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arxiv: 2309.01068 · v2 · submitted 2023-09-03 · 🧮 math.NA · cs.NA· physics.comp-ph

A Cartesian grid-based boundary integral method for moving interface problems

Han Zhou , Shuwang Li , Wenjun Ying This is my paper

Pith reviewed 2026-05-24 06:47 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords boundary integral methodCartesian gridmoving interfaceHele-Shaw flowStefan problemθ-L variablesviscous fingeringdendritic solidification
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The pith

A Cartesian grid-based boundary integral method solves Hele-Shaw flow and Stefan problems by reformulating PDEs and evolving interfaces with θ-L variables.

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

The paper develops a numerical technique that converts elliptic and parabolic equations governing moving interfaces into boundary integral equations. These are solved using a matrix-free GMRES solver where boundary integrals are computed by solving auxiliary problems on a Cartesian grid with fast methods like FFT. The interface is tracked using θ-L variables rather than Cartesian coordinates to avoid mesh quality issues and to allow larger, stable time steps by mitigating curvature-induced stiffness. This approach is demonstrated on viscous fingering and dendritic solidification, suggesting it can handle complex geometries efficiently.

Core claim

The central claim is that elliptic and parabolic PDEs for moving interface problems can be reformulated as boundary integral equations, solved with GMRES, with integrals evaluated via finite difference methods on Cartesian grids using fast solvers, and the interface evolved with θ-L variables to preserve mesh quality and remove stiffness from the curvature term.

What carries the argument

The θ-L representation of the interface, which simplifies mesh quality preservation and enables stable time-stepping schemes that remove curvature stiffness.

If this is right

  • The method can simulate complex viscous fingering patterns in Hele-Shaw flows.
  • It handles dendritic solidification problems in the Stefan model.
  • Fast PDE solvers such as FFT and geometric multigrid can be used for boundary integral evaluation.
  • Matrix-free GMRES solves the boundary integral equations efficiently.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the θ-L approach generalizes beyond these two problems, it may apply to other curvature-driven interface evolutions.
  • The Cartesian grid evaluation could reduce the need for body-fitted meshes in related fluid and heat transfer simulations.

Load-bearing premise

The θ-L variables enable efficient and stable time-stepping schemes that remove the stiffness arising from the curvature term while preserving mesh quality during interface evolution.

What would settle it

A direct comparison showing whether simulations with θ-L variables remain stable and mesh-quality-preserving for longer times or more complex shapes than with x-y variables.

Figures

Figures reproduced from arXiv: 2309.01068 by Han Zhou, Shuwang Li, Wenjun Ying.

Figure 1
Figure 1. Figure 1: A Schematic of a moving interface problem. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) shows the spatial accuracy of the method with di [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Morphologies of the interface from t = 0 to t = 1.2 with a time increment of 0.2. 0 0.2 0.4 0.6 0.8 1 t 0 2 4 6 8 10 value Area Length (a) 0 0.5 1 1.5 2 t 0 2 4 6 8 iteration number (b) [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) shows the time evolutions of the enclosed area and the length of the curve. (b) shows the iteration number [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Detailed morphologies of the curve with di [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical results of long-time computation of the Hele-Shaw flow: (a) morphology histories of the curve; (b) [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Grid refinement analysis of the Stefan problem with di [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Numerical results of the stability test for the Stefan problem with di [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Numerical results of the dendritic growth problem and comparison with the solvability theory: (a) morphology [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Interface morphologies and the temperature field of the dendritic growth problem with four-fold anisotropy. [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Interface morphologies and the temperature field of the dendritic growth problem with six-fold anisotropy. [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Details of dendritic growth histories, temperature fields, and flow fields with di [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution histories of x-components of the left and right tips with different flow velocities. 6.2.6. Dendritic growth with buoyancy-driven flow In the final example, we consider the dendritic growth problem with buoyancy-driven flow. The anisotropic surface tension is chosen as a rotated one, εC(α) = 0.002(8/3 sin4 (2(α − π/4))). The no-slip boundary condition is applied for the fluid equation on the fou… view at source ↗
Figure 14
Figure 14. Figure 14: Details of dendritic growth histories, temperature fields, and flow fields with di [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
read the original abstract

