# Convergence Analysis of Linear Multistep Methods for a Class of Delay Differential-Algebraic Equations

Vu Hoang Linh, Nguyen Duy Truong, M.V. BulatovDelay differential-algebraic equations (DDAEs) can be used for modelling real-life phenomena that involve simultaneously time-delay effect and constraints. It is also known that solving delay DAEs is more complicated than solving non-delay ones because approximation of solutions in the past time is usually needed and discontinuity in higher derivatives of the solutions is typical. Recently, we have proposed and investigated linear multistep (LM) methods for strangeness-free DAEs (without delay). In this paper, we extend the use of LM methods to a class of structured strangeness-free DAEs with constant delay. For the approximation of solutions at delayed time we use polynomial interpolation. Convergence analysis for LM methods is presented. It is shown that, similarly to the non-delay case, the strict stability of the second characteristic polynomial associated with the methods is not required for the convergence if we discretize an appropriately reformulated DDAE instead of the original one. Numerical experiments are also given for illustration.Full text

- Keywords
- delay differential-algebraic equation; strangeness-free; linear multistep method; stability; convergence.
- References
- 1. Baker C.T.H., Paul C.A.H., Tian H. Differential Algebraic Equations with After-Effect. Journal of Computational and Applied Mathematics, 2002, vol. 140, pp. 63-80.

2. Bellen A., Maset S., Zennaro M., Guglielmi N. Recent Trends in the Numerical Solution of Retarded Functional Differential Equations. Acta Numerica, 2009, vol. 18, pp. 1-110.

3. Ha P. Analysis and Numerical Solution of Delay Differential-Algebraic Equations. PhD Thesis. Berlin, 2015.

4. Shampine L.F., Gahinet P. Delay-Differential-Algebraic Equations in Control Theory. Applied Numerical Mathematics, 2006, vol. 56, pp. 574-588.

5. Bellen A., Zennaro M. Numerical Methods for Delay Differential Equations. Oxford, Oxford University Press, 2003.

6. Bellman R., Cooke K.L. Differential-Difference Equations. N.Y., Academic Press, 1963.

7. Ascher U., Petzold L. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Philadelphia, SIAM Society for Industrial and Applied Mathematics, 1998.

8. Hairer E., Wanner G. Solving Ordinary Differential Equation II. Berlin, Springer-Verlag, 1996.

9. Kunkel P., Mehrmann V. Differential-Algebraic Equations Analysis and Numerical Solution. Zurich, EMS Publishing House, 2006.

10. Ha P., Mehrmann V., Steinbrecher A. Analysis of Linear Variable Coefficient Delay Differential-Algebraic Equations. Journal of Dynamics and Differential Equations, 2014, vol. 26, pp. 1-26.

11. Ascher U., Petzold L. The Numerical Solution of Delay-Differential-Algebraic Equations of Retarded and Neutral Type. SIAM Journal on Numerical Analysis, 1995, vol. 32, pp. 1635-1657.

12. Hauber R. Numerical Treatment of Retarded Differential-Algebraic Equations by Collocation Methods. Advances in Computational Mathematics, 1997, vol. 7, pp. 573-592.

13. Hongliang L., Aiguo X. Convergence of Linear Multistep Methods and One-Leg Methods for Index-2 Differential-Algebraic Equations with a Variable Delay. Advances in Applied Mathematics and Mechanics, 2012, vol. 4, no. 5, pp. 636-646.

14. Linh V.H., Mehrmann V. Efficient Integration of Matrix-Valued Non-Stiff DAEs by Half-Explicit Methods. Journal of Computational and Applied Mathematics, 2014, vol. 262, pp. 346-360.

15. Linh V.H., Truong N.D. Stable Numerical Solution for a Class of Structured Differential-Algebraic Equations by Linear Multistep Methods. Submitted for publication, 2018.

16. Bulatov M.V., Linh V.H., Solovarova L.S. On BDF-Based Multistep Schemes for Some Classes of Linear Differential-Algebraic Equations of Index at Most 2. Acta Mathematica Vietnamica, 2016, vol. 41, no. 4, pp. 715-730.

17. Linh V.H., Truong N.D. Runge-Kutta Methods Revisited for a Class of Structured Strangeness-Free DAEs. Electronic transactions on numerical analysis ETNA, 2018, vol. 48, pp. 131-155.

18. Ascher U., Petzold L. Stability of Computational Methods for Constrained Dynamics Systems. SIAM Journal on Scientific Computing, 1993, vol. 14, pp. 95-120.

19. Hairer E., Norsett S.P., Wanner G. Solving Ordinary Differential Equations I. Berlin, Springer-Verlag, 1996.