Stable Explicit p-Laplacian Flows Based on Nonlinear Eigenvalue Analysis

Ido Cohen, Adi Falik and Guy Gilboa SSVM 2019

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<21<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.