Browsing by Author "Outrata, Michal"

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    Domain truncation, absorbing boundary conditions, Schur complements, and Padé approximation
    Gander, Martin J.; Jakabčin, Lukáš; Outrata, Michal (Osterreichische Akademie der Wissenschaften, Verlag, 2024)
    We show for a model problem that the truncation of an unbounded domain by an artificial Dirichlet boundary condition placed far away from the domain of interest is equivalent to a specific absorbing boundary condition placed closer to the domain of interest. This specific absorbing boundary condition can be implemented as a truncation layer terminated by a Dirichlet condition. We prove that the absorbing boundary condition thus obtained is a spectral Padé approximation about infinity of the transparent boundary condition. We also study numerically two improvements for this boundary condition, the truncation with an artificial Robin condition placed at the end of the truncation layer and a Padé approximation about a different point than infinity. Both of these give new and substantially better results compared to using the artificial Dirichlet boundary condition at the end of the truncation layer. We prove our results in the context of linear algebra, using spectral analysis of finite and infinite Schur complements, which we relate to continued fractions. We illustrate our results with numerical experiments.
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    Spectral analysis of implicit 2 stage block Runge-Kutta preconditioners
    Gander, Martin J.; Outrata, Michal (Elsevier, 2023-01-01)
    We analyze the recently introduced family of preconditioners in [15] for the stage equations of implicit Runge-Kutta methods for two stage methods. We give explicit formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general method and use these to give explicit convergence estimates of preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis also allows us to qualitatively predict and explain the main observed features of the GMRES convergence behavior, not only bound it. We illustrate our analysis with numerical experiments. We also consider the direction of numerical optimization for improving the preconditioners performance, as suggested in [15]. We consider two different ways – both distinct to the one introduced in [15] – and numerically optimize these, using the explicit bounds obtained beforehand.