proof for the full solver.

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Pith. "Pith review of A Fast Integral Equation Method for the Two-Dimensional Navier-Stokes Equations." pith.science (2026). https://pith.science/paper/6SC2ZZOR

@misc{pith2026190807392,
  author       = {Pith},
  title        = {Pith review of: A Fast Integral Equation Method for the Two-Dimensional Navier-Stokes Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SC2ZZOR}},
  note         = {Machine review of arXiv:1908.07392}
}
read the original abstract

The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently developed methods for singular quadrature and the solution of PDEs on complex domains, in combination with several more well-established numerical methods. We apply this solver to flow problems on a number of geometries, both simple and challenging, studying its convergence properties and computational performance. This serves as a demonstration that it is now relatively straightforward to develop a robust, efficient, and flexible Navier-Stokes solver, using integral equation methods.

Figures

Figures reproduced from arXiv: 1908.07392 by the authors.

Figure 1
Figure 1. An illustration of the embedded boundary approach. The boundary of the domain is “floating” on top of a uniform [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. (a) We plot a collection of white dots which are well-separated from the black dots. (b) An illustration of the property [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Left: The stresslet functions T1–T3, as defined in (80)–(82). Right: The splits T S 1 –T L 3 , as defined in (85)–(87). Note that the magnitudes of the stresslet functions decay algebraically as z → ∞, even though the magnitudes of their decompositions grow exponentially. It is clear that any singular behavior in the stresslet must come from the functions Ti . Using a standard result [48, §10.31], we can decompose, … view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Illustration of the scheme used to subdivide the source panel when [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Convergence for the homogeneous solver, with [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Left: Convergence for full problem, α = 10. Right: Varying α when N = 800 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png]
Figure 7
Figure 7. Figure 7: Left: Function extension of the first component of ( [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]
Figure 8
Figure 8. Figure 8: Time convergence for spin down problem, ˜u [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]
Figure 9
Figure 9. Figure 9: Simulations of the flow (134), with varying δt and Re. Cases (b) and (d) satisfy the heuristic stability condition (135), and develop a steady state. Snapshots shown are after 200 time steps. On the other hand, cases (a) and (c) violate the stability condition, and dev…
Figure 10
Figure 10. Figure 10: Streamlines and vorticity field of flow past a collection of starfish at [PITH_FULL_IMAGE:figures/full_fig_p028_10.png]
Figure 11
Figure 11. Figure 11: Flow past cylinder, with streaklines drawn in black and field colored by vorticity. At [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: Unsteady flow past cylinder, with a vortex street in the wake. Field colored by vorticity. [PITH_FULL_IMAGE:figures/full_fig_p029_12.png]

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Reference graph

Works this paper leans on

61 extracted references · 27 canonical work pages · cited by 1 Pith paper

  1. [23]

    Fryklund, E

    F. Fryklund, E. Lehto, and A.-K. Tornberg. Partition of unity extension of functions on complex domains. J. Comput. Phys. , 375:57–79, 2018. doi: 10.1016/j.jcp.2018.08.012

    doi:10.1016/j.jcp.2018.08.012 2018

  2. [1]

    D. J. Acheson. Elementary Fluid Dynamics . Oxford University Press, New York, 1990. ISBN 9780198596790

    work page 1990

  3. [2]

    af Klinteberg and T

    L. af Klinteberg and T. Askham. INSE FIEM 2D, 2019. URL https://github.com/ludvigak/ inse-fiem-2d

    work page 2019

  4. [3]

    af Klinteberg and A.-K

    L. af Klinteberg and A.-K. Tornberg. Error estimation for quadrature by expansion in layer potential evaluation. Adv. Comput. Math. , 43(1):195–234, 2017. doi: 10.1007/s10444-016-9484-x

    doi:10.1007/s10444-016-9484-x 2017

  5. [4]

    af Klinteberg and A.-K

    L. af Klinteberg and A.-K. Tornberg. Adaptive Quadrature by Expansion for Layer Potential Evaluation in Two Dimensions. SIAM J. Sci. Comput. , 40(3):A1225–A1249, 2018. doi: 10.1137/17M1121615

    doi:10.1137/17m1121615 2018

  6. [5]

    af Klinteberg, F

    L. af Klinteberg, F. Fryklund, and A.-K. Tornberg. An adaptive kernel-split quadrature method for parameter-dependent layer potentials. arXiv:1906.07713 [math.NA], 2019

    arXiv 1906

  7. [6]

