On convergence of continuous half-explicit Runge-Kutta methods for a class of delay differential-algebraic equations

Abstract

In this paper, we propose and investigate continuous Runge-Kutta methods for solving a class of nonlinear differential-algebraic equations (DAEs) with constant delay. Real-life processes that involve simultaneously time-delay effect and constraints are usually described by delay DAEs. Solving delay DAEs is more complicated than solving non-delay ones since we should focus on both the time-delay and DAE aspects. Recently, we have revisited linear multistep methods and Runge-Kutta methods for a class of nonlinear DAEs (without delay) and shown the advantages of appropriately modified discretizations. In this work, we extend the use of half-explicit Runge-Kutta methods to a similar class of structured strangeness-free DAEs with constant delay. Approximation of solutions at delayed time is obtained by continuous extensions of discrete solution, i.e., continuous output formulas. Convergence analysis for continuous Runge-Kutta methods is presented. It is shown that order reduction that may happen with DAEs is avoided if we discretize an appropriately reformulated delay DAE (DDAE) instead of the original one. Difficulties arising in the implementation are discussed as well. Finally, numerical experiments are given for illustration.