Analysis of numerical methods for integrating high-dimensional stiff systems in SimInTech

Abstract

When modeling complex dynamic processes, it becomes necessary to numerically solve the Cauchy problem for systems of ordinary differential equations (ODEs). The efficiency of the applied numerical methods depends on the degree of stiffness and dimension of the problem [1–3]. Depending on the task class, different methods behave differently.This article provides a comparative analysis of explicit adaptive and diagonal-implicit integration methods implemented in the SimInTech software [4]. The SimInTech software package is designed to simulate complex dynamic processes in systems of various classes. The system supports the ability to develop models in the form of block diagrams, as well as describe systems of differential equations using the built-in programming language and simulate event-driven systems and finite automata.It is shown that the most effective in solving problems of the considered class are the diagonal-implicit Runge-Kutta type integration methods – DIRK2 and DIRK4 from the SimInTech package. The DIRK3 method is inferior due to the large number of calculations of the Jacobian matrix. The preferred method is rather The DIRK2 method is preferable in this case, because it has a greater number of time steps with almost the same performance and does not increase the integration step so much with relatively low accuracy settings of the integration method. Of the explicit methods of the Runge-Kutta type with an adaptive numerical scheme, the “Adaptive-5” method is the most effective for solving problems of this class. For problems of this class, we can recommend the use of explicit integration methods “Adaptive-5”, “Adaptive-1” with a small system dimension.The traditional implicit Gear and Euler integration methods also effectively solve this problem, provided that the algorithm for calculating the Jacobian matrix is effectively implemented.