This paper proposes a Cartesian grid-based boundary integral method for efficiently and stably solving two representative moving interface problems, the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial differential equations (PDEs) are reformulated into boundary integral equations and are then solved with the matrix-free generalized minimal residual (GMRES) method. The evaluation of boundary integrals is performed by solving equivalent and simple interface problems with finite difference methods, allowing the use of fast PDE solvers, such as fast Fourier transform (FFT) and geometric multigrid methods. The interface curve is evolved utilizing the $\theta-L$ variables instead of the more commonly used $x-y$ variables. This choice simplifies the preservation of mesh quality during the interface evolution. In addition, the $\theta-L$ approach enables the design of efficient and stable time-stepping schemes to remove the stiffness that arises from the curvature term. Ample numerical examples, including simulations of complex viscous fingering and dendritic solidification problems, are presented to showcase the capability of the proposed method to handle challenging moving interface problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes a Cartesian grid-based boundary integral method for two moving interface problems: Hele-Shaw flow and the Stefan problem. Elliptic/parabolic PDEs are recast as boundary integral equations solved via matrix-free GMRES; integrals are evaluated by solving auxiliary Cartesian-grid interface problems with finite differences, FFT, and geometric multigrid. The interface is parametrized and evolved in θ-L variables (rather than x-y) to maintain mesh quality and to permit stable explicit or semi-implicit time marching that removes curvature-induced stiffness. Numerical examples of viscous fingering and dendritic solidification are presented to demonstrate the approach.

Significance. If the numerical results and stability claims hold, the work supplies a practical, matrix-free framework that exploits fast Cartesian solvers while addressing two persistent difficulties in moving-boundary computations: expensive integral evaluations and curvature stiffness. The θ-L formulation is a standard device in the literature, but its combination here with grid-based integral evaluation could offer a useful balance of efficiency and robustness for complex geometries.

minor comments (2)
  1. The description of how the θ-L parametrization is discretized and how the curvature term is treated in the time-stepping scheme would benefit from an explicit algorithmic outline or pseudocode (e.g., in §3 or §4).
  2. Several figures showing interface evolution would be clearer if the mesh-quality metric (e.g., point spacing or curvature distribution) were plotted alongside the interface snapshots.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment, including the recommendation for minor revision. The provided summary accurately reflects the core elements of our Cartesian grid-based boundary integral approach, the use of matrix-free GMRES, fast Cartesian solvers, and the θ-L parametrization for stable interface evolution.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a numerical method that reformulates moving-interface PDEs as boundary integral equations solved via GMRES with Cartesian-grid integral evaluation (FFT/multigrid) and evolves the interface in θ-L coordinates to mitigate stiffness and preserve mesh quality. All components are described as combinations of established techniques (boundary integrals, fast PDE solvers, θ-L parametrization) without any claim that a derived quantity equals a fitted input by construction, without load-bearing self-citations that reduce the central result to prior author work, and without renaming or smuggling ansatzes. The derivation chain consists of standard algorithmic steps whose correctness is externally verifiable by implementation and benchmark tests rather than by internal redefinition. No step satisfies the criteria for circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Based solely on the abstract; limited visibility into implementation details. No free parameters or invented entities are explicitly described.

axioms (3)
  • domain assumption Elliptic and parabolic PDEs can be reformulated into boundary integral equations
    Invoked as the starting point for the numerical approach.
  • domain assumption Boundary integrals can be accurately evaluated by solving equivalent interface problems with finite difference methods on Cartesian grids
    Central to the efficiency claim and use of FFT/multigrid solvers.
  • domain assumption θ-L variables remove stiffness from curvature and preserve mesh quality during evolution
    Load-bearing premise for the time-stepping stability.

