# Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

@article{Crandall1996ConvergentDS, title={Convergent difference schemes for nonlinear parabolic equations and mean curvature motion }, author={Michael G. Crandall and Pierre-Louis Lions}, journal={Numerische Mathematik}, year={1996}, volume={75}, pages={17-41} }

Summary.Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal… Expand

#### 81 Citations

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#### References

SHOWING 1-10 OF 18 REFERENCES

Discrete methods for fully nonlinear elliptic equations

- Mathematics
- 1992

This paper exhibits and proves the stability of discrete approximations for the solution of the Dirichlet problem for fully nonlinear, uniformly elliptic partial differential equations in situations… Expand

Approximation schemes for viscosity solutions of Hamilton-Jacobi equations

- Mathematics
- 1985

Abstract Equations of Hamilton-Jacobi type arise in many areas of applications, including the calculus of variations, control theory and differential games. Recently M. G. Crandall and P.-L. Lions… Expand

User’s guide to viscosity solutions of second order partial differential equations

- Mathematics
- 1992

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence… Expand

Two approximations of solutions of Hamilton-Jacobi equations

- Mathematics
- 1984

Abstract : Equations of Hamilton-Jacobi type arise in many areas of application, including the calculus of variations, control theory and differential games. The associated initial-value problems… Expand

Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Final report

- Mathematics
- 1987

New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be… Expand

Motion of level sets by mean curvature. I

- Mathematics
- 1991

We continue our investigation of the “level-set” technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is… Expand

Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations

- Mathematics
- 1989

where Vu is the (spatial) gradiant of u. Here VM/|VW| is a unit normal to a level surface of u, so div(Vw/|Vw|) is its mean curvature unless Vu vanishes on the surface. Since ut/\Vu\ is a normal… Expand

A Simple Proof of Convergence for an Approximation Scheme for Computing Motions by Mean Curvature

- Mathematics
- 1995

We prove the convergence of an approximation scheme recently proposed by Bence, Merriman, and Osher for computing motions of hypersurfaces by mean curvature. Our proof is based on viscosity solutions… Expand

Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations

- Mathematics
- 1984

On etudie l'unicite des solutions viscosite non bornees du probleme stationnaire u+H(x,Du)=0 dans R N et du probleme de Cauchy u t +H(t,x,Du)=0 dans (O,T)×R N , u(o,x)=u o (x) pour x∈R N

Motion of multiple junctions: a level set approach

- Mathematics
- 1994

Abstract A coupled level set method for the motion of multiple junctions is proposed. The new method extends the "Hamilton-Jacobi" level set formulation of Osher and Sethian. It retains the feature… Expand