    U. M. Ascher, S. J. Ruuth, and B. T. R. Wetton. Implicit-Explicit Methods for Time-Dependent Partial Differential Equations. SIAM J. Numer. Anal. , 32(3):797–823, 1995. doi: 10.1137/0732037

    doi:10.1137/0732037 1995

  8. [7]

    T. Askham. Integral-equation methods for inhomogeneous elliptic partial differential equations in complex geometry. PhD thesis, New York University, 2016

    work page 2016

Show all 61 references
  1. [8]

    T. Askham. A Stabilized Separation of Variables Method for the Modified Biharmonic Equation. J. Sci. Comput., 76(3):1674–1697, 2018. doi: 10.1007/s10915-018-0679-9

    2018 doi

  2. [9]

    Askham and A

    T. Askham and A. Cerfon. An adaptive fast multipole accelerated Poisson solver for complex geometries. J. Comput. Phys. , 344:1–22, 2017. doi: 10.1016/j.jcp.2017.04.063

    2017 doi

  3. [10]

    A. H. Barnett. Evaluation of Layer Potentials Close to the Boundary for Laplace and Helmholtz Problems on Analytic Planar Domains. SIAM J. Sci. Comput. , 36(2):A427–A451, 2014. doi: 10.1137/ 120900253

    2014

  4. [11]

    A. H. Barnett, J. F. Magland, and L. af Klinteberg. A parallel non-uniform fast Fourier transform library based on an ”exponential of semicircle” kernel. arXiv:1808.06736 [math.NA], 2018

    2018 arXiv

  5. [12]

    Berrut and L

    J.-P. Berrut and L. N. Trefethen. Barycentric Lagrange Interpolation. SIAM Rev., 46(3):501–517, 2004. doi: 10.1137/S0036144502417715

    2004 doi

  6. [13]

    Bezanson, A

    J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah. Julia: A Fresh Approach to Numerical Computing. SIAM Rev., 59(1):65–98, 2017. doi: 10.1137/141000671

    2017 doi

  7. [14]

    Biros, L

    G. Biros, L. Ying, and D. Zorin. The embedded boundary integral method for the unsteady incompress- ible Navier–Stokes equations. Technical Report TR2003-838, Courant Institute, New York University, 2002

    2002

  8. [15]

    Bj¨ orck and V

    A. Bj¨ orck and V. Pereyra. Solution of Vandermonde Systems of Equations. Math. Comput. , 24(112): 893, 1970. doi: 10.2307/2004623

    1970 doi

  9. [16]

    H. C. Brinkman. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. , A1:27, 1947. doi: 10.1007/BF02120313

    1947 doi

  10. [17]

    Cheng, Z

    H. Cheng, Z. Gimbutas, P. G. Martinsson, and V. Rokhlin. On the Compression of Low Rank Matrices. SIAM J. Sci. Comput. , 26(4):1389–1404, 2005. doi: 10.1137/030602678

    2005 doi

  11. [18]

    Cheng, J

    H. Cheng, J. Huang, and T. J. Leiterman. An adaptive fast solver for the modified Helmholtz equation in two dimensions. J. Comput. Phys. , 211(2):616–637, 2006. doi: 10.1016/j.jcp.2005.06.006. 37

    2006 doi

  12. [19]

    A. Dutt, L. Greengard, and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. Bit Numer. Math. , 40(2):241–266, 2000. doi: 10.1023/A:1022338906936

    doi

  13. [20]

    Fornberg, E

    B. Fornberg, E. Larsson, and N. Flyer. Stable Computations with Gaussian Radial Basis Functions. SIAM J. Sci. Comput. , 33(2):869–892, 2011. doi: 10.1137/09076756X

    2011 doi

  14. [21]

    W. L. Frank. Computing Eigenvalues of Complex Matrices by Determinant Evaluation and by Methods of Danilewski and Wielandt. J. Soc. Ind. Appl. Math. , 6(4):378–392, 1958. doi: 10.1137/0106026

    1958 doi

  15. [22]

    Fryklund and E

    F. Fryklund and E. Lehto. PUXdemo, 2018. URL https://github.com/fryklund/PUXdemo

    2018

  16. [24]

    Fryklund, M

    F. Fryklund, M. C. A. Kropinski, and A.-K. Tornberg. An integral equation based numerical method for the forced heat equation on complex domains. arXiv:1907.08537 [math.NA], 2019

    1907 arXiv

  17. [25]

    Gillman, P

    A. Gillman, P. M. Young, and P.-G. Martinsson. A direct solver with O(N) complexity for inte- gral equations on one-dimensional domains. Front. Math. China , 7(2):217–247, 2012. doi: 10.1007/ s11464-012-0188-3

