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 a k-step formula, we prove that the orders of differential variables and algebraic variables do not exceed k + 1 and k respectively when k is odd and both orders do not exceed k when k is even. In order to achieve the order k + 1, the coefficients in the formula should satisfy some strict conditions. Examples which can achieve the maximum order are given for k = 1, 2, 3. Especially, a class of multistep formula for k = 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.