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Investigation of Convergence Characteristics of the Parareal method for Hyperbolic PDEs using the Reduced Basis Methods Iizuka, Mikio


In this study, we introduce the reduced basis methods (RBMs) to improve a convergence rate of the Parareal method for hyperbolic PDEs. We extract a small subspace consisting of main modes that compose the accurate solution from the data calculated by the fine solver during iterations. Once we got a set of reduced basis, the computational cost of time marching becomes low because of the small subspace, and therefore we can use a fine time step width. Then we can perform a coarse solver with the time step width same as a fine solver. Thus, it is expected that the convergence can be improved and the computation cost can be reduced. However, if the subspace cannot be reduced to small one, the RBMs maybe does not work well, e.g., for the complex phenomena such as the turbulence flow. We need to know what case the RBMs work well for, but currently it is not clear. Therefore, we investigate the convergence characteristics of the Parareal method for hyperbolic PDEs which have different complexity with the RBMs. In this presentation, we discuss about the results of convergence of the Parareal method with the RBMs for the linear advection equation, viscous Burgers' equation and Navier-Stokes equations.

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