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Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    Cao, B.-L

    H.-D. Cao, B.-L. Chen, X.-P. Zhu, Recent developments on the Hamilton’s Ricci Flow, Surveys in Di fferential Geometry 12 (1) (2007) 47–112. doi:10.4310/sdg.2007.v12.n1.a3

    work page doi:10.4310/sdg.2007.v12.n1.a3 2007

  2. [2]

    Chen, Generation and propagation of interfaces for reaction-di ffusion equations, Journal of Di fferential Equa- tions 96 (1) (1992) 116–141

    X. Chen, Generation and propagation of interfaces for reaction-di ffusion equations, Journal of Di fferential Equa- tions 96 (1) (1992) 116–141. doi:10.1016/0022-0396(92)90146-E . URL https://linkinghub.elsevier.com/retrieve/pii/002203969290146E

    work page doi:10.1016/0022-0396(92)90146-e 1992

  3. [3]

    BELLETTINI, M

    G. BELLETTINI, M. PAOLINI, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Mathematical Journal 25 (3) (1996) 537–566. doi:10.14492/hokmj/1351516749. URL https://projecteuclid.org/journals/hokkaido-mathematical-journal/volume-25/ issue-3/Anisotropic-motion-by-mean-curvature-in-the-context-of-Finsler/10.14492/ hokmj/135151...

    work page doi:10.14492/hokmj/1351516749 1996

  4. [4]

    J. M. Hyman, Numerical methods for tracking interfaces, Physica D: Nonlinear Phenomena 12 (1) (1984) 396–407. doi:https://doi.org/10.1016/0167-2789(84)90544-X . URL https://www.sciencedirect.com/science/article/pii/016727898490544X

    work page doi:10.1016/0167-2789(84)90544-x 1984

  5. [5]

    Glimm, D

    J. Glimm, D. Marchesin, O. McBryan, Statistical fluid dynamics: Unstable fingers, Communications in Mathemat- ical Physics 74 (1) (1980) 1–13. doi:10.1007/BF01197574. URL http://link.springer.com/10.1007/BF01197574

    work page doi:10.1007/bf01197574 1980

  6. [6]

    Glimm, D

    J. Glimm, D. Marchesin, O. McBryan, Subgrid resolution of fluid discontinuities, II, Journal of Computational Physics 37 (3) (1980) 336–354. doi:10.1016/0021-9991(80)90041-8 . URL https://linkinghub.elsevier.com/retrieve/pii/0021999180900418

    work page doi:10.1016/0021-9991(80)90041-8 1980

  7. [7]

    W. W. Mullins, Two-dimensional motion of idealized grain boundaries, Journal of Applied Physics 27 (8) (1956) 900–904. doi:10.1063/1.1722511. URL http://aip.scitation.org/doi/10.1063/1.1722511 30

    work page doi:10.1063/1.1722511 1956

  8. [8]

    J. A. Sethian, J. Straint, Crystal growth and dendritic solidification, Journal of Computational Physics 98 (2) (1992) 231–253. doi:10.1016/0021-9991(92)90140-T

    work page doi:10.1016/0021-9991(92)90140-t 1992

  9. [9]

    D. I. Meiron, Selection of steady states in the two-dimensional symmetric model of dendritic growth, Tech. Rep. 4 (1986). doi:10.1103/PhysRevA.33.2704

    work page doi:10.1103/physreva.33.2704 1986

  10. [10]

    W. W. Mullins, R. F. Sekerka, Morphological Stability of a Particle Growing by Diffusion or Heat Flow, Journal of Applied Physics 34 (2) (1963) 323–329. doi:10.1063/1.1702607. URL https://pubs.aip.org/aip/jap/article/34/2/323-329/163811

    work page doi:10.1063/1.1702607 1963

  11. [11]

