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Comparison of 3- and 5-stage multi-implicit Runge-Kutta methods
Slivnik, Boštjan ; Trobec, Roman ; Orel, Bojan
The well known coarse-grain parallelisation of the multi-implicit 3-stage Runge-Kutta method (3-MIRK) with real eigenvalues is compared with the new 5-stage MIRK method. Assume that a problem ...contains ▫$d$▫ spatially discretised domains. It was previously shown that the s-MIRK method gives a system of equations of the dimension ▫$sd$▫ that can be transformed into ▫$s$▫ independed sub-systems of the dimansion ▫$d$▫. Each od sub-systems can be solved in parallel on a separate processor, and consenquently, the amount of work for the solution of the s-MIRK system is ▫$d^3$▫, which is the same order of complexity as for the simple implicit methods. Our hypothesis was that with the same algorithm the greather time step will be allowed in the 5-MIRK method for the same reliability. The amount of sequential calculation is increased by a factor of ▫$(5/3)^3$▫. On the other hand, with the same parallel algorithm the same problem could be solved with the 5-MIRK method on five workstations on a shorter time as with the 3-MIRK method on three workstations. The raason lies in the fact that a more complicated method, which can be performed perfectly in parallel, offers a greater time step. In other words, two additional internal approximations are calculated in parallel, on two additionally workstations, in order to combine better approximation for the next time step. To test our hypothesis, the calculation results of the 3-D heat transfer in the human heart with ▫$32^3$▫ domains is shown.
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