England, Roland; Lamour, René and López-Estrada, Jesús
Multiple shooting using a dichotomically stable integrator for solving differential-algebraic equations.
Applied Numerical Mathematics, 42(1-3)
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 (ODEs), 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 (DAEs) 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 (SYMIRK [R. England, R.M.M. Mattheij, in: Lecture Notes in Math., Vol. 1230, Springer, 1986, pp. 221–234]) has been adapted for the solution of index-one DAEs, and the resulting integrator (SYMDAE) has been inserted into the multiple-shooting code (MSHDAE) 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.
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