Error Analysis of IMEX Runge–Kutta Methods Derived from Differential-Algebraic Systems

In this paper we present an error analysis of the IMEX Runge–Kutta methods when applied to stiff problems containing a nonstiff term and a stiff term, characterized by a small stiffness parameter $\varepsilon$. In this analysis we expand the global error in powers of $\varepsilon$ and show that the coefficients of the error are the global errors of the IMEX Runge–Kutta method applied to a differential-algebraic system. Interesting convergence results of these errors and of the remainder of the expansion allow us to determine sharp error bounds for stiff problems. As a representative example of stiff problems we have chosen the van der Pol equation. We illustrate that the theoretical prediction is confirmed by the numerical test. Specifically, an order reduction phenomenon is observed when the problem becomes increasingly stiff. In particular, making several assumptions, we try to improve global error estimates of several IMEX Runge–Kutta methods existing in the literature.