Vol. 4, No. 9, 2009

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Remarks on the accuracy of algorithms for motion by mean curvature in bounded domains

Simon Cox and Gennady Mishuris

Vol. 4 (2009), No. 9, 1555–1572

Simulations of motion by mean curvature in bounded domains, with applications to bubble motion and grain growth, rely upon boundary conditions that are not necessarily compatible with the equation of motion. Three closed form solutions for the problem exist, governing translation, rotation, and expansion of a single interface, providing the only benchmarks for algorithm verification. We derive new identities for the translation solution. Then we estimate the accuracy of a straightforward algorithm to recover the analytical solution for different values of the velocity V given along the boundary. As expected, for large V the error can reach unacceptable levels especially near the boundary. We discuss factors influencing the accuracy and propose a simple modification of the algorithm which improves the computational accuracy.

motion by mean curvature, grain growth, foam rheology, algorithms, measures of accuracy
Received: 14 January 2009
Revised: 11 March 2009
Accepted: 8 May 2009
Published: 17 January 2010
Simon Cox
Institute of Mathematics and Physics
Aberystwyth University
Aberystwyth SY23 3BZ
United Kingdom
Gennady Mishuris
Institute of Mathematics and Physics and Wales Institute of Mathematical and Computational Sciences
Aberystwyth University
Aberystwyth SY23 3BZ
United Kingdom