# Error analysis of BDF compound-fast multirate method for differential-algebraic equations

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department ofMathematics and Computer Science CASA-Report 06-10 April 2006 Error analysis ofBDF compound-fast multirate method for differential-algebraic equations by A. Verhoeven, T.GJ. Beelen, A. EI Guennouni, E.J.W. ter Maten, RM.M. Mattheij, B. Tasic Centre for Analysis, Scientific computing and Applications Department ofMathematics and Computer Science Eindhoven University ofTechnology P.O. Box 513 5600 MB Eindhoven, The Netherlands ISSN: 0926-4507 Error analysis of BDF Compound-Fast multirate method for differential-algebraic equations A. Verhoeven!, T.G.J. Beelen2 , A. El Guennouni3 , E.J.W. ter Maten1,2, R.M.M. Mattheij!, B. Tasic2 1 Technische Universiteit Eindhoven averhoevalllin. tue. nl 2 Philips Research Laboratories 3 Magma Design Automation 1 Introduction Analogue electrical circuits are usually modeled by differential-algebraic equations of the following type: ~ [q(t, x)] + j(t, x) = 0, (1) (2) (3) where x E ~d represents the state of the circuit. A common analysis is the transient analysis, which computes the solution x(t) of this non-linear DAE along the time interval [0, T] for a given initial state. In the classical circuit simulators, this Initial Value Problem is solved by means of implicit inte- gration methods, like the BDF-methods. Each iteration, all equations are discretized by means of the same stepsize. Often, parts of electrical circuits have latency or multirate behaviour. Latency means that parts of the circuit are constant or slowly time-varying during a certain time interval. Multirate behaviour means that some variables are slowly time-varying compared to other variables. In both cases, it would be attractive to integrate these parts with a larger timestep. 1.1 Partition of the system For a multirate method it is necessary to partition the variables and equations into an active (A) and a latent (L) part. This can be done by the user or automatically. Let BA E ~dA xd and B L E ~dL xd with dA +

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