    Y . Li, D. Jeong, J.-i. Choi, S. Lee, J. Kim, Fast local image inpainting based on the Allen–Cahn model, Digital Signal Processing 37 (1) (2015) 65–74. doi:10.1016/j.dsp.2014.11.006. URL http://dx.doi.org/10.1016/j.dsp.2014.11.006https://linkinghub.elsevier.com/ retrieve/pii/S1051200414003418

    work page doi:10.1016/j.dsp.2014.11.006 2015

  12. [12]

    Bene ˇs, V

    M. Bene ˇs, V . Chalupeck´y, K. Mikula, Geometrical image segmentation by the Allen-Cahn equation, Applied Nu- merical Mathematics 51 (2-3) (2004) 187–205. doi:10.1016/j.apnum.2004.05.001

    work page doi:10.1016/j.apnum.2004.05.001 2004

  13. [13]

    DEGREGORIA, L

    A. DEGREGORIA, L. SCHW ARTZ, A boundary-integral method for two-phase displacement in hele-shaw cells, Dynamics of Curved Fronts (1988) 201–218doi:10.1016/B978-0-08-092523-3.50022-8

    work page doi:10.1016/b978-0-08-092523-3.50022-8 1988

  14. [14]

    T. Y . Hou, J. S. Lowengrub, M. J. Shelley, Removing the sti ffness from interfacial flows with surface tension, Journal of Computational Physics 114 (2) (1994) 312–338. doi:10.1006/jcph.1994.1170

    work page doi:10.1006/jcph.1994.1170 1994

  15. [15]

    S. Li, J. S. Lowengrub, P. H. Leo, A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell, Journal of Computational Physics 225 (1) (2007) 554–567. doi:10.1016/j.jcp. 2006.12.023

    work page doi:10.1016/j.jcp 2007

  16. [16]

    Games and Economic Behavior 12(1), pp

    D. Juric, G. Tryggvason, A front-tracking method for dendritic solidification, Journal of Computational Physics 123 (1) (1996) 127–148. doi:10.1006/jcph.1996.0011

    work page doi:10.1006/jcph.1996.0011 1996

  17. [17]

    Strain, A boundary integral approach to unstable solidification, Journal of Computational Physics 85 (2) (1989) 342–389

    J. Strain, A boundary integral approach to unstable solidification, Journal of Computational Physics 85 (2) (1989) 342–389. doi:10.1016/0021-9991(89)90155-1 . URL https://www.sciencedirect.com/science/article/pii/0021999189901551https: //linkinghub.elsevier.com/retrieve/pii/0021999189901551

    work page doi:10.1016/0021-9991(89)90155-1 1989

  18. [18]

    P. Zhao, J. C. Heinrich, D. R. Poirier, Fixed mesh front-tracking methodology for finite element simulations, In- ternational Journal for Numerical Methods in Engineering 61 (6) (2004) 928–948. doi:https://doi.org/10. 1002/nme.1098. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.1098

    work page doi:10.1002/nme.1098 2004

  19. [19]

    Schmidt, Computation of Three Dimensional Dendrites with Finite Elements, Journal of Computational Physics 125 (2) (1996) 293–312

    A. Schmidt, Computation of Three Dimensional Dendrites with Finite Elements, Journal of Computational Physics 125 (2) (1996) 293–312. doi:10.1006/jcph.1996.0095. URL https://linkinghub.elsevier.com/retrieve/pii/S0021999196900959

    work page doi:10.1006/jcph.1996.0095 1996

  20. [20]

    S. Chen, B. Merriman, S. Osher, P. Smereka, A simple level set method for solving stefan problems, Tech. Rep. 1 (1997). doi:10.1006/jcph.1997.5721

    work page doi:10.1006/jcph.1997.5721 1997

  21. [21]

