Linearly implicit quantization-based integration methods for stiff ordinary differential equations

Abstract

In this paper, new integration methods for stiff ordinary differential equations (ODEs) are developed. Following the idea of quantization-based integration (QBI), i.e., replacing the time discretization by state quantization, the proposed algorithms generalize the idea of linearly implicit algorithms. Also, the implementation of the new algorithms in a DEVS simulation tool is discussed. The efficiency of these new methods is verified by comparing their performance in the simulation of two benchmark problems with that of other numerical stiff ODE solvers. In particular, the advantages of these new algorithms for the simulation of electronic circuits are demonstrated.

Ann–th Order Generalization of the Activity Measure for Continuous Systems

“…Besides QSS1, the Backward QSS (BQSS) [4], Centered QSS (CQSS) [4] and Linearly Implicit QSS (LIQSS1) [5] perform first order approximations. They differ from QSS1 in the definition of the quantization function given by Eq.…”

“…02001-p. 5 The quantization schemes presented above are those employed within QSS methods that shall be described in the next section.…”

“…Besides QSS2 and QSS3, there exist linearly implicit QSS methods of orders 2 (LIQSS2) and 3 (LIQSS3) which are particularly suitable to simulate stiff systems [5], while having definitions very similar to those of QSS2 and QSS3.…”

“…Zero-order quantization functions like that of Fig.1 are used in some Quantized State Systems (QSS) numerical integration algorithms, such as the first order accurate QSS1 method [3], the Backward QSS (BQSS) method [4], and the first order accurate Linearly Implicit QSS (LIQSS1) algorithm [5].…”

“…Evidently, this difference is not captured by Eq. (1) Secondly, it is known that in second order accurate methods like QSS2 the number of steps varies with the square root of the quantum ΔQ i [5,6]. However, Equation (2) shows a linear dependence, which is clearly wrong.…”