    2012

  18. [26]

    Gimbutas and L

    Z. Gimbutas and L. Greengard. FMMLIB2D, 2017. URL https://github.com/zgimbutas/fmmlib2d

    2017

  19. [27]

    Greengard and S

    L. Greengard and S. Jiang. A new mixed potential representation for the equations of unsteady, incom- pressible flow. arXiv:1809.08442 [math.NA], 2018

    2018 arXiv

  20. [28]

    Greengard and M

    L. Greengard and M. C. Kropinski. An Integral Equation Approach to the Incompressible Navier– Stokes Equations in Two Dimensions. SIAM J. Sci. Comput. , 20(1):318–336, 1998. doi: 10.1137/ S1064827597317648

    1998

  21. [29]

    Greengard and V

    L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. J. Comput. Phys. , 73(2): 325–348, 1987. doi: 10.1016/0021-9991(87)90140-9

    1987 doi

  22. [30]

    Greengard and V

    L. Greengard and V. Rokhlin. A new version of the Fast Multipole Method for the Laplace equation in three dimensions. Acta Numer., 6:229, 1997. doi: 10.1017/S0962492900002725

    1997 doi

  23. [31]

    Greengard, S

    L. Greengard, S. Jiang, and J. Wang. On the accurate evaluation of unsteady Stokes layer potentials in moving two-dimensional geometries. 10010:1–17, 2018

    2018

  24. [32]

    S. Hao, A. H. Barnett, P. G. Martinsson, and P. Young. High-order accurate methods for Nystr¨ om discretization of integral equations on smooth curves in the plane. Adv. Comput. Math., 40(1):245–272,

  25. [33]

    J. Helsing. Integral equation methods for elliptic problems with boundary conditions of mixed type. J. Comput. Phys., 228(23):8892–8907, 2009. doi: 10.1016/j.jcp.2009.09.004

    2009 doi

  26. [34]

    Helsing and A

    J. Helsing and A. Holst. Variants of an explicit kernel-split panel-based Nystr¨ om discretization scheme for Helmholtz boundary value problems. Adv. Comput. Math. , 41(3):691–708, 2015. doi: 10.1007/ s10444-014-9383-y

    2015

  27. [35]

    Helsing and S

    J. Helsing and S. Jiang. On Integral Equation Methods for the First Dirichlet Problem of the Biharmonic and Modified Biharmonic Equations in NonSmooth Domains. SIAM J. Sci. Comput. , 40(4):A2609– A2630, 2018. doi: 10.1137/17M1162238

    2018 doi

  28. [36]

    Helsing and R

    J. Helsing and R. Ojala. On the evaluation of layer potentials close to their sources. J. Comput. Phys. , 227(5):2899–2921, 2008. doi: 10.1016/j.jcp.2007.11.024. 38

    2008 doi

  29. [37]

    Helsing and R

    J. Helsing and R. Ojala. Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning. J. Comput. Phys. , 227(20):8820–8840, 2008. doi: 10.1016/j.jcp.2008.06.022

    2008 doi

  30. [38]

    K. Ho, S. Olver, T. Kelman, and R. M. Slevinsky. LowRankApprox.jl, 2018. URL https://zenodo. org/record/1481098

    2018

  31. [39]

    Jiang, M

    S. Jiang, M. C. A. Kropinski, and B. D. Quaife. Second kind integral equation formulation for the modified biharmonic equation and its applications. J. Comput. Phys. , 249:113–126, 2013. doi: 10.1016/ j.jcp.2013.04.034

    2013

  32. [40]

    Kim and S

    S. Kim and S. J. Karrila. Microhydrodynamics : principles and selected applications . Butterworth- Heinemann series in chemical engineering. Butterworth-Heinemann, Boston, 1991. ISBN 0750691735

    1991

  33. [41]

    Kl¨ ockner, A

    A. Kl¨ ockner, A. Barnett, L. Greengard, and M. O’Neil. Quadrature by expansion: A new method for the evaluation of layer potentials. J. Comput. Phys. , 2013. doi: 10.1016/j.jcp.2013.06.027

    2013 doi

  34. [42]

    M. C. A. Kropinski and B. D. Quaife. Fast integral equation methods for Rothe’s method applied to the isotropic heat equation. Comput. Math. with Appl. , 61(9):2436–2446, 2011. doi: 10.1016/j.camwa. 2011.02.024

    2011 doi

  35. [43]