    Y .-T. Kim, N. Goldenfeld, J. Dantzig, Computation of dendritic microstructures using a level set method, Physical Review E 62 (2) (2000) 2471–2474. doi:10.1103/PhysRevE.62.2471. URL https://link.aps.org/doi/10.1103/PhysRevE.62.2471

    work page doi:10.1103/physreve.62.2471 2000

  22. [22]

    Gibou, R

    F. Gibou, R. Fedkiw, R. Caflisch, S. Osher, A Level Set Approach for the Numerical Simulation of Dendritic Growth, Tech. Rep. 1-3 (2003). doi:10.1023/A:1025399807998

    work page doi:10.1023/a:1025399807998 2003

  23. [23]

    Boledi, B

    L. Boledi, B. Terschanski, S. Elgeti, J. Kowalski, A level-set based space-time finite element approach to the modelling of solidification and melting processes, Journal of Computational Physics 457 (2022) 111047. arXiv: 2105.09286, doi:10.1016/j.jcp.2022.111047. URL https://doi.org/10.1016/j.jcp.2022.111047

    work page doi:10.1016/j.jcp.2022.111047 2022

  24. [24]

    Limare, S

    A. Limare, S. Popinet, C. Josserand, Z. Xue, A. Ghigo, A hybrid level-set / embedded boundary method applied to solidification-melt problems, Journal of Computational Physics 474 (2023) 111829. arXiv:2202.08300, doi:10.1016/j.jcp.2022.111829. URL http://arxiv.org/abs/2202.08300https://linkinghub.elsevier.com/retrieve/pii/ S0021999122008920

    work page doi:10.1016/j.jcp.2022.111829 2023

  25. [25]

    Scardovelli, S

    R. Scardovelli, S. Zaleski, DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW, Annual Review of Fluid Mechanics 31 (1) (1999) 567–603. doi:10.1146/annurev.fluid.31.1.567. URL https://www.annualreviews.org/doi/10.1146/annurev.fluid.31.1.567

    work page doi:10.1146/annurev.fluid.31.1.567 1999

  26. [26]

    C. Hirt, B. Nichols, V olume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics 39 (1) (1981) 201–225. doi:10.1016/0021-9991(81)90145-5 . URL https://www.sciencedirect.com/science/article/pii/0021999181901455https: //linkinghub.elsevier.com/retrieve/pii/0021999181901455

    work page doi:10.1016/0021-9991(81)90145-5 1981

  27. [27]

    Karma, W

    A. Karma, W. J. Rappel, Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys- ical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 57 (4) (1998) 4323–4349. doi:10.1103/PhysRevE.57.4323. 31

    work page doi:10.1103/physreve.57.4323 1998

  28. [28]

    H. Wang, R. Li, T. Tang, Efficient computation of dendritic growth with r-adaptive finite element methods, Journal of Computational Physics 227 (12) (2008) 5984–6000. doi:10.1016/j.jcp.2008.02.016

    work page doi:10.1016/j.jcp.2008.02.016 2008

  29. [29]

    X. Hu, R. Li, T. Tang, A multi-mesh adaptive finite element approximation to phase field models, Communications in Computational Physics 5 (5) (2009) 1012–1029

    work page 2009

  30. [30]

    J. Zhu, X. Chen, T. Y . Hou, An e fficient boundary integral method for the Mullins-Sekerka problem, Journal of Computational Physics 127 (2) (1996) 246–267. doi:10.1006/jcph.1996.0173

    work page doi:10.1006/jcph.1996.0173 1996

  31. [31]

    Cristini, J

    V . Cristini, J. Lowengrub, Three-dimensional crystal growth—I: linear analysis and self-similar evolution, Journal of Crystal Growth 240 (1-2) (2002) 267–276. doi:10.1016/S0022-0248(02)00831-X . URL https://linkinghub.elsevier.com/retrieve/pii/S002202480200831X

    work page doi:10.1016/s0022-0248(02)00831-x 2002

  32. [32]

