A Time-Stepping Algorithm for Parallel Computers

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Abstract
Parabolic and hyperbolic differential equations are often solved numerically by time-stepping algorithms. These algorithms have been regarded as sequential in time; that is, the solution on a time level must be known before the computation of the solution at subsequent time levels can start. While this remains true in principle, it is demonstrated that it is possible for processors to perform useful work on many time levels simultaneously. Specifically, it is possible for processors assigned to “later” time levels to compute a very good initial guess for the solution based on partial solutions from previous time levels, thus reducing the time required for solution. The reduction in the solution time can be measured as parallel speedup. This algorithm is demonstrated for both linear and nonlinear problems. In addition, the convergence properties of the method based on the convergence properties of the underlying iterative method are discussed, and an accurate performance model from which the speedup and other quantities can be estimated is presented.

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