General Linear Methods for Ordinary Differential Equations

Two-step almost collocation methods for ordinary differential equations

“…is the generalization of preconsistency conditions for TSRK methods (1.3), compare [13]. This condition implies that θ,θ , u j andũ j appearing in (1.3) satisfy the conditions θ +θ = 1, u j +ũ j = 1, j = 1, 2, .…”

Continuous two-step Runge–Kutta methods for ordinary differential equations

“…TSRK methods were introduced by Jackiewicz and Tracogna [18] and further investigated in [1,3,8,9,15,17,20,25], Conte et al (unpublished manuscript) and [26]. Continuous methods (1.1) provide an approximation to the solution y(t) of (1.2) on the whole interval of integration, and not only in the gridpoints {t n } as in the case of discrete TSRK methods (1.3).…”

Practical implementation of extended Kalman filtering in chemical systems with sparse measurements

“…Its inefficiency is clearly shown for chemical systems with sparse measurement information, below. This is quite obvious because the theory of numerical methods for ODEs [5,10,11,14] says that small discretization errors are ensured only for sufficiently small sizes of the sampling period δ, and this is not the case for chemical systems with long waiting times. On the other hand, Soroush [39] explains that such an inaccurate numerical solution seriously limits the applied potential of the traditional EKF method because it does not succeed in chemical models (1.1), (1.2) with infrequent measurements, and the short sampling periods may be technically (or by any other reason) impossible or too expensive in practice.…”