Eigenvalue Dependence of Numerical Oscillations in Parabolic Partial Differential Equations
For computational scientists solving parabolic PDEs, this work provides more precise conditions to avoid numerical oscillations that can cause infeasible results.
This paper derives oscillation-free stability conditions for finite difference schemes solving linear parabolic PDEs, showing that these conditions depend on eigenvalues and are not guaranteed by standard stability. The conditions are verified numerically for a range of mesh sizes.
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.