The actual hardware advancement is not sufficient to meet the requirements as they occur in large-scale problems. A natural approach for giving a positive answer to the need of faster solvers consists in the use of parallel computers. The main problem is effectively exploiting this huge potential of computer power since there is very little software available for these machines. In order to be efficient, a such software should be based on algorithms that are well turned to the new architectures.

Although parallel computers are available now for quite a few years, it is remarkable that parallel algorithms for solving initial value problems for ordinary differential equations (IVPs for ODEs) have received only marginal attention in the literature compared to the enormous work devoted to parallel algorithms for linear algebra. It is not a coincidence that only the last few years some significant results have started to appear. The last eight years have shown an increased interest in solving the IVP on the parallel computers. A possible explanation may be that the integration of an IVP by a step-by step process is sequentially in nature and thus offers little scope to exploit parallelism.

Parallelism can be exploited in many ways. For example, if solutions are needed in real times, if there is a large period of integration, if parameters fitting is to be performed which requires repeated integrations, and if function evaluations are expensive.

Another source of parallelism arises if the problem to be solved is stiff. Stiffness is a very difficult property to characterize explicitly, but problems that are stiff typically arise from models which have widely differing time components. In particular, the technique of the method of lines applied to parabolic partial differential equations (parabolic PDEs) usually results in a stiff system if the spatial grid is moderately fine-grained. If a problem is stiff and nonlinear, the explicit methods are generally not suited because of unnatural restrictions on the stepsize and so implicit methods must be used. This means the solution of large systems of nonlinear equations at each time step. Hence codes for solving linear system of equations in a parallel environment can be exploited at this stage.

Very large systems of ODEs arise in solution methods for time dependent PDEs. When greater accuracy is required, the spatial grid needs to be refined and this leads to even larger systems of equations. Such problems are not solvable in a reasonable time on a serial machine because of this magnitude.

Compared to parallel methods in the field of numerical linear algebra, there exist only few parallel procedures for the practical solution of initial value problems (IVPs) of systems of ordinary differential equations (ODEs). This fact can be explained primarily by the inherently sequential nature of any forward step method for the numerical solution of such problems.

The numerical integration of the initial value problem for an ODE by finite differences is a sequential calculation. By this we mean that the approximation to the solution of an ODE, obtained by methods like linear multistep or Runge-Kutta process involves one point at a time the solution at each new mesh point, is a prescribed function of the values of the solution at certain previous mesh points.

Massive parallelism in this field either requires a priori knowledge of a special structure of the respective problem or a great number of redundant calculations. In this way the potential power of a parallel computer system is normally utilized only to a modest extent. Most existing methods are efficient only for a low degree of parallelism (fully utilizing only 2-5 processor elements).

The International Conference on Parallel Methods for ODEs, the state-of-the-art, held in Grado, Italy (10-13 September 1991) emerged that the biggest chances could lie in the development of methods which should be essentially new and whose analytic and numerical structure should take both the deep nature of the particular underlying problem and the characteristics of the target parallel architecture strictly into account.

The introduction of parallel computers may lead to the use of genuinely newer types of ODE methods. We say this because onestep and straight multistep methods have little potential for parallelism and true multivalue methods have more to offer, as long as the number of saved values exceeds the degree of parallelism.

In many papers only formulas are presented and analyzed with respect to their speed-up. Such analysis can be used as a basis for the selection of potentially interesting methods. However, the actual behaviour of a piece of software produced on the basis of one of these methods may be impaired by many factors. A realistic assessment is usually only possible by an evaluation based on modeling of the respective algorithms and target computers and/or extensive numerical experiments on parallel computers.

Most of the work to date on the parallel solution of IVPs can be considered preliminary research, in that it concentrates on developing potentially useful numerical schemes, rather than their effective implementation, comparisons of methods, or the development of reliable, robust, and (it is hoped) portable mathematical software.