State event location in differential-algebraic models

ABSTRACT

An efficient discontinuity handling algorithm for initial value problems in differential-algebraic equations is presented. The algorithm supports flexible representation of state conditions in propositional logic, and guarantees the location of all state events in strict time order. The algorithm consists of two phases:(1) event detection and(2) consistent event location. In the event detection phase, the entire integration step is searched for the state event by solving the interpolation polynomials for the discontinuity functions generated by the BDF method. An efficient hierarchical polynomial root-finding procedure based upon interval arithmetic guarantees detection of the state event even if multiple state condition transitions exist in an integration step, in which case many existing algorithms may fail. As a second phase of the algorithm, a consistent even location calculation is developed that accurately locates the state event detected earlier while completely eliminating incorrect reactivation of the same state event immediately after the consistent initialization calculation that may follow. This numerical phenomenon has not been explained before and is termed discontinuity sticking. Results from various test problems are presented to demonstrate the correctness and efficiency of the algorithm.

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INDEX TERMS

Primary Classification:
  I. Computing Methodologies
  I.6 SIMULATION AND MODELING
      I.6.8 Types of Simulation
          Subjects: Combined

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.1 Interpolation
          Subjects: Interpolation formulas
      G.1.7 Ordinary Differential Equations
          Subjects: Multistep and multivalue methods; Initial value problems

  I. Computing Methodologies
  I.1 SYMBOLIC AND ALGEBRAIC MANIPULATION
      I.1.2 Algorithms
          Subjects: Algebraic algorithms
  I.6 SIMULATION AND MODELING

General Terms:
Algorithms

Keywords:
combined discrete/continuous simulation, differential algebraic equation, discontinuities, interval arithmetic, state event