Convergent Numerical Solutions for Unsteady Regular or Chaotic Differential Equations

TL;DR

This paper discusses the conditions necessary for convergent numerical solutions of unsteady differential equations, emphasizing that stability alone is insufficient for transient problems and additional considerations are needed.

Contribution

It clarifies the limitations of von Neumann's stability criterion for unsteady problems and highlights the need for extra conditions to ensure convergence.

Findings

Stability is necessary but not sufficient for unsteady problems.

Additional criteria are required for accurate unsteady solutions.

The paper emphasizes the distinction between steady and unsteady problem stability.

Abstract

Von Neumann established that discretized algebraic equations must be consistent with the differential equations, and must be stable in order to obtain convergent numerical solutions for the given differential equations. The "stability" is required to satisfactorily approximate a differential derivative by its discretized form, such as a finite-difference scheme, in order to compute in computers. His criterion is the necessary and sufficient condition only for steady or equilibrium problems. It is also a necessary condition, but not a sufficient condition for unsteady transient problems; additional care is required to ensure the accuracy of unsteady solutions.