document: On Implicit Euler for high-order high-index DAEs The Implicit Euler method is seldom used to solve differential-algebraic equations -LRB- DAEs -RRB- of differential index r > or = 3 , since the method in general fails to converge in the first r - 2 steps after a change of stepsize . However , if the differential equation is of order d = r - 1 > or = 1 , an alternative variable-step version of the Euler method can be shown uniformly convergent . For d = r - 1 , this variable-step method is equivalent to the Implicit Euler except for the first r - 2 steps after a change of stepsize . Generalization to DAEs with differential equations of order d > r - 1 > or = 1 , and to variable-order formulas is discussed
doc_bio_tags: O O O O O O O O B I I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: implicit euler method differential index
abstractive_keyphrases: differential-algebraic equations convergence stepsize change variable-step method variable-order formulas linear multistep method backward differentiation formula initial value problem