It is known that when we apply a linear multistep method to differential-algebraic equations (DAEs), usually the strict stability of the second characteristic polynomial is required for the zero stability. In this paper, we revisit the use of linear multistep discretizations for a class of structured strangeness-free DAEs. Both explicit and implicit linear multistep schemes can be used as underlying methods. When being applied to an appropriately reformulated form of the DAEs, the methods have the same convergent order and the same stability property as applied to ordinary differential equations (ODEs). In addition, the strict stability of the second characteristic polynomial is no longer required. In particular, for a class of semi-linear DAEs, if the underlying linear multistep method is explicit, then the computational cost may be significantly reduced. Numerical experiments are given to confirm the advantages of the new discretization schemes.
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The authors would like to thank the anonymous referee for the very helpful comments and suggestions that led to the improvements of this paper.
This work was supported by the Nafosted Project No. 101.02-2017.314.
Dedicated to Professor Pham Ky Anh on the occasion of his 70th birthday
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Linh, V.H., Truong, N.D. Stable Numerical Solution for a Class of Structured Differential-Algebraic Equations by Linear Multistep Methods. Acta Math Vietnam 44, 955–976 (2019). https://doi.org/10.1007/s40306-018-00310-5
- Differential-algebraic equations
- Strangeness-free form
- Linear multistep methods
Mathematics Subject Classification (2010)