Improved structural methods for nonlinear differential-algebraic equations via combinatorial relaxation†

Differential-algebraic equations (DAEs) are widely used for modelling dynamical systems. In the numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing steps prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, structural methods fail if the DAE has a singular system Jacobian matrix. For such DAEs, methods have been proposed to modify them to other DAEs to which structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. Our methods are implemented as a MATLAB library using MuPAD, and through its application to practical DAEs, we show that our methods can be used as a promising preprocessing of DAEs that the index reduction procedure in MATLAB cannot handle.