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

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