    Cristini, J

    V . Cristini, J. Lowengrub, Three-dimensional crystal growth - II: Nonlinear simulation and control of the Mullins- Sekerka instability, Journal of Crystal Growth 266 (4) (2004) 552–567. doi:10.1016/j.jcrysgro.2004.02. 115

    work page doi:10.1016/j.jcrysgro.2004.02 2004

  33. [33]

    C. S. Peskin, Numerical analysis of blood flow in the heart, Journal of Computational Physics 25 (3) (1977) 220–

    work page 1977

  34. [34]

    URL https://linkinghub.elsevier.com/retrieve/pii/0021999177901000

    doi:10.1016/0021-9991(77)90100-0 . URL https://linkinghub.elsevier.com/retrieve/pii/0021999177901000

    work page doi:10.1016/0021-9991(77)90100-0

  35. [35]

    C. S. Peskin, The immersed boundary method, Acta Numerica 11 (2002) 479–517. doi:10.1017/ S0962492902000077. URL https://www.cambridge.org/core/product/identifier/S0962492902000077/type/journal_ article

    work page 2002

  36. [36]

    Taira, T

    K. Taira, T. Colonius, The immersed boundary method: A projection approach, Journal of Computational Physics 225 (2) (2007) 2118–2137. doi:10.1016/j.jcp.2007.03.005

    work page doi:10.1016/j.jcp.2007.03.005 2007

  37. [37]

    R. J. Leveque, Z. Li, Immersed interface method for elliptic equations with discontinuous coe fficients and singular sources, SIAM Journal on Numerical Analysis 31 (4) (1994) 1019–1044. doi:10.1137/0731054. URL https://doi.org/10.1137/0731054

    work page doi:10.1137/0731054 1994

  38. [38]

    Li, Immersed interface methods for moving interface problems, Numerical Algorithms 14 (4) (1997) 269–293

    Z. Li, Immersed interface methods for moving interface problems, Numerical Algorithms 14 (4) (1997) 269–293. doi:10.1023/A:1019173215885. URL https://doi.org/10.1023/A:1019173215885

    work page doi:10.1023/a:1019173215885 1997

  39. [39]

    R. J. LeVeque, Z. Li, Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension, SIAM Journal on Scientific Computing 18 (3) (1997) 709–735. doi:10.1137/S1064827595282532. URL http://epubs.siam.org/doi/10.1137/S1064827595282532

    work page doi:10.1137/s1064827595282532 1997

  40. [40]

    Z. Li, M. C. Lai, The immersed interface method for the navier-stokes equations with singular forces, Journal of Computational Physics 171 (2) (2001) 822–842. doi:10.1006/jcph.2001.6813

    work page doi:10.1006/jcph.2001.6813 2001

  41. [41]

    R. P. Fedkiw, T. Aslam, B. Merriman, S. Osher, A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method), Journal of Computational Physics 152 (2) (1999) 457–492.doi:10.1006/jcph. 1999.6236

    work page doi:10.1006/jcph 1999

  42. [42]

    R. P. Fedkiw, T. Aslam, S. Xu, The Ghost Fluid Method for Deflagration and Detonation Discontinuities, Journal of Computational Physics 154 (2) (1999) 393–427. doi:10.1006/jcph.1999.6320

    work page doi:10.1006/jcph.1999.6320 1999

  43. [43]

    X. D. Liu, R. P. Fedkiw, M. Kang, A Boundary Condition Capturing Method for Poisson’s Equation on Irregular Domains, Journal of Computational Physics 160 (1) (2000) 151–178. doi:10.1006/jcph.2000.6444

    work page doi:10.1006/jcph.2000.6444 2000

  44. [44]

    D. Q. Nguyen, R. P. Fedkiw, M. Kang, A boundary condition capturing method for incompressible flame disconti- nuities, Journal of Computational Physics 172 (1) (2001) 71–98. doi:10.1006/jcph.2001.6812

    work page doi:10.1006/jcph.2001.6812 2001

  45. [45]

