Highly Parallel Schwarz Preconditioning for Space‐Time Implicit Runge–Kutta Method for Parabolic Problems

The classical implicit Runge‐Kuta (IRK) method is a powerful method with some desirable accuracy and stability properties but is rarely used in practical applications because of its high computational cost and difficulties in preconditioning. In this paper, we introduce a space‐time coupled IRK scheme that offers high‐order accuracy and stability, and also has a high degree of parallelism in both space and time when used with two‐level tensor‐structure‐preserving overlapping Schwarz preconditioners. The convergence of the proposed method is studied numerically and we show that the convergence rate depends only mildly on the mesh size, the time step size, the number of processors, and the window size. We compare the parallel performance of the proposed method with the classical method in terms of the strong scalability, and the window‐size‐scaled weak scalability. The numerical results indicate that the proposed method outperforms the classical method when the number of processors is large and the space‐only parallelization of the classical method is a limiting factor. Moreover, in terms of the total compute time, we show numerically that higher order space‐time IRK outperforms the lower order space‐time IRK when a suitable window size is chosen for solving the problem in the entire space‐time domain with similar accuracy.