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Single Residuals

These are computed in prediction and as needed during correction. Multiple residuals are computed when forming the finite-difference Jacobian. Single residuals are computed repetitively in each process column, whereas the multiple residuals of a Jacobian computation are computed uniquely in the process columns.

Here, we consider the single residual computation required by the integration computations just described. Given a state vector , and approximation for , we need to evaluate . The exploitable concurrency available in this step is strictly a function of the model equations. As defined, there are N equations in this system, so we expect to use at best N computers for this step. Practically, there will be interprocess communication between the process rows, corresponding to the connectivity among the equations. This will place an upper limit on (the number of row processes) that can be used before the speed will again decrease: We can expect efficient speedup for this step provided that the cost of the interprocess communication is insignificant compared to the single-equation grain size. As estimated in [Skjellum:90a], the granularity for the Symult s2010 multicomputer is about fifty, so this implies about 450 floating-point operations per communication in order to achieve 90% concurrent efficiency in this phase.

Guy Robinson
Wed Mar 1 10:19:35 EST 1995