An efficient family of strongly A-stable Runge–Kutta collocation methods for stiff systems and DAEs. Part II: Convergence results

The convergence properties of a newly defined uniparametric family of collocation Runge–Kutta methods on non-stiff systems, stiff semi-linear problems and Differential-Algebraic Equations are analysed. For each s⩾3, the so-called family of SAFERK methods is based on interpolatory quadrature rules of order 2s−3, and they have the same number of implicit stages as the s-stage LobattoIIIA and (s−1)-stage RadauIIA methods. Since s-stage SAFERK methods possess the same algebraic order as the (s−1)-stage RadauIIA method, the corresponding principal terms of local error are compared, and it is shown that those SAFERK methods with positive weights and nodes located in the integration interval possess, in the l2-norm, a smaller principal term of local error. Moreover, it is proved that SAFERK methods are convergent when integrating relevant classes of stiff problems and Differential-Algebraic Equations. In particular, they possess a higher stiff order than the RadauIIA method of the same algebraic order, due to a higher stage order of the new methods. On the other hand, because of the strong A-stability, SAFERK methods can be competitive with LobattoIIIA methods, particularly for stiff problems on large time intervals.