Embedded diagonally implicit Runge-Kutta algorithms on parallel computers

Abstract

This paper investigates diagonally implicit Runge-Kutta methods in which the implicit relations can be solved in parallel and are singly diagonal-implicit on each processor. The algorithms are based on diagonally implicit iteration of fully implicit Runge-Kutta methods of high order. The iteration scheme is chosen in such a way that the resulting algorithm is A ( α ) A(\alpha ) -stable or L ( α ) L(\alpha ) -stable with α \alpha equal or very close to π / 2 \pi /2 . In this way, highly stable, singly diagonal-implicit Runge-Kutta methods of orders up to 10 can be constructed. Because of the iterative nature of the methods, embedded formulas of lower orders are automatically available, allowing a strategy for step and order variation.