Abstract
Implementation of nonlinear flows by explicit schemes can be very convenient, due to their simplicity and low-computational cost per time step. A well known drawback is the small time step bound, referred to as the CFL condition, which ensures a stable flow. For p-Laplacian flows, with \(1< p <2\), explicit schemes without gradient regularization require, in principle, a time step approaching zero. However, numerical implementations show explicit flows with small time-steps are well behaved. We can now explain and quantify this phenomenon.
In this paper we examine explicit p-Laplacian flows by analyzing the evolution of nonlinear eigenfunctions, with respect to the p-Laplacian operator. For these cases analytic solutions can be formulated, allowing for a comprehensive analysis. A generalized CFL condition is presented, relating the time step to the inverse of the nonlinear eigenvalue. Moreover, we show that the flow converges and formulate a bound on the error of the discrete scheme. Finally, we examine general initial conditions and propose a dynamic time-step bound, which is based on a nonlinear Rayleigh quotient.
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Acknowledgements
This project is supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 777826. We acknowledge support by the Israel Science Foundation (grant No. 718/15). This work was supported by the Technion Ollendorff Minerva Center.
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Cohen, I., Falik, A., Gilboa, G. (2019). Stable Explicit p-Laplacian Flows Based on Nonlinear Eigenvalue Analysis. In: Lellmann, J., Burger, M., Modersitzki, J. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2019. Lecture Notes in Computer Science(), vol 11603. Springer, Cham. https://doi.org/10.1007/978-3-030-22368-7_25
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