A fully discrete curve-shortening polygonal evolution law for moving boundary problems

We consider the numerical integration of moving boundary problems with the curve-shortening property, such as the mean curvature flow and Hele-Shaw flow. We propose a fully discrete curve-shortening polygonal evolution law. The proposed evolution law is fully implicit, and the key to the derivation is to devise the definitions of tangent and normal vectors and tangential and normal velocities at each vertex in an implicit manner. Numerical experiments show that the proposed method allows the use of relatively large time step sizes and also captures the area-preserving or dissipative property in good accuracy.

[1]  G. Quispel,et al.  A new class of energy-preserving numerical integration methods , 2008 .

[2]  J. M. Sanz-Serna,et al.  Numerical Hamiltonian Problems , 1994 .

[3]  O. Gonzalez.  Time integration and discrete Hamiltonian systems , 1996 .

[4]  W. Mullins.  Two‐Dimensional Motion of Idealized Grain Boundaries , 1956 .

[5]  D. Ševčovič,et al.  Evolution of plane curves with a curvature adjusted tangential velocity , 2010, 1009.2588.

[6]  D. Furihata,et al.  Finite Difference Schemes for ∂u∂t=(∂∂x)αδGδu That Inherit Energy Conservation or Dissipation Property , 1999 .

[7]  M. Grayson.  The heat equation shrinks embedded plane curves to round points , 1987 .

[8]  G. J. Cooper.  Stability of Runge-Kutta Methods for Trajectory Problems , 1987 .

[9]  M. Kimura,et al.  Accurate Numerical Scheme for the Flow by Curvature , 1994 .

[10]  Polygonal Hele?Shaw problem with surface tension , 2013 .

[11]  R. Nürnberg,et al.  The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute , 2011 .

[12]  David Cohen,et al.  Linear energy-preserving integrators for Poisson systems , 2011 .

[13]  T. Hou,et al.  Removing the stiffness from interfacial flows with surface tension , 1994 .

[14]  Shengliang Pan,et al.  Gage’s original normalized CSF can also yield the Grayson theorem , 2016 .

[15]  N. Požár.  On the self-similar solutions of the crystalline mean curvature flow in three dimensions , 2018 .

[16]  G. Quispel,et al.  Solving ODEs numerically while preserving a first integral , 1996 .

[17]  E. Hairer.  Energy-preserving variant of collocation methods 1 , 2010 .

[18]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[19]  Ernst Hairer,et al.  Energy-diminishing integration of gradient systems , 2014 .

[20]  K. Mikula,et al.  Evolution of curves on a surface driven by the geodesic curvature and external force , 2006 .

[21]  Daisuke Furihata,et al.  Discrete Variational Derivative Method , 2010 .

[22]  D. Ševčovič,et al.  On a Gradient Flow of Plane Curves Minimizing th e Anisoperimetric Ratio , 2012, 1203.2238.

[23]  M. Gage,et al.  The heat equation shrinking convex plane curves , 1986 .

[24]  K. Mikula,et al.  A direct method for solving an anisotropic mean curvature flow of plane curves with an external force , 2004 .

[25]  T. E. Hull,et al.  Comparing Numerical Methods for Ordinary Differential Equations , 1972 .

[26]  Takayasu Matsuo.  Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations , 2008 .

[27]  H. S. Hele-Shaw,et al.  The Flow of Water , 1898, Nature.

[28]  Elena Celledoni,et al.  Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method , 2012, J. Comput. Phys..

[29]  Koya Sakakibara,et al.  Structure-preserving numerical scheme for the one-phase Hele-Shaw problems by the method of fundamental solutions , 2019, Comput. Math. Methods.

[30]  B. Gustafsson,et al.  Conformal and Potential Analysis in Hele-Shaw Cells , 2006 .

[31]  H. J.,et al.  Hydrodynamics , 1924, Nature.

[32]  Jian Zhang,et al.  Numerical Studies of Discrete Approximations to the Allen--Cahn Equation in the Sharp Interface Limit , 2009, SIAM J. Sci. Comput..

[33]  T. Kemmochi.  Energy dissipative numerical schemes for gradient flows of planar curves , 2017 .

[34]  Masato Kimura,et al.  Numerical analysis of moving boundary problems using the boundary tracking method , 1997 .

[35]  M. Gurtin.  Thermomechanics of Evolving Phase Boundaries in the Plane , 1993 .

[36]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[37]  Yuto Miyatake,et al.  A Characterization of Energy-Preserving Methods and the Construction of Parallel Integrators for Hamiltonian Systems , 2015, SIAM J. Numer. Anal..

[38]  Hans Z. Munthe-Kaas,et al.  Topics in structure-preserving discretization* , 2011, Acta Numerica.

[39]  Yuto Miyatake,et al.  A general framework for finding energy dissipative/conservative $$H^1$$H1-Galerkin schemes and their underlying $$H^1$$H1-weak forms for nonlinear evolution equations , 2014 .

[40]  M. Gage,et al.  The Curve Shortening Flow , 1987 .

[41]  M. Gage.  On an area-preserving evolution equation for plane curves , 1986 .

[42]  Daniel Sevcovic,et al.  Numerical Solution of Constrained Curvature Flow for Closed Planar Curves , 2015, ENUMATH.