A Successive Linear Programming Approach for Initialization and Reinitialization after Discontinuities of Differential-Algebraic Equations

Determination of consistent initial conditions is an important aspect of the solution of differential-algebraic equations (DAEs). Specification of inconsistent initial conditions, even if they are only slightly inconsistent, often leads to a failure in the initialization problem. In this paper, we present a successive linear programming (SLP) approach for the solution of the DAE derivative array equations for the initialization problem. The SLP formulation handles roundoff errors and inconsistent user specifications, among other things, and allows for reliable convergence strategies that incorporate variable bounds and trust region concepts. A new consistent set of initial conditions is obtained by minimizing the deviation of the variable values from the specified ones. For problems with discontinuities caused by a step change in the input functions, a new criterion is presented for identifying the subset of variables which are continuous across the discontinuity. The SLP formulation is then applied to determine a consistent set of initial conditions for further solution of the problem in the domain after the discontinuity. Numerous example problems are solved to illustrate these concepts.

[1]  J. Barros-Neto.  An introduction to the theory of distributions , 1973 .

[2]  W. E. Stewart,et al.  Sensitivity analysis of initial value problems with mixed odes and algebraic equations , 1985 .

[3]  Leon S. Lasdon,et al.  An Improved Successive Linear Programming Algorithm , 1985 .

[4]  L. Petzold,et al.  Numerical methods and software for sensitivity analysis of differential-algebraic systems , 1986 .

[5]  J. Nocedal,et al.  The use of linear programming for the solution of sparse sets of nonlinear equations , 1987 .

[6]  C. W. Gear,et al.  Differential-algebraic equations index transformations , 1988 .

[7]  C. W. Gear,et al.  Approximation methods for the consistent initialization of differential-algebraic equations , 1988 .

[8]  C. Pantelides.  The consistent intialization of differential-algebraic systems , 1988 .

[9]  A. W. Westerberg,et al.  A proposed numerical algorithm for solving nonlinear index problems , 1990 .

[10]  R. Bachmann,et al.  ON METHODS FOR REDUCING THE INDEX OF DIFFERENTIAL ALGEBRAIC EQUATIONS , 1990 .

[11]  Yonsoo Chung.  Solving index and near index problems in dynamic simulation , 1991 .

[12]  L. Brüll,et al.  On Consistent Initialization of Differential-Algebraic Equations with Discontinuities , 1991 .

[13]  Paul I. Barton,et al.  Equation-oriented dynamic simulation current status and future perspectives , 1993 .

[14]  S. Campbell,et al.  Progress on a general numerical method for nonlinear higher index DAEs II , 1994 .

[15]  S. Dirkse,et al.  A Comparison of Algorithms for Large Scale Mixed Complementarity Problems , 1995 .

[16]  E. Gilles,et al.  Reinitialization of DAE's after discontinuities , 1995 .

[17]  W. Marquardt,et al.  Structural analysis of differential-algebraic equation systems—theory and applications , 1995 .

[18]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[19]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[20]  L. Biegler,et al.  Nonsmooth dynamic simulation with linear programming based methods , 1997 .