document: Multiple shooting using a dichotomically stable integrator for solving differential-algebraic equations In previous work by the first author , it has been established that a dichotomically stable discretization is needed when solving a stiff boundary-value problem in ordinary differential equations -LRB- ODEs -RRB- , when sharp boundary layers may occur at each end of the interval . A dichotomically stable implicit Runge-Kutta method , using the 3-stage , fourth-order , Lobatto IIIA formulae , has been implemented in a variable step-size initial-value integrator , which could be used in a multiple-shooting approach . In the case of index-one differential-algebraic equations -LRB- DAEs -RRB- the use of the Lobatto IIIA formulae has an advantage , over a comparable Gaussian method , that the order is the same for both differential and algebraic variables , and there is no need to treat them separately . The ODE integrator has been adapted for the solution of index-one DAEs , and the resulting integrator -LRB- SYMDAE -RRB- has been inserted into the multiple-shooting code -LRB- MSHDAE -RRB- previously developed by R. Lamour for differential-algebraic boundary-value problems . The standard version of MSHDAE uses a BDF integrator , which is not dichotomically stable , and for some stiff test problems this fails to integrate across the interval of interest , while the dichotomically stable integrator SYMDAE encounters no difficulty . Indeed , for such problems , the modified version of MSHDAE produces an accurate solution , and within limits imposed by computer word length , the efficiency of the solution process improves with increasing stiffness . For some nonstiff problems , the solution is also entirely satisfactory
doc_bio_tags: B I O O B I 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 B I 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 B I 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 B I 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 B I 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
extractive_keyphrases: multiple shooting ordinary differential equations lobatto iiia formulae dichotomically stable integrator
abstractive_keyphrases: stiff boundary-value problem implicit runge-kutta method initial-value integrator differential-algebraic equations