The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations

TL;DR

This paper introduces and analyzes the cutoff method for numerically computing nonnegative solutions of parabolic PDEs, demonstrating its effectiveness and broad applicability through theoretical convergence results and practical applications.

Contribution

The paper provides a convergence analysis for the cutoff method applied to finite difference schemes and demonstrates its effectiveness for anisotropic diffusion and lubrication-type equations.

Findings

Convergence of the cutoff method is established for linear parabolic equations.

Numerical results align with theoretical predictions and existing literature.

The method is adaptable to various discretization techniques and positive solutions.

Abstract

The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations. Two applications are investigated, linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication-type equations for which it is unclear if the convergence analysis applies. The numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature. The convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions. Cutoff can also be used with other discretization methods such as collocation, finite volume, finite element, and spectral methods and for the computation of positive solutions.