Symbolic-Numeric Methods for Improving Structural Analysis of DAEs

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 the Σ method’s failures, and presents two symbolic-numeric conversion methods for fixing these failures. These methods convert a DAE on which the Σ -method fails to a DAE on which this structural analysis may succeed.