Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations
Abstract
This paper investigates oscillation-free stability conditions of numerical methods for linear parabolic partial differential equations with some example extrapolations to nonlinear equations. Not clearly understood, numerical oscillations can create infeasible results. Since oscillation-free behavior is not ensured by stability conditions, a more precise condition would be useful for accurate solutions. Using Von Neumann and spectral analyses, we find and explore oscillation-free conditions for several finite difference schemes. Further relationships between oscillatory behavior and eigenvalues is supported with numerical evidence and proof. Also, evidence suggests that the oscillation-free stability condition for a consistent linearization may be sufficient to provide oscillation-free stability of the nonlinear solution. These conditions are verified numerically for several example problems by visually comparing the analytical conditions to the behavior of the numerical solution for a wide range of mesh sizes.
Cite
@article{arxiv.1701.04798,
title = {Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations},
author = {R. Corban Harwood and Mitch Main},
journal= {arXiv preprint arXiv:1701.04798},
year = {2017}
}
Comments
12 pages, 6 figures, 1 table