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We analyze an extension of backward differentiation formulas, used as boundary value methods, that generates a class of methods with nice stability and convergence properties. These methods are obtained starting from the boundary value GBDFs class, and are in the class of EBDF-type methods. We discuss different ways of using these linear multistep...
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Citations
... Other approaches that start from a boundary-value problem point of view are Generalized BDF methods (GBDF) [18], but these can be applied to initial-value problems as well. Parallelism for Generalized BDF methods (GBDF) was considered in ref. [19]. The FEM is the standard procedure for solving ordinary or partial differential equations (ODEs or PDEs) when dealing with continuum field problems. ...
In this paper, a novel algorithm for a transient linear circuit analysis based on the Finite Element Technique (FET) was established. The FET procedure allows a straightforward solution for complex electric circuits since it is based on the Finite Element Method (FEM) approach. The developed algorithm allows us to select various types of time integration schemes when forming a local system of equations for a coupled circuit finite element. To illustrate the basic principle of the developed FET-based algorithm and to perform transient analysis, a random coupled linear circuit was analyzed. Numerical solutions obtained using Heun’s and the generalized trapezoidal rule (ϑ-method) integration schemes were compared to the solution obtained by Matlab Simulink software. It was shown that the accuracy of results depends on the employed time integration scheme when performing the FET procedure. It was shown that using Heun’s method as the time integration scheme yields more accurate results than using the ϑ-method time integration scheme.
... Different applications of the parallelism across steps approach for classic numerical methods are presented and studied in [6,31,32,33,34]. ...
This work introduces novel parallelization techniques for Quantized State System (QSS) simulation of continuous time and hybrid systems and their implementation on a multi-core architecture. Exploiting the asynchronous nature of QSS algorithms, the novel methodologies are based on the use of non–strict synchronization between logical processes. The fact that the synchronization is not strict allows to achieve large speedups at the cost of introducing additional numerical errors that, under certain assumptions, are bounded depending on some given parameters.
... Other approaches, that start from a boundary-value problem point of view, are Generalized BDFmethods (GBDF) [5], but these can be applied to initial-value problems as well. Parallelism for this last case is considered in [22]. ...
Recent developments of new methods for simulating electric circuits are described. Emphasis is put on methods that fit existing datastructures for backward differentiation formulae methods. These methods can be modified to apply to hierarchically organized datastructures, which allows for efficient simulation of large designs of circuits in the electronics industry.
In this paper a one parameter family of s-stage singly implicit two-step peer (SIP) methods with order \((s-1)\) that are L-stable for some values of this parameter addressed for the numerical solution of stiff IVPs has been developed. General peer methods are multistage two-step methods for solving IVPs where all stages possess essentially the same accuracy and stability properties. In particular a s-stage SIP requires at each step the solution of s-implicit non-linear systems of equations of the same type in a similar way to the singly implicit Runge–Kutta methods. Here for each \( s \ge 3\) a family of one parameter s-stage SIP methods with order \((s-1)\) that are optimally zero-stable for arbitrary step size sequences is derived. For \( s \le 8\) intervals of values of this parameter that ensure their L-stability are obtained. Hence for \( s \le 8\), L-stable methods with order \((s-1)\) and a computational cost per step equivalent to s one-step backward Euler methods with the same Jacobian matrix are obtained. Further it is shown that under some restriction on the parameter each s-stage SIP method can be formulated as a cyclic multistep method of order \((s-1)\) and this implies that the Dahlquist barrier of second-order for A-stable linear multistep methods can be broken with suitable s-stage cyclic methods of these families.
Publisher Summary
This chapter discusses the modeling aspect of differential-algebraic equations (DAEs). In computational engineering, the network modeling approach forms the basis for computer-aided analysis of time-dependent processes in multibody dynamics, process simulation, or circuit design. Its principle is to connect compact elements via ideal nodes, and to apply some kind of conservation rules for setting up equations. The mathematical model, a set of so-called network equations, is generated automatically by combining network topology with characteristic equations describing the physical behavior of network elements under some simplifying assumptions. Interconnects and semiconductor devices (i.e., transistors) are modeled by multi-terminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities. Interconnects and semiconductor devices (i.e., transistors) are modeled by multiterminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities.
