Parallel Jacobi iteration in implicit step-by-step methods

Abstract

An iteration scheme is descibed to solve the implicit relations that result from the application of an implicit integration method to an initial value problem (IVP). In this iteration scheme the amount of implicitness is still free so as to comprise a large variety of methods, running from fully explicit (functional iteration) to fully implicit (Newton's method). In the intermediate variants (the so-called Jacobi-type methods), the influence of the Jacobian matrix of the problem is gradually increased. Special emphasis is placed on the 'stage value-Jacobi' iteration which uses only the diagonal of the Jacobian matrix. Therefore, the convergence of this method crucially depends on the diagonally dominance of the Jacobian. Another characteristic of this scheme is that it allows for massive parallelism: for a d dimensional IVP, d uncoupled systems of dimensions have to be solved, where sis the number of stages in the underlying implicit method (e.g., an s-stage Runge-Kutta method). Hence, on a parallel architecture with d processors (d>>l), we may expect an efficient process (for high-dimensional problems). the transformed problem is indicative for the convergence factor corresponding to the origmal problem. We illustrate tliis for the IVP for ODEs. Let