Design and implementation of DIRK integrators for stiff systems Original Research Article

This paper presents new four-stage diagonally implicit Runge–Kutta integration formulas for stiff initial value problems, designed to be L-stable and have optimal order of accuracy. The design makes estimation of local error and interpolation of the solution possible without additional stages. It also enables improved prediction of stage values in a software implementation. Correctly monitoring the convergence rate of simplified Newton iterations improves the implementation too, by providing a sound basis for deciding on Jacobian updates.