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Coarse-grain parallelisation of multi-implicit Runge-Kutta methods
Trobec, Roman ; Orel, Bojan ; Slivnik, Boštjan
A parallel implementation for a multi-implicint Runge-Kutta method (MIRK) with real eigenvalues is decribed. The parallel method is analysed and the algorithm is devised. For the problem with ▫$d$▫ ...domains, the amount of work within the ▫$s$▫-stage MIRK method, associated with the solution of system, is proportional to ▫$(sd^)2$▫, in contrast to the simple implicit finite difference method (IFD) where the amount of work is proportional to ▫$d^3$▫. However, it is shown that s-stage MIRK admits much greater time steps for the same order of error. Additionally, the proposed parallelisation transforms the system of the dimension ▫$sd$▫ to ▫$s$▫ independent sub-systems of dimension ▫$d$▫. The amount of work for the sequential solution of such systems is proportional to ▫$sd^3$▫. The described parallel algorithm enables the solving of each of the s subsystems on a separate processor; finally, the amount of work is again ▫$d^3$▫, but the profit of a larger time step still remains. To test the theory, a comparative example of the 3-D heat transfer in a human heart with 64 3 domains is shown and numerically calculated by 3-stage MIRK.
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