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Steplength thresholds for invariance preserving of three types ofdiscretization methods on a polyhedron are considered. For Taylor approximationtype discretization methods we prove that a valid steplength threshold can beobtained by finding the first positive zeros of a finite number of polynomialfunctions. Further, a simple and efficient algorithm is proposed to numericallycompute the steplength threshold. For rational function type discretizationmethods we derive a valid steplength threshold for invariance preserving, whichcan be computed by using an analogous algorithm as in the first case. Therelationship between the previous two types of discretization methods and theforward Euler method is studied. Finally, we show that, for the forward Eulermethod, the largest steplength threshold for invariance preserving can becomputed by solving a finite number of linear optimization problems.

Steplength Thresholds for Invariance Preserving of Discretization Methods of Dynamical Systems on a Polyhedron