An Algorithm to Solve Systems of Nonlinear Differential-Algebraic Equations With Extraordinary Efficiency Even at High Demanded Precisions

There are many situations when systems of non-linear differential algebraic equations need to be solved with extraordinary precision. A steady-state analysis (determining the steady-state period of a system after a transient) is a typical case because a vector of unknown variables should be exactly the same after a numerical integration on the period-long interval. Therefore, we need to develop such kinds of numerical algorithms that are computationally effective, even at very high requirements on the accuracy of the results. In the paper, an efficient and reliable algorithm for solving systems of algebraic-differential nonlinear equations is characterized first. Unlike in other cases, the procedure is based on a sophisticated arrangement of the Newton interpolation polynomial (i.e., not the Lagrange one). This feature provides greater flexibility in rapidly changing interpolation step sizes and orders during numerical integration. At the end of the paper, two complicated examples are presented to demonstrate that the algorithm’s computational requirement is quite low, even at very high demands on the accuracy of results.