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The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations. (English) Zbl 1297.65097

Summary: 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 (PDEs). 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations

Software:

MOVCOL4
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References:

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