The choice of parameters in parallel general linear methods for stiff problems
This paper studies the possibilities of some diagonally implicit multistage integration (DIMSI) methods for the numerical solution of stiff initial value problems in a parallel environment. The methods under consideration, which are a subclass of general linear methods, were proposed by the first author [Appl. Numer. Math. 11, No. 5, 347-363 (1993; Zbl 0773.65046)] in a more general setting. Here the authors discuss some particular choices of the available parameters in these methods taking into account not only the standard stability and order properties but also the implementation details. More specifically, a DIMSI method defined by the matrices \((A,U,B,V)\) advances the numerical solution from \(t_n\) to \(t_{n+1}=t_n+h\) by means of a formula of type \(y^{[n+1]}= V y^{[n]}+ h B F(Y)\) where \(y^{[n]}\) is the external stage vector at time \(t_n\) which collects the numerical information on the solution at time \(t_n\) (typically this information is given by a Nordsieck vector \((y_n, h y_n'', \ldots, h^p y_n^{(p)}))\) and \(Y\) is the internal stage vector that satisfies the algebraic equation \(Y = h A F(Y)+ U y^{[n]}\). In the first part of the paper the authors assume \( A = \lambda I\) and \(V\) a rank one matrix. In this context they derive a family of A-stable methods with different orders and some available parameters, but its main drawback is that they possess large error constants. In the second part of the paper \(A\) is assumed to be a diagonal matrix with non equal diagonal elements and \(U\) a rank two matrix. In this more general setting several methods with good stability properties and smaller error constants are derived. Finally the results of some numerical experiments are presented to compare the behaviour of the new (parallel) methods with other standard sequential solvers with some test problems.
diagonally implicit multistage integration methods
scientific article; zbMATH DE number 1455716