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Von Neumann established that discretized algebraic equations must beconsistent with the differential equations, and must be stable in order toobtain convergent numerical solutions for the given differential equations. The"stability" is required to satisfactorily approximate a differential derivativeby its discretized form, such as a finite-difference scheme, in order tocompute in computers. His criterion is the necessary and sufficient conditiononly for steady or equilibrium problems. It is also a necessary condition, butnot a sufficient condition for unsteady transient problems; additional care isrequired to ensure the accuracy of unsteady solutions.

Convergent Numerical Solutions for Unsteady Regular or Chaotic Differential Equations