##
**A convergent monotone difference scheme for motion of level sets by mean curvature.**
*(English)*
Zbl 1070.65082

There has been a great deal of interest in motion by mean curvature, in terms of both theory and computation. The equation for motion of level sets by mean curvature (MC) in \(\mathbb{R}^n \) is given by
\[
u_t = \Delta _1 u \equiv |Du|\text{ div}\left(\frac{Du}{|Du|}\right) \equiv \sum_{i = 1}^n {u_{x_i x_i }} - \frac{1}{|Du|^2} \sum_{i,j = 1}^n {u_{x_i x_j} u_{x_i} u_{x_j}}.
\]
This equation arises from the well-known level set method, which gives the normal velocity, \(v_n\), of the level set of the function \(u(x)\) as \(v_n = \frac{u_t}{|Du|}\). The unit normal of the level set is given by the direction of the gradient, \(\frac{Du}{|Du|}\), and the mean curvature is the divergence of the unit normal, resulting in (MC). The level set method has been successful both as a framework for the theoretical study and as a numerical method for the simulation of the motion.

Applications of motion by mean curvature, and of related geometric motions, can be found in many areas, including differential geometry, fluid dynamics, combustion, front propagation and image processing. Despite the great deal of interest, no practical, provably convergent numerical scheme has been proposed for the equation for motion of level sets by mean curvature.

An obvious approach to building a difference scheme for (MC) is to simply combine centered finite differences for each of the terms in the equation. The author presents a numerical experiment which demonstrates that this approach is not convergent.

An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented in the paper. The scheme is defined on a Cartesian grid, using neighbors arranged approximately in a circle. The accuracy of the scheme, which depends on the radius of the circle, \(dx\), and on the angular resolution, \(d\theta \), is formally \(O(dx^2 + d\theta)\). The scheme is explicit and nonlinear: the update involves computing the median of the values at the neighboring grid points.

In the first section of the paper the simplest version of the scheme is presented. Despite its limited accuracy, this version of the scheme may be of practical use. A possibility is in applications to image processing, where the data itself has a natural discrete structure. This version of scheme is presented with an arbitrary spatial resolution, but with a nine point stencil. The scheme is monotone: increasing any of the neighboring values will lead to an increase (or no change) in the value of the solution.

In the second section it is shown that the scheme can be implemented in general (structured, unstructured) geometries, in two or more dimensions. For concreteness, the author first addresses the scheme on a uniform grid in two dimensions. In this context, a detailed convergence proof is given. The requirements for the choice of neighbors specified so that the scheme may easily be adapted to an unstructured grid. The extension to higher dimensions is men addressed.

A numerical consistency is checked in the third section. This is followed by testing the accuracy of the numerical solution against an exact solution, using different stencils and grid sizes. Finally, an example which demonstrates the fattening phenomena is presented.

Applications of motion by mean curvature, and of related geometric motions, can be found in many areas, including differential geometry, fluid dynamics, combustion, front propagation and image processing. Despite the great deal of interest, no practical, provably convergent numerical scheme has been proposed for the equation for motion of level sets by mean curvature.

An obvious approach to building a difference scheme for (MC) is to simply combine centered finite differences for each of the terms in the equation. The author presents a numerical experiment which demonstrates that this approach is not convergent.

An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented in the paper. The scheme is defined on a Cartesian grid, using neighbors arranged approximately in a circle. The accuracy of the scheme, which depends on the radius of the circle, \(dx\), and on the angular resolution, \(d\theta \), is formally \(O(dx^2 + d\theta)\). The scheme is explicit and nonlinear: the update involves computing the median of the values at the neighboring grid points.

In the first section of the paper the simplest version of the scheme is presented. Despite its limited accuracy, this version of the scheme may be of practical use. A possibility is in applications to image processing, where the data itself has a natural discrete structure. This version of scheme is presented with an arbitrary spatial resolution, but with a nine point stencil. The scheme is monotone: increasing any of the neighboring values will lead to an increase (or no change) in the value of the solution.

In the second section it is shown that the scheme can be implemented in general (structured, unstructured) geometries, in two or more dimensions. For concreteness, the author first addresses the scheme on a uniform grid in two dimensions. In this context, a detailed convergence proof is given. The requirements for the choice of neighbors specified so that the scheme may easily be adapted to an unstructured grid. The extension to higher dimensions is men addressed.

A numerical consistency is checked in the third section. This is followed by testing the accuracy of the numerical solution against an exact solution, using different stencils and grid sizes. Finally, an example which demonstrates the fattening phenomena is presented.

Reviewer: Leonid B. Chubarov (Novosibirsk)

### MSC:

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

35J70 | Degenerate elliptic equations |

35K65 | Degenerate parabolic equations |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

### Keywords:

motion of level sets by mean curvature; degenerate elliptic equation; regularity of solutions; differential geometry; fluid dynamics; combustion; front propagation; image processing; explicit convergent finite difference scheme; monotonicity; directional resolution; lattice points; rotation invariance; stencil; accuracy; stability; consistency; convergence; degenerate nonlinear parabolic equation; numerical experiment
Full Text:
DOI

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