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arXiv · 1701.04798

Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations

R. Corban Harwood·Mitch Main
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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.

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R. Corban Harwood, Mitch Main. 2017-01-17. Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations. https://arxiv.org/abs/1701.04798

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