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.