Discrete Maximum Principle and Adequate Discretizations of Linear Parabolic Problems

Authors:
Istva´n Farago´;Ro´bert Horva´th
Affiliations:
-;-
Venue:
SIAM Journal on Scientific Computing
Year:
2006

Citing 0
Cited 13

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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

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Linearly Implicit Approximations of Diffusive Relaxation Systems

Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications

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Abstract

In this paper, we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with Dirichlet-type boundary condition. First we formulate the qualitative properties for the differential equations and shed light on their relations. Then we show how the well-known discretization schemes can be written in the form of a one-step iterative process. We give necessary and sufficient conditions of the main qualitative properties of these iterations. We apply the results to the finite difference and Galerkin finite element solutions of linear parabolic problems. In our main result we show that the nonnegativity preservation property is equivalent to the maximum-minimum principle and they imply the maximum norm contractivity. In one, two, and three dimensions, we list sufficient a priori conditions that ensure the required qualitative properties. Finally, we demonstrate the above results on numerical examples.