Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations

TL;DR

This paper explores the relationship between eigenvalues and oscillations in numerical solutions of parabolic PDEs, proposing conditions to ensure oscillation-free stability for finite difference schemes.

Contribution

It introduces spectral and Von Neumann analysis-based conditions for oscillation-free stability, extending insights to nonlinear equations and verifying them numerically.

Findings

Eigenvalue-based conditions predict oscillation-free behavior.

Numerical evidence supports the sufficiency of linearization conditions.

Conditions are validated across various mesh sizes and problems.

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.