    E. Larsson. RBF-QR, 2015. URL http://www.it.uu.se/research/scientific_computing/ software/rbf_qr

    2015

  36. [44]

    Liberty, F

    E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin, and M. Tygert. Randomized algorithms for the low-rank approximation of matrices. Proc. Natl. Acad. Sci. , 104(51):20167–20172, 2007. doi: 10.1073/ pnas.0709640104

    2007

  37. [45]

    G. R. Marple, A. Barnett, A. Gillman, and S. Veerapaneni. A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape. SIAM J. Sci. Comput., 38(5):B740–B772, 2016. doi: 10.1137/15M1043066

    2016 doi

  38. [46]

    Martinsson and V

    P. Martinsson and V. Rokhlin. A fast direct solver for boundary integral equations in two dimensions. J. Comput. Phys. , 205(1):1–23, 2005. doi: 10.1016/j.jcp.2004.10.033

    2005 doi

  39. [47]

    M. L. Minion. Semi-implicit spectral deferred correction methods for ordinary differential equations. Commun. Math. Sci. , 1(3):471–500, 2003. doi: 10.4310/CMS.2003.v1.n3.a6

    2003 doi

  40. [48]

    Digital Library of Mathematical Functions

    NIST. Digital Library of Mathematical Functions. Release 1.0.16, 2017. URL http://dlmf.nist.gov/

    2017

  41. [49]

    Ojala and A.-K

    R. Ojala and A.-K. Tornberg. An accurate integral equation method for simulating multi-phase Stokes flow. J. Comput. Phys. , 298:145–160, 2015. doi: 10.1016/j.jcp.2015.06.002

    2015 doi

  42. [50]

    C. S. Peskin. The immersed boundary method. Acta Numer., 11(2002):479–517, 2002. doi: 10.1017/ S0962492902000077

    2002

  43. [51]

    Pozrikidis

    C. Pozrikidis. A singularity method for unsteady linearized flow. Phys. Fluids A Fluid Dyn. , 1(9): 1508–1520, 1989. doi: 10.1063/1.857329

    1989 doi

  44. [52]

    Pozrikidis

    C. Pozrikidis. Boundary Integral and Singularity Methods for Linearized Viscous Flow . Cambridge University Press, Cambridge, 1992. ISBN 9780511624124. doi: 10.1017/CBO9780511624124. URL http://ebooks.cambridge.org/ref/id/CBO9780511624124

    1992 doi

  45. [53]

    Rachh and K

    M. Rachh and K. Serkh. On the solution of Stokes equation on regions with corners. arXiv:1711.04072 [math-ph], 2017

    2017 arXiv

  46. [54]

    Saad and M

    Y. Saad and M. H. Schultz. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsym- metric Linear Systems. SIAM J. Sci. Stat. Comput. , 7(3):856–869, 1986. doi: 10.1137/0907058. 39

    1986 doi

  47. [55]

    Serkh and V

    K. Serkh and V. Rokhlin. On the solution of elliptic partial differential equations on regions with corners. J. Comput. Phys. , 305:150–171, 2016. doi: 10.1016/j.jcp.2015.10.024

    2016 doi

  48. [56]

    D. B. Stein, R. D. Guy, and B. Thomases. Immersed Boundary Smooth Extension (IBSE): A high-order method for solving incompressible flows in arbitrary smooth domains. J. Comput. Phys. , 335:155–178,

  49. [57]

    Tafuni, J

    A. Tafuni, J. Dom´ ınguez, R. Vacondio, and A. Crespo. A versatile algorithm for the treatment of open boundary conditions in Smoothed particle hydrodynamics GPU models. Comput. Methods Appl. Mech. Eng., 342:604–624, 2018. doi: 10.1016/j.cma.2018.08.004

    2018 doi

  50. [58]

    Wala and A

    M. Wala and A. Kl¨ ockner. A fast algorithm for Quadrature by Expansion in three dimensions. J. Comput. Phys., 388:655–689, 2019. doi: 10.1016/j.jcp.2019.03.024

    2019 doi

  51. [59]

    Wang and L

    J. Wang and L. Greengard. Hybrid asymptotic/numerical methods for the evaluation of layer heat po- tentials in two dimensions. Adv. Comput. Math., 45(2):847–867, 2019. doi: 10.1007/s10444-018-9641-5. 40

    2019 doi

  52. [2014]

    doi: 10.1007/s10444-013-9306-3

    doi

  53. [2017]

    doi: 10.1016/j.jcp.2017.01.010

    2017 doi

Pith tools

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