On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations
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Lai Wang and Meilin Yu. "On the parallel implementation and performance study of high-order Rosenbrock-type implicit Runge-Kutta methods for the FR/CPR solutions of the Navier-Stokes equations", 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1095) https://doi.org/10.2514/6.2018-1095
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The Rosenbrock-type implicit Runge-Kutta (ROIRK) methods only require one Jaco-bian matrix evaluation per time step rather than per stage as other types of implicit Runge-Kutta (IRK) methods need. This feature makes ROIRK attractive for numerical simulations using implicit methods. We present the parallel implementation of several matrix-based ROIRK methods with ﬂux reconstruction/correction procedure re-construction (FR/CPR) formulations for solving the 3D Navier-Stokes equations. In this study, METIS has been utilized to partition the mesh in the preprocessing. The complex-step derivative approximation is employed to evaluate the Jacobi matrix, ac-curate to machine zero. The GMRES solver in the PETSc library is used to iteratively solve the linear system. The ROIRK methods have demonstrated high order of ac-curacy in numerical simulations. The scalability study reveals that the matrix-based ROIRK methods have good parallel eﬃciency. With the block Jacobi preconditioner, it is observed that the linear systems resulting from ROIRK3-3 are stiﬀer than those from ROIRK2-2 and ROIRK4-6. This makes the scalability of ROIRK3-3 worse than ROIRK2-2 and ROIRK4-6 taking the number of stages into account.