Convergent Numerical Solutions for Unsteady Regular or Chaotic Differential Equations
This work addresses a fundamental gap in numerical analysis for unsteady problems, providing a more complete framework for ensuring solution accuracy in computational simulations.
The paper identifies that von Neumann's stability criterion is necessary but not sufficient for unsteady differential equations, and proposes additional conditions to ensure accurate convergent numerical solutions for both regular and chaotic transient problems.
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.