Symbolic-Numeric Methods for Improving Structural Analysis of Differential-Algebraic Equation Systems

Systems of differential-algebraic equations (DAEs) are generated routinely by simulation and modeling environments, such as MapleSim and those based on the Modelica language. Before a simulation starts and a numerical method is applied, some kind of structural analysis is performed to determine which equations to be differentiated, and how many times. Both Pantelides’s algorithm and Pryce’s Σ-method are equivalent in the sense that, if one method succeeds in finding the correct index and producing a nonsingular Jacobian for a numerical solution procedure, then the other does also. Such a success occurs on many problems of interest, but these structural analysis methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates Σ-method’s failures and presents two symbolic-numeric conversion methods for fixing them. Both methods convert a DAE on which the Σ-method fails to a DAE on which this SA may succeed.