    H.-J. Jou, P. Leo, J. Lowengrub, Microstructural evolution in inhomogeneous elastic media, Journal of Computa- tional Physics 131 (1997) 109–148. doi:10.1006/jcph.1996.5581. URL https://linkinghub.elsevier.com/retrieve/pii/S0021999196955813

    work page doi:10.1006/jcph.1996.5581 1997

  46. [47]

    T. Y . Hou, Z. Shi, An efficient semi-implicit immersed boundary method for the Navier-Stokes equations, Journal of Computational Physics 227 (20) (2008) 8968–8991. doi:10.1016/j.jcp.2008.07.005

    work page doi:10.1016/j.jcp.2008.07.005 2008

  47. [48]

    Cristini, J

    V . Cristini, J. Lowengrub, Q. Nie, Nonlinear simulation of tumor growth, Journal of Mathematical Biology 46 (3) (2003) 191–224. doi:10.1007/s00285-002-0174-6 . URL http://link.springer.com/10.1007/s00285-002-0174-6

    work page doi:10.1007/s00285-002-0174-6 2003

  48. [49]

    S. Li, J. S. Lowengrub, P. H. Leo, V . Cristini, Nonlinear theory of self-similar crystal growth and melting, Journal of Crystal Growth 267 (3-4) (2004) 703–713. doi:10.1016/j.jcrysgro.2004.04.002. URL https://linkinghub.elsevier.com/retrieve/pii/S0022024804004014

    work page doi:10.1016/j.jcrysgro.2004.04.002 2004

  49. [50]

    S. Li, J. S. Lowengrub, P. H. Leo, V . Cristini, Nonlinear stability analysis of self-similar crystal growth: con- trol of the Mullins–Sekerka instability, Journal of Crystal Growth 277 (1-4) (2005) 578–592. doi:10.1016/j. jcrysgro.2004.12.042. URL https://linkinghub.elsevier.com/retrieve/pii/S0022024804020196 32

    work page doi:10.1016/j 2005

  50. [51]

    S. Li, J. S. Lowengrub, P. H. Leo, Nonlinear morphological control of growing crystals, Physica D: Nonlinear Phenomena 208 (3-4) (2005) 209–219. doi:10.1016/j.physd.2005.06.021. URL https://linkinghub.elsevier.com/retrieve/pii/S0167278905002691

    work page doi:10.1016/j.physd.2005.06.021 2005

  51. [52]

    W. Ying, C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, Journal of Computational Physics 227 (2) (2007) 1046–1074. doi:10.1016/j.jcp.2007.08.021. URL https://linkinghub.elsevier.com/retrieve/pii/S0021999107003774

    work page doi:10.1016/j.jcp.2007.08.021 2007

  52. [53]

    W. Ying, W. C. Wang, A kernel-free boundary integral method for implicitly defined surfaces, Journal of Compu- tational Physics 252 (2013) 606–624. doi:10.1016/j.jcp.2013.06.019. URL http://dx.doi.org/10.1016/j.jcp.2013.06.019

    work page doi:10.1016/j.jcp.2013.06.019 2013

  53. [54]

    Y . Xie, W. Ying, A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions, Journal of Computational Physics 415 (2020) 109526. doi:10.1016/j.jcp.2020.109526. URL https://doi.org/10.1016/j.jcp.2020.109526

    work page doi:10.1016/j.jcp.2020.109526 2020

  54. [55]

    W. Ying, W. C. Wang, A kernel-free boundary integral method for variable coe fficients elliptic pdes, Communica- tions in Computational Physics 15 (4) (2014) 1108–1140. doi:10.4208/cicp.170313.071113s

    work page doi:10.4208/cicp.170313.071113s 2014

  55. [56]

