Improving the accuracy of BDF methods for index 3 differential-algebraic equations

Partial Eigenvalue Assignment for High-Order Linear Systems in a Time Delayed System

“…PEAP arises from many practical situations such as electrical circuit simulation and acoustic system, vibration analysis of structural mechanical, fluid mechanics, finite element model updating in automobile industries and aerospace (Arévalo and Lötstedt, 1995;De Boor and Kreiss, 1986;Sand, 2002;Mackey et al, 2006). PEAP for high-order linear systems without time delay introduced in many paper (Wang and Zhang, 2014;Ramadan and El-Sayed, 2010).…”

A derivative array approach for linear second order differential-algebraic systems

“…For DAEs, however, the classical order reduction approach has to be performed with great care, since it may lead to a number of mathematical difficulties as has been discussed in several publications; see [1,5,12,14,15]. Moreover, the numerical solution of DAEs usually requires the reformulation of higher index DAEs as an equivalent system of lower index to be able to use standard integration methods suited for DAEs; see [2,10,11].…”

“…= f 2 −f 3 + tf 2 + 2ḟ 2 f 3 − (1 + t)f 2 +f 3 − tf 2 − 2ḟ 2 , and x 1 is the unique solution of the second order ordinary differential equation tẍ 1 + x 1 + x 1 = f 1 for some given initial values x 1 (t 0 ) = x 1,0 andẋ 1 (t 0 ) =ẋ 1,0 . Hence, the minimum requirement for the existence of a continuous solution is that f 1 is continuous, and f 2 and f 3 are twice continuously differentiable (corresponding to a strangeness index of µ = 2).…”

“…However, since the standard way to obtain a strangeness-free first order formulation-first introducing new variables for the derivatives to transform the system into a first order system and then applying the usual index reduction procedures to the first order system-can fail due to a possible increase in the index, at first an index reduction of the higher order system should be used, which is followed by an appropriate order reduction to obtain a suitable strangeness-free first order formulation. Recently, it has been shown in [14,17] that also the direct discretization of the second order system may yield better numerical results and is able to prevent certain numerical difficulties as the failure of numerical methods; see also [1,2,16].…”

Numerical Treatment of Second Order Differential‐Algebraic Systems

“…However, it has been discussed in several publications, see e.g. [1,4], that for differential-algebraic equations (DAEs) this classical approach has to be performed with great care, since it may lead to a number of mathematical difficulties. In [3,5] several examples show that the classical approach of introducing new variables may lead to higher smoothness requirements for the inhomogeneity, corresponding to an increase in the index of the DAE, or may result in a loss of structures in the system.…”