Optimally zero stable explicit peer methods with variable nodes
The authors develop explicit peer-to-peer methods for solving unstable initial problems, in which some of the values of the stage are copies of the values of the stage from the previous steps. They argue that this reduces the number of function calls per step and can be interpreted as a generalization of the ``first as last'' principle, known for Runge-Kutta methods, which states that the last stage is evaluated at the same point as the first stage of the next stage. \par The article presents a special class of explicit peer-to-peer methods with shifted stages. This allows one to save the estimates of the functions. In addition, the constructed methods are superconvergent and converge faster than expected. \par The authors focus on optimally zero stable methods, which give excellent results. The order conditions are also derived and it is proved that under some natural conditions numerical approximations can be calculated for all step-size sequences. Concrete optimally zero stable methods are constructed up to order 8. \par At the end of the work, some numerical experiments are given. They confirm that the new methods are more effective than the built-in functions of Matlab, such as ode23, ode45 and ode113.
scientific article; zbMATH DE number 6894727
scientific article