    Peyret, Spectral Methods for Incompressible Viscous Flow, V ol

    R. Peyret, Spectral Methods for Incompressible Viscous Flow, V ol. 148 of Applied Mathematical Sciences, Springer New York, New York, NY , 2002, pp. XII, 434.doi:10.1007/978-1-4757-6557-1 . URL http://link.springer.com/10.1007/978-1-4757-6557-1

    work page doi:10.1007/978-1-4757-6557-1 2002

  56. [57]

    G. C. Hsiao, W. L. Wendland, Boundary Integral Equations, Springer International Publishing, Cham, 2021, pp. 25–94. doi:10.1007/978-3-030-71127-6_2 . URL https://doi.org/10.1007/978-3-030-71127-6_2

    work page doi:10.1007/978-3-030-71127-6_2 2021

  57. [58]

    Saad, Iterative Methods for Sparse Linear Systems, 2nd Edition, Society for Industrial and Applied Mathematics,

    Y . Saad, Iterative Methods for Sparse Linear Systems, 2nd Edition, Society for Industrial and Applied Mathematics,

    work page

  58. [59]

    doi:10.1137/1.9780898718003

    work page doi:10.1137/1.9780898718003

  59. [60]

    Beale, A

    T. Beale, A. Layton, On the accuracy of finite di fference methods for elliptic problems with inter- faces, Communications in Applied Mathematics and Computational Science 1 (1) (2006) 91–119. doi:10.2140/camcos.2006.1.91. URL http://www.physics.nyu.edu/$\sim$fh417/ensemble-sampler-with-affine-invariance. pdfhttp://msp.org/camcos/2006/1-1/p05.xhtml

    work page doi:10.2140/camcos.2006.1.91 2006

  60. [61]

    Brandt, N

    A. Brandt, N. Dinar, Multigrid Solutions to Elliptic Flow Problems, in: S. V . PARTER (Ed.), Numerical Meth- ods for Partial Di fferential Equations, Academic Press, 1979, pp. 53–147. doi:https://doi.org/10.1016/ B978-0-12-546050-7.50008-3 . URL https://www.sciencedirect.com/science/article/pii/B9780125460507500083

    work page 1979

  61. [62]

    Brandt, I

    A. Brandt, I. Yavneh, On multigrid solution of high-reynolds incompressible entering flows, Journal of Computa- tional Physics 101 (1) (1992) 151–164. doi:https://doi.org/10.1016/0021-9991(92)90049-5 . URL https://www.sciencedirect.com/science/article/pii/0021999192900495

    work page doi:10.1016/0021-9991(92)90049-5 1992

  62. [63]

    Pr ¨uss, G

    J. Pr ¨uss, G. Simonett, Moving interfaces and quasilinear parabolic evolution equations, V ol. 105, Springer, 2016

    work page 2016

  63. [64]

    Sakakibara, Y

    K. Sakakibara, Y . Miyatake, A fully discrete curve-shortening polygonal evolution law for moving boundary prob- lems, Journal of Computational Physics 424 (2021) 109857. doi:10.1016/j.jcp.2020.109857. URL https://doi.org/10.1016/j.jcp.2020.109857

    work page doi:10.1016/j.jcp.2020.109857 2021

  64. [65]

    H. Chen, C. Min, F. Gibou, A numerical scheme for the stefan problem on adaptive cartesian grids with supralinear convergence rate, Journal of Computational Physics 228 (16) (2009) 5803–5818. doi:10.1016/j.jcp.2009. 04.044

    work page doi:10.1016/j.jcp.2009 2009

  65. [66]

    H. Zhou, W. Ying, A Dimension Splitting Method for Time Dependent PDEs on Irregular Domains, Journal of Scientific Computing 94 (1) (2023) 20. doi:10.1007/s10915-022-02066-5 . URL https://doi.org/10.1007/s10915-022-02066-5https://link.springer.com/10.1007/ s10915-022-02066-5 33

    work page doi:10.1007/s10915-022-02066-5 2023