On dichotomy and well conditioning in BVP
SIAM Journal on Numerical Analysis
Boundary value problems and dichotomic stability
SIAM Journal on Numerical Analysis
Numerical solution of boundary value problems in differential-algebraic systems
SIAM Journal on Scientific and Statistical Computing
The conditioning of boundary value problems in transferable differential-algebraic equations
SIAM Journal on Numerical Analysis
Conditioning and dichotomy in differential algebraic equations
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Collocation software for boundary value differential-algebraic equations
SIAM Journal on Scientific Computing
A Shooting Method for Fully Implicit Index-2 Differential Algebraic Equations
SIAM Journal on Scientific Computing
Hi-index | 0.00 |
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.