Runge-Kutta methods for the numerical solution of stiff semilinear systems (Q1592688)

This paper is concerned with the convergence properties of implicit Runge-Kutta methods applied to stiff semilinear systems \[ y'(t) = J(t) y(t) + g(t,y(t)), \] \[ y(0) = y_0 \in R^m, \qquad t \in [0, T] \] with the variable coefficient linear part containing all the stiffness. Two different classes of stiff systems are defined by requiring suitable properties for the relative variation of the matrix \( J(t) \). For each of these classes a family of stable and convergent Runge-Kutta methods which define a unique numerical solution is characterized. As a consequence it is possible to identify Runge-Kutta methods which are not algebraically stable but are suitable for the numerical integration of semilinear problems. The obtained results extend previous ones by \textit{K. Burrage, W. H. Hundsdorfer} and \textit{J. G. Verwer} [Computing 36, 17-34 (1986; Zbl 0572.65053)] on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with constant coefficient linear part.