Highest order multistep formula for solving index-2 differential-algebraic equations

In this paper, the maximum order of linear multistep methods (LMM) for solving semi-explict index-2 differential-algebraic equations (DAEs) is discussed. For ak-step formula, we prove that the orders of differential variables and algebraic variables do not exceedk+1 andk respectively whenk is odd and both orders do not exceedk whenk is even. In order to achieve the orderk+1, the coefficients in the formula should satisfy some strict conditions. Examples which can achieve the maximum order are given fork=1,2,3. Especially, a class of multistep formula fork=3, not appearing in the literature before, are proposed. Further, a class of predictor-corrector methods are constructed to remove the restriction of the infinite stability. They give the same maximum order as that for solving ODEs. Numerical tests confirm the theoretical results. This work was partially supported by the National Natural Science Foundation of China.