Embedded diagonally implicit Runge-Kutta algorithms on parallel computers

Abstract

Abstract. This paper investigates diagonally implicit Runge-Kutta methods in which the implicit relations can be solved in parallel and are singly diagonalimplicit 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)-stable or Z,(a)-stable with a equal or very close to n/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.

Analysis of parallel diagonally implicit iteration of Runge-Kutta methods

“…The stage vector equation in (2.1) is solved by applying the diagonal iteration method studied in [11,12]. Let y<J.t> denote the successive iterates, then we may define the (highly parallel) iteration process…”

“…In [11,12] the performance of PDIRK methods was studied in the case where in each of the m outer iterations the inner iteration method was continued until convergence before starting the next outer iteration (this iteration strategy is also used in conventional DIRK methods). However, this strategy may be rather expensive if many iterations arc needed to get the inner iteration converged.…”

“…Several families of methods constructed according to the fixed-number-of-iterations option were already considered in [12]. An interesting family considered in that paper possesses the stability functions investigated by Wolfbrandt [13] and uses constant iteration parameters 8; determined by these stability functions.…”

“…We shall employ the diagonally implicit iteration-type methods proposed in [11,12]. These methods are designed in such a way that a large number of the implicit systems to be solved can be processed in parallel, so that the number of systems that have to be solved sequentially is substantially reduced when implemented on multi-processor computers.…”

“…Such RK methods possess both a large step-point order and a large stage order. Furthermore, by a suitable choice of the collocation parameters, these RK methods are unconditionally stable for any order of accuracy.We shall employ the diagonally implicit iteration-type methods proposed in [11,12]. These…”